PHYS102 Study Guide

Site: Saylor Academy
Course: PHYS102: Introduction to Electromagnetism
Book: PHYS102 Study Guide
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Date: Thursday, March 28, 2024, 10:31 PM

Navigating this Study Guide

Study Guide Structure

In this study guide, the sections in each unit (1a., 1b., etc.) are the learning outcomes of that unit. 

Beneath each learning outcome are:

  • questions for you to answer independently;
  • a brief summary of the learning outcome topic;
  • and resources related to the learning outcome. 

At the end of each unit, there is also a list of suggested vocabulary words.

 

How to Use this Study Guide

  1. Review the entire course by reading the learning outcome summaries and suggested resources.
  2. Test your understanding of the course information by answering questions related to each unit learning outcome and defining and memorizing the vocabulary words at the end of each unit.

By clicking on the gear button on the top right of the screen, you can print the study guide. Then you can make notes, highlight, and underline as you work.

Through reviewing and completing the study guide, you should gain a deeper understanding of each learning outcome in the course and be better prepared for the final exam!

Unit 1: Mechanical Vibrations and Waves in Extended Objects

 

1a. State Hooke's Law

  • How can Hooke's Law be used to analyze the motion of a system?
  • How can the magnitude and direction of a restoring force be found?

Simple harmonic motion (SHM) arises when a restoring force appears in a system when the system is disturbed from equilibrium, and when the force is proportional to and has the opposite direction of the displacement. A typical example of this type of system is a point mass attached to the end of a massless spring. In this case, Hooke's Law states that the restoring force is F_{x}=-kx, where x is the displacement of the mass and is equal to the length of the spring is stretched or compressed. k is the spring constant, determined by the material of the spring and how tightly it is wound. The negative sign indicates the force is in the direction opposite to the displacement.

Review the basic behavior of springs with some examples in Spring Force.

 

1b. Describe the properties of simple harmonic motion and provide examples

  • What kind of force causes the objects to undergo simple harmonic motion?
  • How do position, velocity, and acceleration depend on time in SHM? What is their relationship?

Simple harmonic motion (SHM) is a unique kind of motion in which the position is a sinusoidal function of time:

x=x_{max}sin(2\pi ft).

Here, x_{max} is the amplitude, and f is the frequency of the oscillations. We can calculate the period of the oscillations as T=\frac{1}{f}. In general, x does not have to be a position; it could be any variable describing a system, such as the angle of a pendulum.

The reason we begin the course with this topic is that it introduces abstract concepts such as frequency, and later energy, in a context that is more tangible than the phenomena of electricity and magnetism that make up the bulk of the material you will encounter later. In mechanical oscillations, you can see the effect of the restoring force directly by measuring the acceleration of the oscillating object. Oscillation is a repeating type of motion where the velocity constantly changes in magnitude and even in direction. That is acceleration.

Whenever the oscillating object reaches its largest deviation from equilibrium, it feels the largest force and therefore the largest acceleration. When the object returns to its equilibrium position, it feels no force and therefore has no acceleration. But without a force, there's nothing to bring the mass to a halt, and so it overshoots its equilibrium and keeps going. To explain oscillation, it is important to understand the difference between velocity and acceleration: you can have a large velocity with zero acceleration, but you need a large acceleration to change velocity rapidly.

Review simple harmonic motion in The Simple Pendulum. Review the mathematical description and properties of simple harmonic motion as applied to a system of a mass on a spring in Simple Harmonic Motion: A Special Periodic Motion.

 

1c. Describe the forms energy is contained in oscillatory motion

  • Which two forms of energy are in a continuous interplay when a harmonic oscillation takes place?
  • At what stage of an oscillation does an object have its largest potential energy?
  • When an object oscillates, at what points does its velocity change direction? When is its speed largest?

Energy is an overarching principle in physics because we can use it to make predictions about many different types of motion. The main ingredient needed to make the concept of energy useful in oscillatory motion is the potential energy which depends on the deviation from equilibrium and on the spring constant. When you add potential energy to the kinetic energy of the oscillating mass, their sum stays constant over time, in the ideal case of simple harmonic motion. This total amount of energy depends only on how the oscillation was originally launched. Knowing this, we can predict the speed of the oscillating object just based on how far away it happens to be from equilibrium.

To check your understanding, identify which of the pictures in Fig. 16.16 (a) – (d) in Energy and the Simple Harmonic Oscillator show the maximum potential energy. Hint: there are two of them, and they correspond to zero velocity.

 

1d. Define resonance

  • Does the total potential and kinetic energy of a pendulum stay constant over time if the motion is damped?
  • To harvest almonds from a tree, farmers use a machine that shakes the tree back and forth. This works best if the shaking is not too slow and not too fast. Why?
  • What determines the resonance frequency of an oscillator that is driven by a periodic external force?

When you leave an oscillating mass alone, it shows a specific, "natural" frequency at which it likes to oscillate. Think of a vibrating guitar string that produces a certain pitch no matter how you pluck it. This natural frequency is also the frequency when it becomes easiest to supply external energy to the oscillating object. The reason you need an external force in the first place is that all realistic oscillations lose energy over time, which is called damping.

An example is how you have to push a playground swing at just the right rate to match the rate at which it would swing by itself, to keep it going. This phenomenon is called resonance, and the optimal frequency for energy transfer is the resonance frequency. You could also try to push the oscillator at a different frequency, but that would lead to less transferred energy per second. You can see the effect of the transferred energy in the amplitude of the oscillator. The amplitude is largest at the resonance frequency, and smaller if the driving frequency is too low or too high.

Be sure to understand the horizontal axis of Fig. 16.27 in Forced Oscillations and Resonance. It is the frequency with which the external force changes. The peak in the figure corresponds to the natural frequency, f_{0}.

 

1e. Define terms related to wave motion: frequency, wavelength, diffraction, and interference

  • What are the properties of wave motion?
  • How is wave motion different from particle motion?
  • What is superposition?
  • What is the relationship between frequency, wavelength, and velocity in a periodic wave?

Waves surrounding us have so many different types of manifestations – ripples on a surface of water, sound, and light – that many do not realize they all have the same underlying properties. The main property that distinguishes wave motion from the motion of matter particles is that waves can pass through each other without affecting each other's motion. Waves combine when more than one wave is present at the same place. We call this interference of waves superposition.

We characterize periodic waves by their wavelength – the distance the wave travels during one period (the time it takes for the oscillation to go through a full cycle). The relationship between the wavelength and period is \lambda=vT, where v is the speed of the wave propagation. Alternatively, wavelength can be related to frequency. Frequency is the reciprocal of period, and is the number of cycles the wave goes through in one second: f=\frac {1}{\tau}. Thus, \lambda = \frac {v}{\tau}.

Review Superposition and Interference for more on waves. You can interpret the example of standing waves on a string as the interference of a traveling wave: another traveling wave is reflected at one end, and then propagates in the opposite direction.

 

1f. Solve problems using simple harmonic motion

  • What are the necessary conditions for a system to undergo simple harmonic motion?
  • What quantities are necessary to calculate the angular frequency, frequency, and period of oscillations?
  • What quantities are necessary to calculate the total energy of oscillations at a given time?
  • What is the relationship between displacement, velocity, and acceleration of a system at a given time? What is the relationship between the maximum values of these variables?

Many application problems that involve simple harmonic motion focus on the relationship between the maximum displacement (amplitude) and maximum velocity of a system. Many of these questions ask you to calculate the frequency, period, and energy of oscillations.

Any harmonic oscillator has a characteristic constant k that we usually call the "spring constant" or stiffness. This is the proportionality constant between the restoring force and the displacement from equilibrium according to Hooke's Law, F=-kx. We can calculate the oscillation frequency f from this spring constant and the mass of the object that is moving back and forth, according to the formula:

f=\frac{1}{2 \pi}\sqrt{\frac{k}{m}}.

You can also solve this for the spring constant:

k=m(d\pi f)^{2}

This allows you to find the spring constant if you know the mass and the oscillation frequency f.

Let's say you would like to know the acceleration of the oscillating mass at a given displacement x, but you are only given the frequency and not the spring constant. You can calculate the answer by inserting the last equation into Hooke's Law and combining it with Newton's Second Law, f=ma. The result is:

a=-(2\pi f)^{2}x

Review the basic formula for the oscillation frequency with an example in Simple Harmonic Motion: A Special Periodic Motion.

 

Unit 1 Vocabulary

You should be familiar with these terms to complete the final exam.

  • acceleration
  • amplitude
  • cycle
  • displacement
  • energy (potential and kinetic)
  • equilibrium
  • force constant (same as spring constant)
  • frequency
  • Hooke's Law
  • oscillation
  • natural frequency
  • period
  • phase
  • resonance
  • restorative (or restoring) force
  • simple harmonic motion
  • superposition
  • velocity
  • vibration
  • wave
  • wavelength

Unit 2: Electrostatics

 

2a. State Coulomb's law and identify the units of the physical quantities contained in the law

  • How do two point charges interact with one another? Make a sketch that demonstrates three cases: two positive charges, one positive and one negative charge, and two negative charges. Label all relevant quantities and draw and label force vectors on both charges.
  • What is the expression for Coulomb's force acting on a charge?
  • How does the sign of the product of two charges determine the direction of the force?
  • What are the units of the constant k and the electric permittivity of vacuum \varepsilon _{0}?

Electric charge is a basic property of the particles that make up matter. There are two types of charge: positive and negative. Electrostatic force, the force between charges, is one of the four fundamental forces of nature. Many macroscopic forces that we can easily observe, such as the normal force and friction, result from the electric interactions between charged particles on the microscopic level.

Coulomb's Law states that the expression for the electrostatic force between two point charges, q_{1} and q_{2} (measured in units of Coulombs), is:

F=k\frac{\left| q_{1}q_{2} \right|}{r^{2}}

k=8.99\times 10^{9} Nm2/C2 is the electrostatic constant, while \tau is the distance between the centers of the two charges (measured in meters).

Notice that the formula only gives you the magnitude of the force, but not its direction. To figure out the direction, it is best to draw a sketch of the two charges; then decide for which of the charges you want to find the force. If the two charges have the same sign (known as "like" charges), they repel, and therefore the force on the charge you are looking at points away from the other charge. If the two charges have opposite signs, they attract, and the force on your charge points toward the other. The strength of the force F is the same for both of the charges.

If you remember Newton's Third Law, then you will recognize that it is in complete agreement with the rules for the force directions in Coulomb's Law.

Review this material in Coulomb's Law.

 

2b. Solve problems involving electric forces, electric fields, and electric potentials

  • What information is necessary to calculate the electric field or potential of a charge distribution? Explain how the vector sum of fields of several charges differs from the scalar sum of potentials of several charges.
  • What information is necessary to calculate the force on a charge in the electric field or the electric potential energy of a charge in the electric field?
  • In a parallel-plate capacitor, is the electric field or the electric potential constant?

The electric field is a would-be electric force, and the electric potential is would-be potential energy.

This means you get the force on a charge Q by multiplying the field by Q, and you get the potential energy of that charge by multiplying the potential by Q. Electric field and potential are created by other charges that are already there, and the charge Q is an additional object that you place in the vicinity of those existing charges.

Keep this distinction in mind to avoid getting confused: When you are asked to calculate the force on a charge Q, you do not need to find the electric field created by that charge; instead, you need the electric field created by the other charges in its vicinity (for example on the plates of a capacitor).

The following formulas pertain to problems involving electric forces, fields, and potential:

  • Coulomb's Law: F=k\frac{\left| q_{1}1_{2} \right|}{r^{2}}. This is a force between two point or two spherical charges. In the case of the spherical charges, r is the distance between their centers.

  • The electric field outside of a uniform spherical charge q has the strength E=k\frac{\left| q \right|}{r^{2}}. For a positive charge q, it points radially outward; for a negative charge, it points radially inward.

  • The superposition principle states that the electric field of several point charges, measured at any given location, is a vector sum of the fields that each charge creates at that location

  • The potential at a point in the field of a point charge Q is V=k\frac{Q}{r}. Potential is a scalar, an electric field is a vector. We obtain the potential energy of a different charge q by multiplying q with the electric potential V that exists at the location of q.

  • For the case of a uniform electric field pointing in the x-direction, the electric potential is V=-E_{x}x. The electric field points in the direction of decreasing potential, and electric field lines are perpendicular to the equipotential surfaces at every point.

There are several common types of problems involving electric forces, fields, and potential. You should use superposition to calculate the electric field or force on a charge due to several point charges. Review the examples in Electric Field Lines: Multiple Charges. You should also know how to calculate the electric potential of a point charge, a charged conducting sphere, and a parallel-plate capacitor. Review this material in Electric Field: Concept of a Field Revisited, Electric Field Lines: Multiple Charges, and Electric Potential Energy: Potential Difference.

 

2c. Compare and contrast the electric potential and the electric field

  • How do the concepts of electric potential energy and electric potential arise from the calculation of work performed by an electric field on a charge placed in the field?
  • How are field lines and the direction of a field related to the location of its points of equal potential?

We can describe the motion of a particle in terms of the forces acting on it, or in terms of its potential and kinetic energy. In the case of a charged particle placed in an electric field, the force is an electric force, and its potential energy is the energy associated with the electric field.

One thing that electric fields and potential have in common is that they exist in an abstract way at every point in the space surrounding an electric charge. The point you are looking at can be completely empty space. The field and potential tell you the value of the force and potential energy that an object would have if you were to put it at that point.

We describe the force of a charge at one point in time, whereas electric potential and electric fields usually indicate when a charge moves from one location to another.

Due to energy conservation, any change in kinetic energy must be balanced by an opposite change in electrical potential energy. The kinetic energy lets you figure out the speed of an object. But you only ever need to know the change in potential energy to do this. In practice, we do not need to know the electric potential, but only the change in electric potential. This has its own name: voltage. Voltage (V) between two points is the difference in electric potential between those points. Sometimes we denote voltage by \Delta V, rather than V, to distinguish it from the electric potential.

Review this material in Electric Potential in a Uniform Electric Field. Review Potential vs. Voltage if you are unsure about the difference between potential and voltage.

 

2d. Solve problems involving the motion of charged particles in an electric field

  • What is the direction of the force on a charged particle placed in an electric field? How does it depend on the charge of the particle?
  • Consider a charged particle entering a region with a uniform electric field. What will the particle's acceleration be? Draw the trajectory of a positively charged particle in the field for the cases when initial velocity is zero, parallel to the field, or perpendicular to the field. What will change if the particle is negatively charged?

The motion of a charged particle in a uniform electric field is similar to the motion of a massive object near the surface of the Earth, which is projectile motion. In both cases, the acceleration is constant. The acceleration of a charged particle in a uniform electric field is a=\frac {qE}{m}, where E is the field, q is the charge, and m is the mass of the particle. The acceleration always points in the direction of the electric field if the charge is positive, and in the opposite direction if the charge is negative.

We can calculate the trajectory by determining the direction of the initial velocity of the particle. If the initial velocity is zero or parallel to the field, the particle will move in a straight line. The electric force will cause it to accelerate or decelerate. However, if an angle exists between the initial velocity and the field, the trajectory will be parabolic, like the trajectory of a projectile in free fall.

Review the analogy between a uniform electric field and gravity in Electric Potential Energy: Potential Difference. To calculate the acceleration from the electric force, you need to remember Newton's Second Law. Review example 18.5 in Applications of Electrostatics.

 

2e. Define capacitance and describe the factors that determine capacitance

  • When is a parallel-plate capacitor considered ideal (the electric field is uniform between its plates and zero outside)?
  • Consider an ideal parallel-plate capacitor with surface charge density +\sigma on one plate and -\sigma on another. Assume we know the area of the plates and the distance between them. What is the voltage between the plates? How would you use that to find the capacitance of the parallel-plate capacitor?
  • If you have two capacitors, how would you connect them to a battery so they connect in a series? What will the relationship between the charges on each capacitor be in this case? What will the relationship between the voltage on each capacitor and the voltage supplied by the battery be? Use these considerations to determine the equivalent capacitance of two capacitors connected in series.
  • If you have two capacitors, how would you connect them to the battery so they connect in parallel? What will the relationship between the charges on each capacitor be in this case? What will the relationship between the voltage on each capacitor and the voltage supplied by the battery be? Use these considerations to determine the equivalent capacitance of two capacitors connected in parallel.

If you put a charge on the surface of a conductor or combination of conductors, the resultant electric voltage is proportional to that charge: Q=CV. The proportionality constant C between the potential and the charge is called capacitance, and it depends only on the geometry of the conductor. For a parallel-plate capacitor without any material in the space separating the plates, C=\frac{\varepsilon_{0}A}{d}, where \varepsilon_{0} is the dielectric constant (also called electric permittivity) of vacuum, A is the area of the plates, and d is the distance between the plates. The distance must be much smaller than the size of the plates so that the electric field inside the capacitor is uniform.

When two or more capacitors are connected in series:

  • They have the same charge, because charge is conserved.

  • The sum of voltages on each capacitor equals the voltage supplied by the battery.

  • The equivalent capacitance (C_{eq}) is determined by the formula \frac{1}{C_{eq}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\cdots.

When two or more capacitors are connected in parallel:

  • They have the same voltage, which also equals the voltage supplied by the battery

  • The sum of charges on each capacitor equals to total charge, proportional to the voltage supplied by the battery

  • The equivalent capacitance is determined by the formula \frac{1}{C_{eq}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\cdots

Review these explanations in Capacitors in Series and Parallel.

 

2f. Describe the effect of a dielectric material in a capacitor

  • What are the differences between conductors and dielectrics? How are charged particles in each material affected when the material is placed in an electric field?
  • How is the electric field inside a dielectric material related to the external electric field? Considering this, how is the capacitance affected when the dielectric is placed inside a capacitor?

When you place dielectric material inside an electric field, the positive and negative charges inside the material experience forces that act in opposite directions. As a result, the positive charges move toward the direction of the field, while the negative charges move in the direction opposite to the field. The charges remain bound to molecules, so they cannot move far. One says the molecules become polarized.

Since there are a huge number of molecules, the separation of the charges creates an internal electric field in the dielectric material, which is directed opposite to the external field. Therefore (according to the superposition principle), the magnitude of the net field inside the dielectric is smaller than the external field by a factor K, which is called the dielectric constant of the material. As a result of the decrease in the electric field when a capacitor is filled with a dielectric, its capacitance increases by a factor of K.

Review this material in Capacitors and Dielectrics.

 

2g. Define electric potential energy and describe how capacitors can be used to store energy

  • How much work is performed by an external agent to charge a capacitor to a given charge?
  • What are different ways to express the energy stored by a capacitor using its capacitance, charge, and voltage?

When an electrically-charged object feels an electric force due to some other object, Newton's Second Law says it will accelerate. This also applies to a charged particle (like an electron) if we place it between the plates of a charged capacitor. Then the electron's negative charge will make it accelerate toward the positive capacitor plate. This means the electric force is doing work on the electron, increasing its speed and therefore its kinetic energy. The gain in kinetic energy under an electric force can be attributed to an equal but opposite loss in electric potential energy.

But if you instead want to move the electron against the electric force, from the positive plate to the negative plate of a capacitor, then you need an additional, external force to overcome the electric force. Going in that direction, the electron would gain electric potential energy, and your external force is the source that supplied that energy by doing work. This is analogous to lifting a weight against the pull of gravity.

Since work must be performed to charge a capacitor, the charged capacitor stores energy equal to that work. The energy stored in a capacitor with the charge Q is \frac{Q^{2}}{2C}.

Review the general concept of work in the presence of an electric field in Electric Potential in a Uniform Electric Field. Review the specific case of how a capacitor stores energy in Energy Stored in Capacitors.

 

Unit 2 Vocabulary

You should be familiar with these terms to complete the final exam.

  • capacitance
  • capacitor
  • charge
  • charge density (linear, surface, and volume)
  • Coulomb's Law
  • dielectric constant
  • dielectric material
  • electrostatic (Coulomb's) force
  • electric field
  • electric permittivity
  • electric potential
  • electric potential energy
  • field line
  • parallel connection
  • series connection
  • superposition principle
  • voltage

Unit 3: Electronic Circuit Theory

 

3a. State Ohm's Law in words

  • What happens when you apply a potential difference to the ends of a conductor? What physical quantity measures the rate of the flow of charge?
  • What is the definition of the resistance of a conductor? What is the relationship between resistance, current, and potential difference between the ends of a conductor? When is this relationship called Ohm's Law?

If you put a conductor in an electric field (for example, if its ends are connected to a battery, so there is a voltage between each end), the free charges inside the conductor will begin to move. Current is the rate of the flow of charge. The ratio between the applied voltage (also called potential difference) and the current is defined as resistance: R=\frac{V}{I}. For some materials, this ratio is constant, and the conductor is said to obey Ohm's Law. These materials are known as Ohmic materials. In them, the potential difference between the ends of the conductor is proportional to the current through the conductor. Ohm's Law is V=IR.

Review this material in Ohm's Law.

 

3b. Apply Ohm's Law to simple circuits

  • Sketch an example of a simple circuit containing a battery and a resistor. What is the relationship between the current in the circuit and the voltage supplied by the battery?

According to Ohm's Law, the current established in a circuit with one battery supplying a voltage V is I=\frac{V}{R}. Here, R can be the resistance of the only resistor in the circuit, or the equivalent resistance of the network of several resistance in the circuit.

Make sure you understand Example 20.4 in Ohm's Law: Resistance and Simple Circuits.

 

3c. Calculate effective resistance of a network of resistors

  • Let's say you have two resistors. How can you connect them to a battery so they are connected in series? What is the relationship between the current through each resistor? What is the relationship of the potential differences between the ends of each resistor and the voltage supplied by the battery? Use these considerations to determine the equivalent resistance of two resistors connected in series.
  • If you have two resistors, how can you connect them to a battery so they are connected in parallel? What is the relationship between the current through each resistor? What is the relationship of the potential differences between the ends of each resistor and the voltage supplied by the battery? Use these considerations to determine the equivalent resistance of two resistors connected in parallel.

When two or more resistors are connected in series:

  • They have the same current going through them (this follows from the conservation of charge)

  • The sum of the potential differences between the ends of each resistor equals the voltage supplied by the battery.

  • The equivalent resistance is determined by the formula R_{eq}=R_{1}+R_{2}+\cdots.

When two or more resistors are connected in parallel:

  • They have the same potential difference between the ends, which also equals the voltage supplied by the battery.

  • The sum of currents through each resistor equals the total current in the circuit, or the current drawn from the battery.

  • The equivalent resistance is determined by the formula \frac{1}{R_{eq}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\cdots.

Review this material in Resistors in Series and Parallel. Review a more advanced example in Combo Circuits.

 

3d. Determine the resistance of a cylindrical wire

  • How do conducting materials resist the flow of a current, at the microscopic level?
  • How does the resistance of a wire depend on the resistivity of the material and the length and cross-sectional area of the wire?

Resistivity is a characteristic of a conducting material and describes its ability to allow charges to flow. Resistivity depends on a variety of factors, including the density of atoms in the material and the material's temperature. The resistance of a wire made out of a material with resistivity \rho is proportional to the wire's length (the longer the charges have to travel, the greater the resistance), and inversely proportional to its cross-sectional area (the greater the area, the more pathways for the charges to travel; hence, less resistance), and can be described with R=\rho \frac{l}{A}.

Review this material in Resistance and Resistivity.

 

3e. Compare and contrast voltage and current

  • What conditions are necessary for a current to flow?
  • What are some possible ways to create voltage?

Current is the amount of charge that passes through the cross-section of a conductor during a unit of time: I=\frac{\Delta q}{\Delta t}, or rate of flow of charge. Voltage is the potential difference between two points in space, or between the ends of a conductor. Current describes the motion of charged particles, whereas voltage measures the energy per unit of charge acquired or lost by these particles.

A good way to internalize the difference between current and voltage is to see how the two quantities are measured in practice. Review this in Ammeter and Voltmeter and DC Voltmeters and Ammeters.

 

3f. Use the junction and loop rules to analyze basic circuits

  • What is the junction rule? Explain in terms of conservation of charge.
  • What is the loop rule? Explain in terms of conservative forces and equipotential surfaces.

For some circuits, it is impossible to find an equivalent resistance network of resistors. We can use Kirchhoff's Rules (the Junction Rule and the Loop Rule) to determine the current that flows through each resistor in a circuit. We can apply these to any circuit.

The Junction Rule states that the sum of all currents entering a junction equals the sum of all currents leaving a junction.

The Loop Rule states that the algebraic sum of all changes in electric potential due to electromotive forces of the batteries and the voltage drops across the resistors equals zero for any closed loop of a circuit.

Review the drawings in Kirchhoff's Rules to understand these rules. Try to reproduce the drawings in the examples and make sure you can follow how they are labeled.

 

3g. Explain how a battery works

  • Why is a battery a necessary component of an electrical circuit?
  • What is an electromotive force?

A typical battery is an electrochemical cell. The chemical reaction inside the cell separates its positive and negative ions and makes them move in opposite directions. This results in a potential difference between the two ends of an electrical circuit. In this way, the battery converts chemical energy to electrical energy, and supplies that energy to the circuit. Electromotive force (EMF) is the amount of energy per unit of charge supplied by the battery. Despite the name, it is not a force, but rather work per unit charge, which is measured in volts. For an ideal battery with negligible internal resistance, electromotive force equals the output voltage.

Review batteries and electromotive force in Electromotive Force: Terminal Voltage.

 

3h. Calculate the power in a DC circuit

  • How would you define power using the concepts of work and energy?
  • What are the different ways to calculate the power dissipated by a resistor in DC (direct current) circuits in terms of current, resistance, and voltage?

In electromagnetism, we define power the same way as in mechanics: power is the rate at which work is performed or the rate at which energy is dissipated. As charges move through a conductor, they lose energy, which gets converted into thermal energy or light. The power dissipated by resistor R with current I going through it can be calculated as P=I^{2}R. Alternatively, since the voltage across the resistor is V=IR, the power can be calculated as P=VI=\frac{V^{2}}{R}.

Review the explanation for these formulas in Electric Power and Energy.

 

3i. Calculate the power in an AC circuit

  • What is the difference between direct and alternating current, DC and AC?
  • What is the difference between rms voltage and peak voltage?
  • How does the average power delivered to a lightbulb relate to its resistance and the voltage?

Alternating current periodically changes direction. When connected to a lightbulb or some other resistor, this means the voltage must also change periodically, because Ohm's law states that current (I) and voltage (V) are proportional to each other. In a household AC supply, this periodic reversal occurs at a frequency of 50 or 60 Hertz. To determine how bright a lightbulb gets, you need to know the average power delivered to it, compounded over an entire cycle. Even though the current averages out to zero because it reverses periodically, the power does not average out to zero. This is because the power at every moment in time is P=VI, and if both reverse signs, then their product does not.

To calculate the average power, you have to use values for V and I that best represent the ever-changing currents and voltages. These are called the rms voltage (V_{rms}) and rms current (I_{rms}). Here, "rms" stands for "root-mean-square", which describes a way of forming the average where negative signs are dropped so that they do not lead to everything canceling out to zero. They are related to the peak values V_{0} and I_{0} by

V_{rms}=\frac{V_{0}}{\sqrt{2}} and I_{rms}=\frac{I_{0}}{\sqrt{2}}.


Then the average power is P_{ave}=V_{rms}I{rms}

Review Alternating Current vs. Direct Current for examples of how to calculate the rms values of current and voltage, and how to obtain the average power.

 

Unit 3 Vocabulary

You should be familiar with these terms to complete the final exam.

  • average power
  • battery
  • current
  • Electromotive Force (EMF)
  • Kirchhoff's Rules: Junction Rule and Loop Rule
  • Ohm's Law
  • parallel connection
  • power
  • resistance
  • resistivity
  • resistor
  • rms voltage and current
  • series connection
  • voltage

Unit 4: Magnetism

 

4a. Describe the magnetic field associated with a moving charge; a magnetic dipole; a long, straight current-carrying wire; a wire loop; and, a solenoid

  • Can you sketch the magnetic field lines of a bar magnet? Label the north and south poles and the direction of magnetic field inside and outside of the magnet.
  • Can you sketch the magnetic field lines of a long, straight current-carrying wire for two cases: when the wire is vertical and when the wire is coming in or out of the page? Label the direction of current and the direction of magnetic field.
  • Can you sketch the magnetic field lines of a wire loop for two cases: when the loop is in the plane of the page and when the loop is horizontal? Label the direction of current in the loop and the direction of the magnetic field.
  • Can you sketch the magnetic field lines of a solenoid? Label the direction of the current in the solenoid and the direction of the magnetic field lines inside and outside of the solenoid. Compare this magnetic field to the one of the bar magnet. What would the field look like for an ideal solenoid?

The magnetic field inside of a bar magnet is directed from its south pole to its north pole. As the field lines leave the magnet on the north side, they loop around outside and enter the magnet on the south side. A magnetic dipole is a small bar magnet with north and south poles close together. Any bar magnet looks like a dipole from far away. You can recognize a dipole by its characteristic field-line pattern, shaped somewhat like the wings of a butterfly when viewed from the side.

The magnetic field lines of a long, straight, current-carrying wire form concentric circles around the wire in a plane that is perpendicular to the wire. The direction of the field is determined by the right-hand rule: if you curve the fingers of your right hand in the direction of the field, your thumb will point in the direction of the current, and vice versa. The magnitude of the field is inversely proportional to the distance from the wire: B=\frac{\mu_{0}I}{2\pi r}. Here, I is the current, r is the distance to the wire, and \mu _{0}=4 \pi \times 10^{-7}T \cdot m/A is the magnetic permeability of vacuum. This is the simplest version of Ampere's Law.

Magnetic field lines of a wire loop are perpendicular to the plane of the loop inside the loop. They are nearly straight lines near the center, and have greater curvature near the circumference of the loop. The direction of the field is determined by the right-hand rule: if you curve the fingers of your right hand in the direction of the current in the loop, your thumb will point in the direction of the field near the center, and vice versa. The magnetic field at the center of the loop is B=\frac{\mu _{0}I}{2R}, where R is the radius of the loop.

The magnetic field of a solenoid is the vector sum of the fields of several coaxial wire loops. The field lines are nearly straight inside the solenoid, running parallel to the cylinder axis of the coil, and loop around far away from the solenoid. They are similar to the field lines of a permanent magnet. For an ideal, infinitely-long solenoid, the magnetic field is zero on the outside and uniform inside: B=\mu_{0}nI. Here, the number of loops per unit length of the solenoid is n=\frac{N}{L}.

Review the characteristic "butterfly" shape of the field lines for a magnetic dipole in Figure 22.15 in Magnetic Fields and Magnetic Field Lines. Review this introduction to Ampere's Law in Magnetic Fields Produced by Currents: Ampere's Law.

 

4b. Find the force exerted by a magnetic field on a moving charged particle

  • What quantities does the force on a particle moving in a magnetic field depend on? How can you determine the direction of the force?
  • What quantities does the magnetic force on a current-carrying wire depend on? How can you determine the direction of the force?
  • Describe the interaction between the two long current-carrying wires. When do they repel, and when do they attract?

You can determine the magnetic force on a moving charged particle, also called the Lorentz force, by the charge and velocity of the particle, and the magnetic field. The force equals the charge of the particle times the vector product of the velocity of the particle and the magnetic field. From the definition of the vector product, this means that the magnitude of the force is F=qvB\ sin\theta. Here, \theta is the angle between the velocity and the magnetic field. The direction of the force, for the positively-charged particle, is determined by the right-hand rule and is perpendicular to both vectors. For the negatively-charged particle, the force will be in the opposite direction.

The magnetic force on a current-carrying wire can be found as the sum of the forces on all moving charges creating the current in the wire. For a straight wire of length L, the magnitude of this force is F=ILB, where I is the current. The direction of the force is determined using the right-hand rule: as you curve the fingers of your right hand from the direction of the current toward the magnetic field vector, your thumb will point in the direction of the force.

The Lorentz force is the real reason why the concept of magnetic field exists in the first place. Review this in Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field.

 

4c. Explain the fundamental difference between a magnetic and a non-magnetic material

  • Why are objects made of ferromagnetic materials attracted to bar magnets and electromagnets?
  • How can they be turned into permanent magnets?

Some objects, especially those made out of iron and its alloys, are attracted to permanent magnets. All materials experience changes when placed in a magnetic field. For most materials, however, these effects are weak and we cannot observe them directly.

Paramagnetic materials (such as magnesium and lithium) tend to slightly increase external magnetic fields. The majority of materials are diamagnetic, which tend to slightly decrease external magnetic fields (due to the electromagnetic induction).

Ferromagnetic materials, such as iron or neodymium, become magnetized in external magnetic fields. This means that the external field causes the randomly oriented magnetic moments of atoms within the material to align in the same direction as the field, increasing the total field inside the material. When we remove the external field, there is a delay in the return of these microscopic magnetic moments to their original state. Under certain conditions, an object made out of ferromagnetic material can be magnetized permanently. This is how permanent magnets are made.

Review the section in the text that examines the magnetic properties of materials in Ferromagnets and Electromagnets. It also discusses electromagnets, which use these materials to enhance their magnetic field strength.

 

4d. State Ampere's law for the force between two wires

  • Can you sketch the magnetic field vectors of a current-carrying wire?
  • The unit of electric current is called Ampere. What phenomenon occurs between two wires when they carry a current?
  • How does the magnetic field strength of a current-carrying wire change if you move twice as far away from it?

Ampere's Law for magnetic fields is somewhat analogous to Coulomb' Law for electric fields. It states that the magnetic field around a straight wire is proportional to the net current in the wire. If two such wires are placed parallel to each other, a different version of Ampere's law also specifies the force per unit length between those wires.

In the particular case of a long straight wire with current I_{1} in the field of another long straight wire with current I_{2}, the force between the two wires per unit length has the magnitude F=\frac{\mu_{0}I_{1}I_{2}}{2\pi d}, where d is the distance between the wires. This is the version of Ampere's Law that gave rise to the definition of the unit of current, which is named after Ampere himself. Note that like currents (those running in the same direction) attract, while currents in opposite directions repel. This is in contrast to electrostatic force, which is attractive between unlike charges and repulsive between like charges.

Review this experiment in Force between Parallel Wires With Parallel Currents and Force between Parallel Wires With Anti-parallel Currents.

 

4e. Solve problems involving the motion of a charged particle in a magnetic field

  • What factors determine the magnitude and direction of the force on a charged particle entering a magnetic field?
  • Consider a charged particle entering a region with a uniform magnetic field. What will the particle's acceleration be? Draw the trajectory of a positively charged particle in the field for the cases when initial velocity is parallel to the field, perpendicular to the field, or at an angle to the field lines. What would change if the particle was negatively charged?

Magnetic force on a moving charged particle is given by the Lorentz force.

This means that the magnitude of the force is F=qvB\ sin \theta, where \theta is the angle between the velocity and the magnetic field. Alternatively, this can be written as F=qv_{\bot}B, where v_{\bot}=v\ sin \theta is the component of the velocity of the particle perpendicular to the magnetic field. The force is perpendicular to both velocity and magnetic field vectors.

If a charged particle enters a region with a magnetic field at a velocity perpendicular to the field, the magnetic force will accelerate the particle perpendicular to the velocity, and the particle will move in a circular trajectory in the plane perpendicular to the field. You can find the radius of the trajectory from Newton's Second Law, R=\frac {mv}{qB}.

If a charged particle enters a region with a magnetic field at a velocity parallel to the field, then the perpendicular component of the velocity is zero, so there will be no force on the particle and it will pass through the field undeflected. However, if the velocity has both perpendicular and parallel components, the particle will undergo circular motion in the plane perpendicular to the field, while moving in the original direction parallel to the field. Its trajectory will be a helix.

Review applications of the Lorentz force in Force on a Moving Charge in a Magnetic Field: Examples and Applications.

 

4f. Calculate the torque on a current loop

  • What is the magnitude of the torque exerted on a current-carrying loop placed in a magnetic field?
  • What orientation of the loop relative to the field results in maximum torque?

The Lorentz force tells us that a straight current-carrying wire experiences a force when placed in a magnetic field. But when a wire is bent into a loop, the charges inside the wire move in opposite directions on opposite sides of the loop, so the Lorentz forces on the entire loop should cancel out. Indeed, there is no overall net force on the loop in a uniform magnetic field. But the Lorentz forces still have a twisting action on the loop.

Torque is the twisting action of a force. The torque on a current loop with N windings and cross-sectional area A, carrying a current I and tilted relative to the magnetic field B by an angle \theta is \tau=NIAB\ sin\theta.

The torque is zero when \theta=0, that is, when the plane of the loop is perpendicular to the field. The torque is maximum when \theta is 90° (when the plane of the loop is parallel to the field). As the torque causes the loop to rotate, the electrical energy of the current then converts into the mechanical energy of rotation. This is how motors work.

As you review Torque on a Current Loop: Motors and Meters, make sure you understand the relation between Lorentz force and torque.

 

Unit 4 Vocabulary

You should be familiar with these terms to complete the final exam.

  • Ampere's Law
  • ferromagnetic material
  • magnetic field
  • magnetic force
  • magnetic dipole
  • magnetic moment
  • magnetic permeability
  • magnetization
  • motor
  • Lorentz force
  • paramagnetic material
  • permanent magnet
  • solenoid
  • torque

Unit 5: Electromagnetic Induction

 

5a. Define flux

  • When a circular wire loop is placed in a uniform magnetic field, how does the tilt angle of the loop affect the magnetic flux enclosed by the loop?
  • Can you change the magnetic flux enclosed by a circular loop if you squish the loop into an ellipse?
  • When the magnetic field in a solenoid is doubled, what happens to the magnetic flux through the solenoid?
  • Can you illustrate the concept of magnetic flux through a surface bounded by a loop by drawing an example?

Magnetic flux is a quantity that is important in the phenomenon of magnetic induction, which is how electric generators work. You can visualize magnetic flux by drawing a field line picture of the magnetic field, e.g. with equally spaced lines for a uniform field. Then imagine placing a loop of wire in the picture. The magnetic flux \phi is proportional to the number of field lines that penetrate through the cross-section of the loop.

Essentially, flux measures the "exposure" of the loop to the field. Think of a person trying to catch falling rain with a bucket. If the opening of the bucket points up, you catch more water than if the same bucket is tipped sideways. Likewise, the more field lines a loop catches, the larger the flux is. Mathematically, the magnetic flux through a loop of cross-sectional area A in a uniform magnetic field B has the value \phi = BA\cos \theta, where \theta is the tilt angle of the loop.

So you can change the flux through a loop either by changing the magnetic field, or by changing the shape of the loop such that its area changes, or by orienting the loop at a different angle relative to the field.

Review magnetic flux and its connection to magnetic induction in Induced Emf and Magnetic Flux.

 

5b. State Faraday's and Lenz's Laws

  • What quantity does the induced electromotive force (EMF) in a wire loop depend on? Consider all possible ways that this quantity can be made to be non-zero.
  • How does the sign of the induced EMF (and the direction of the resultant induced current) depend on the sign of the rate of change of magnetic flux? In the example you drew, what will the direction of the induced current be if the flux increases? What will the direction of the induced current be if the flux decreases?

Faraday's Law states that if there is a change of flux through a surface bounded by a loop, there will be an induced electromotive force (EMF) in the loop equal to the rate of change of the flux:

EMF=\frac{-N \Delta \phi}{\Delta t}.

If the loop is made of conducting material, there will be an induced current. This is called magnetic induction.

There are three instances where the rate of change of magnetic flux might not be zero:

  • The field changes. An electromotive force will be induced when the magnetic field is not constant over time.

  • The area enclosed by the loop changes. An electromotive force will be induced if the loop changes shape, such as by stretching or compressing.

  • The angle between the field and the loop changes. An electromotive force will be induced if the loop turns or rotates in the magnetic field. This is a basic principle of operation of a generator: the rotational kinetic energy of a loop converts into electric energy via an induced current.

The negative sign in Faraday's Law indicates that the induced EMF has a sign opposite to that of the rate of change of flux. This leads to Lenz's Law: the induced EMF corresponds to the induced magnetic field that will counteract the change in the flux. That is, if flux increases, the magnetic field of the induced current will be in the opposite direction of the external field. If the flux decreases, the magnetic field of the induced current will be in the same direction as the external field.

Review applications of Lenz's and Faraday's Laws in Faraday's Law of Induction: Lenz's Law.

 

5c. Solve problems using Faraday's Law

  • What attributes of a circuit or wire loop determine the magnitude of the EMF induced in the circuit or loop?
  • What is the relationship between induced current and induced EMF? What determines the direction of the current?

Typical problems involving Faraday's Law include:

  1. Motional EMF problems. In these problems, a conducting rod moves along conducting rails to form a closed circuit, and these are placed in a magnetic field. Due to the motion of the rod, the area of the circuit changes; thus, a motional EMF and a current are induced. If the rod moves with a constant velocity v, the motional EMF is Blv. Here, B is the magnetic field and l is the length of the moving rod.

    The induced current is related to the induced EMF according to Ohm's Law: I=\frac{E}{R}=\frac{Blv}{R}, where R is the total resistance of the circuit. The force required to maintain the rod's motion with a constant velocity once the current is established has to equal the force exerted by the magnetic field on the rod: F=BIl=\frac{B^{2}l^{2}v}{R}.

    Lenz's Law determines the direction of the current: the magnetic field of the induced current is in the same direction as the external field if the area of the circuit decreases, and it is in the opposite direction if the area of the circuit increases.

  2. Problems involving a loop turning or continuously rotating in an external magnetic field. An electromotive force is then induced due to the change of the angle between the surface of the loop and the field lines. If the loop is turned once, the magnitude of the average induced EMF is calculated as change between the initial and final magnetic flux divided by the time during which the turn took place: \left| \frac{\phi_{f}-\phi_{i}}{\Delta t} \right|.

    If the loop rotates with a constant frequency, the flux depends on time as a sine or cosine function (\phi = BA\cos(2\pi ft), where f is the frequency), and the induced EMF is 2\pi BAf\sin(2\pi ft). Keep in mind that when calculating the flux through a coil of wire consisting of several loops, the flux has to be multiplied by the number of loops N, as each loop contributes to the total area.

  3. Problems where the magnetic field through a circuit or wire loop changes due to some external agent. In this case, the magnetic field or its rate of change will be given as a function of time. Sometimes, the change in the magnetic field is due to a time-dependent current through a different wire loop or solenoid.

    In this case, the magnetic field and its rate of change has to be calculated. The magnitude of the induced EMF is then found as \left| \frac{\Delta B}{\Delta t} \right|A, assuming the field is perpendicular to the circuit. Again, the area would have to be multiplied by the number of the loops of wire in the coil if there are several.

You will know you have mastered the idea behind Faraday's Law and magnetic flux if you understand the discussion in Motional Emf and Eddy Currents and Magnetic Damping.

 

5d. Define inductance and explain how it affects the change of current in a circuit

  • How can an element of a circuit (such as a coil of wire or a solenoid) provide resistance to a change of current in a circuit? Explain using Faraday's Law and Lenz's Law.

If the current through a wire loop or a solenoid changes, the magnetic field created by the current also changes. According to Faraday's Law and Lenz's Law, this will cause an electromotive force to be induced in the wire loop or solenoid that is opposite in direction to the electromotive force of the battery supplying the original current. This induced electromotive force will resist the change in current. Inductance is the tendency of a part of the circuit to provide this resistance. Sometimes the EMF is generated in a part of the circuit that is separate from the part where the magnetic field is generated. Transformers are an example: one coil feels the magnetic field generated by another coil.

Sometimes it is important to account for the fact that even a single coil can create a magnetic field and also feel its own magnetic field. When the current through such a coil changes, this creates a changing magnetic flux through the coil itself, and this, in turn, causes magnetic induction. The result is an induced EMF in the coil, in response to the changing current through that same coil. Self-Inductance (L) is the proportionality coefficient between the induced EMF and the rate of change of current:

EMF=-\frac{L\Delta I}{\Delta t},

This is because the induced EMF is equal to the rate of change of magnetic flux, and the flux is proportional to the current. The self-inductance depends on the geometry and magnetic properties of the part of the circuit, but is independent of the current.

For example, the self-inductance of a solenoid with a number of turns N, length l, and cross-sectional area A is L=\frac{\mu_{0}N^{2}A}{l}. The part of the circuit that provides resistance to a change in current is called an inductor. Inductance is sometimes called an electrical inertia, since it measures resistance to the change in current similarly to the way mass, or mechanical inertia, measures resistance to change in velocity.

Inductors are coils purposely designed to have a specific inductance. The property of inductance as related to a magnetic field is analogous to capacitance. Just like how a capacitor can contain an electric field and store electric field energy, an inductor can contain a magnetic field and store magnetic field energy. The magnetic energy stored inside an inductor with inductance L and current I is \frac{LI^{2}}{2}. Notice that this expression is similar in form to the one for kinetic energy (K=\frac{mv^{2}}{2}) and the one for electric energy stored inside a capacitor (\frac{Q^{2}}{2C}).

Review the role of induction in transformers in Transformers. Be sure to review Inductance so you can understand the next section.

 

5e. Analyze RC, RL, and RCL circuits

  • Can you sketch a circuit with a resistor, a capacitor, and a battery connected in series? How will the charge on the capacitor depend on time in this circuit, assuming the capacitor is initially uncharged? What happens to the charge when the battery is shorted out of the circuit?
  • Can you sketch a circuit with a resistor, an inductor, and a battery connected in series? How will the current in this circuit behave? What will happen to the current if the battery is shorted out of the circuit?
  • Can you sketch a circuit with a resistor, a capacitor, and an inductor connected in series to a source of AC (alternating) current? How does the current depend on time in this circuit? What is the natural frequency of the oscillations? What AC frequency will result in resonance (maximum current in the circuit)?

In an RC circuit, as a battery is connected to an uncharged capacitor through a resistor, the charge on the capacitor will build up exponentially until it reaches its maximum value Q_{max}=CV, where V is the voltage supplied by the battery: Q=Q_{max}(1-e^{-t/\tau}).

Here, the constant \tau=RC is called the time constant of the RC circuit. It indicates the time when the charge on the capacitor reaches about two-thirds of its final, or maximum, value. If the battery is shorted out of the circuit, the capacitor will discharge through the resistor and its charge will exponentially approach zero: Q=Q_{max}(1-e^{-t/\tau}).

In an RL circuit, as a battery is connected, the current will tend to increase, but its growth will be slowed down by the inductor. The current will rise exponentially until it reaches its maximum, constant value of I_{max}=\frac{V}{R}, where V is the voltage supplied by the battery: I=I_{max}(1-e^{-t/\tau}).

Here, the constant \tau =\frac{L}{R} is called the time constant of the RL circuit. It indicates the time when the current in the circuit rises to about two thirds of its final value. If the battery is shorted out of the circuit, the current will exponentially decay to zero: {max}(1-e^{-t/\tau}).

In an ideal LC circuit (consisting of an inductor and a capacitor with negligible resistance), the charge on the capacitor will oscillate, as will the current in the circuit. This current is the electronic equivalent of a mechanical system that undergoes simple harmonic motion, such as a mass on a spring. If at t=0 the capacitor is fully charged to charge Q_{max}, the charge will depend on time with a cosine function: Q=Q_{max}\cos(2\pi ft). The frequency of the oscillations (the natural frequency of the system) is f=\frac{1}{2\pi \sqrt{LC}}. The energy in this oscillating circuit is conserved, as it is periodically converted from electric energy by the capacitor to magnetic energy by the inductor.

Since all real circuit elements have resistance, circuits used in real-world applications are RLC circuits. Here, energy dissipates in the resistor, and the amplitudes of voltage and current gradually decrease. If, however, a circuit is connected to an AC voltage source, that source will supply the energy to keep the oscillations going. The maximum amplitude of the current in the circuit is achieved when the frequency of the AC voltage source (the driving frequency) equals the natural frequency of the system, f=\frac{1}{2\pi \sqrt{LC}}. This phenomenon is known as resonance.

Review Charging and Discharging a Capacitor in a Circuit, DC Circuits Containing Resistors and Capacitors, and RL Circuits for a side-by-side comparison of RC and RL circuits. Compare Figure 21.38 for the capacitor with Figure 23.44(b) for the inductor. The difference lies in the variable that is plotted on the vertical axis. It indicates that current and voltage "switch roles" between RL and RC circuits. This is also reflected in the formulas above. Review resonance in a combined circuit containing capacitors and inductors in RLC Series AC Circuits.

 

5f. Compare and contrast electromagnetic generators and motors

  • What are the basic physical principles involved in the operation of a generator?
  • What are the basic physical principles involved in the operation of a motor?

Electromagnetic generators are wire loops that rotate in an external magnetic field that is usually created using bar magnets. As the angle between the surface of the loop and the magnetic field lines changes, the magnetic flux through the loop changes, which induces an electromotive force and current in the loop. The rotational kinetic energy of the loop is thus converted to electrical energy and generates an AC current of the same frequency as the frequency of the rotation of the loop. To produce DC current, a commutator is required, which changes the direction of the EMF every half-period.

In a motor, the torque of an external magnetic field (usually provided by bar magnets) rotates the current-carrying loop of wire, thus converting electric energy into mechanical (kinetic rotational energy). If the current is constant, the torque will simply flip the loop only once, so there has to be a way to change the direction of the current: supply alternating current to the loop, or connect the loop to a commutator which will change the direction of the current periodically.

Then, continuous rotation will be established. However, there will be electromotive force induced in the loop rotating in the magnetic field, which will tend to slow down the rotation because it reduces the total voltage and with it the total current going through the motor; this is called back EMF. Additional modifications are required in order to counteract that effect.

Review generators Electric Generators and the undesirable effects of back EMF in Back Emf.

 

Unit 5 Vocabulary

You should be familiar with these terms to complete the final exam.

  • AC (alternating) current
  • driving frequency
  • Faraday's Law
  • generator
  • frequency
  • induced current
  • induced electromotive force
  • induced magnetic field
  • inductance
  • inductor
  • LC Circuit
  • Lenz's Law
  • magnetic induction
  • magnetic flux
  • motor
  • natural frequency
  • oscillating circuit
  • RC Circuit
  • RL Circuit
  • RLC Circuit
  • resonance
  • self-inductance
  • time constant
  • transformer

Unit 6: Electromagnetic Waves

 

6a. State the information contained in each of Maxwell's equations in words

  • Can electric field lines form loops?
  • Can magnetic field lines emanate from a single point?
  • Is an electric current the only way to create a magnetic field?

There are four Maxwell's equations. Each describes possible sources for electric and magnetic fields:

  1. Gauss' Law for electricity states that the electric field lines begin at positive charges and terminate at negative charges.

  2. Ampere-Maxwell's Law states that a magnetic field can be created by a current or a moving charge, and by the presence of a changing electric field. To understand how a changing electric field would create a magnetic field, think of dielectric material between the plates of a capacitor. When the capacitor charges, a current flows onto one plate and off of the other. In the gap between the plates, there is a movement of charges, too: it is the individual molecules getting polarized. Although there is no way for a charge to cross all the way from one side of the gap to the other, the charges inside each molecule can move slightly toward or away from the plates. Collectively, this movement of charge within the molecules is enough to create a magnetic field – Maxwell called this the displacement current.

  3. Faraday's Law states that an electric field can be created by the presence of a changing magnetic field. The lines of the electric field created in this way form closed loops, and the strength of the field is proportional to the magnetic flux through those loops.

  4. Gauss' Law for magnetism states that there is no such a thing as a magnetic charge: all sources of magnetic fields, such as bar magnets and currents, contain north and south poles. Separating them and creating a magnetic monopole (another term for a hypothetical magnetic charge) is impossible. Magnetic field lines always form closed loops; they do not have starting or ending points, which would be necessary to have a magnetic charge.

Review Maxwell's Equations: Electromagnetic Waves Predicted and Observed.

 

6b. Explain how Maxwell's equations resulted in the prediction of electromagnetic waves and the realization that light was an electromagnetic wave

  • What mathematical form does the solution of Maxwell's equations take that indicates that electric and magnetic fields can propagate?
  • What combination of quantities indicates the speed of propagation? Calculate the value of this combination of quantities to show that it equals the known speed of light.

Maxwell's addition to Ampere's law predicted that a changing electric field can create a magnetic field; on the other hand, Faraday's law states that a changing magnetic field can create a magnetic field. This means that electric and magnetic fields can create each other. This is analogous to how a water wave appears on the surface of a lake when you throw in a pebble: the water level at the impact point changes, and this change creates a slope in the surface. This slope makes the water level at neighboring points change, which in turn changes the slope of the surface next to them, in a repeating chain of events that we call a wave, propagating outward from the center.

Because electric and magnetic fields influence each other in a similar way, they can also create a wave. This is called an electromagnetic wave because neither electric nor magnetic fields alone can create it.

The frequency f and wavelength \lambda depend on the interplay between the inertia of the medium in which the wave is traveling, and the stiffness of the connections between different points in the medium. In all wave phenomena, one finds that the wave speed is related to wavelength and frequency by the same relationship: v=f \lambda.

In an electromagnetic wave, the medium is empty space itself, but even this emptiness has two characteristic quantities that are contained in Maxwell's equations: the dielectric permittivity and magnetic permeability, v=\frac{1}{\sqrt {\varepsilon_{0}\mu_{0}}}.

The speed of propagation v v from Maxwell's equations is v=\frac{1}{\varepsilon_{0}\mu_{0}}. Plugging in the values for \varepsilon_{0} and \mu _{0} results in v=3\times 10^{B}\ m/s. One conventionally uses the symbol c instead of v for this special speed.

At the time, when Maxwell and others were developing the theory of electromagnetism, the speed of light was already measured and known to be equal to this value. Therefore, the fact that electric and magnetic fields propagate with this speed served to indicate that light is an electromagnetic wave. Hertz, who showed that light has the properties that electromagnetic waves are expected to have, confirmed this experimentally.

Review how antennas produce and interact with electromagnetic waves in Production of Electromagnetic Waves.

 

6c. List several examples of electromagnetic phenomena

  • What wavelength and frequency do radio waves exist within? How are radio waves used? What are some common phenomena involving radio waves? How are radio waves produced?
  • What wavelength and frequency do microwaves exist within? How are microwaves used? How are microwaves produced?
  • What wavelength and frequency does visible light exist within? On which side of the visible light spectrum (lower or higher frequency) are infrared and ultraviolet radiation? What are some phenomena that involve infrared and ultraviolet radiation? How is visible light produced?
  • What wavelength and frequency do x-rays exist within? What are some applications of x-rays? How are x-rays produced?
  • What wavelength and frequency do gamma rays exist within? What are some phenomena involving gamma rays? How are gamma rays produced?

Different types of electromagnetic waves are classified according to their frequency or wavelength. Each can be obtained from the other by using the formula \lambda = \frac{v}{f}, where v is the speed of propagation of the wave and f is frequency. Since electromagnetic waves propagate at the speed of light c=3\times 10 ^{8} \ m/s, their wavelength and frequency are related as \lambda = \frac{3\times 10^{8}}{f}. Together, different kinds of electromagnetic waves form a continuous electromagnetic spectrum.

Radio waves are on the longest wavelength/lowest frequency end of the electromagnetic spectrum. They have wavelengths between several centimeters and thousands of kilometers, which correspond to frequencies from 100 Hz to 10^{10} Hz. Devices that operate on radio waves include television and cell phones. The common way to produce, transmit, and receive radio waves is via an antenna: a conductor connected to a circuit with an AC current. Astronomical events also produce radio waves naturally in space.

Microwaves overlap with radio waves but have shorter wavelengths and higher frequencies. Their wavelength is between several millimeters and several centimeters, which correspond to frequencies from 10^{8} to 10^{10} Hz. This is the highest possible frequency of electromagnetic waves that electronic circuits can produce. Molecules can also produce and absorb microwaves, which results in thermal agitation (rotational and vibrational motion). Thus, microwaves can be used to increase the internal energy (and therefore temperature) of various objects. The microwave ovens in our kitchens also use this property.

Visible light is the most familiar and the narrowest part of the electromagnetic spectrum: it ranges from red light, with a wavelength of about 700 nm (7\times 10^{-7} m) and frequency of about 4\times 10^{14} Hz, to violet light, with a wavelength of about 400 nm (4\times 10^{-7} m) and frequency of about 7\times 10^{14} Hz. Visible light is produced by atomic vibrations and electronic transitions within atoms and molecules. Infrared radiation has a frequency just below that of red light (3\times 10^{14} Hz, or wavelength of 800 nm), and is produced by the thermal motion of atoms and molecules. It is also known as thermal radiation, which is one of the ways heat transfers.

Ultraviolet (UV) light, as the name implies, has a higher frequency than violet light (it has wavelengths of 10^{-10} to 10^{-7} m and frequencies of 10^{15} to 10^{18} Hz). Electronic transitions also produce UV light. It can damage skin and affect vision.

X-rays have a wavelength of 10^{-12} to 10^{-10} m and a frequency of 10^{18} to 10^{20} Hz. They overlap with high-frequency UV waves and low-frequency gamma rays. X-rays are produced by high-energy electronic transitions. They are widely known for their applications in medical imaging.

Gamma rays are emitted as a result of nuclear decay and transitions between the energy states of nuclei. Their wavelengths are smaller than 10^{-15} m, which corresponds to frequencies higher than 10^{23} Hz. Nuclear medicine is based on the properties of gamma rays, and we use them in many of the same applications as x-rays.

Review the wide range of phenomena that involve electromagnetic waves in The Electromagnetic Spectrum.

 

6d. Solve problems involving properties of electromagnetic waves

  • What is the relationship between the wavelength and frequency of electromagnetic waves?
  • What does the intensity of electromagnetic radiation depend on?
  • What is the relationship between the magnitude of the electric and magnetic fields in an electromagnetic wave?

Typically, problems involving the properties of electromagnetic waves include:

  1. Calculating wavelength \lambda and frequency f of the wave. They are related to the speed of light in vacuum, c=3\times 10^{8}\ m/s: \lambda=\frac{c}{f}

  2. The electric and magnetic field vectors in an electromagnetic wave are always perpendicular to each other, and the direction of the propagation of the wave is perpendicular to both vectors. The E(t) and B(t) functions are in phase (reach their maxima and minima at the same times) and they have a proportional relationship: E=cB.

  3. The intensity of radiation is measured using power per unit area: I=\frac{P}{A}. For a source of spherical radiation, which spreads out uniformly in all directions, the intensity at distance R away from the source will be I=\frac{P}{4\pi R^{2}}; thus, intensity is inversely proportional to the distance away from the source. The average intensity for propagating an electromagnetic wave is proportional to the product of the amplitudes of the electric or magnetic field: I_{ave}=\frac{E_{max}B_{max}}{2\mu _{0}}.

This can be expressed in terms of the amplitude of electric or magnetic field alone, by using E=cB and c=\frac{1}{\sqrt{\varepsilon_{0}\mu_{0}}}:

I_{ave}=\frac{cB^{2}_{max}}{2\mu_{0}}=\frac{c\varepsilon_{0}E^{2}_{max}}{2}.

Review how energy, intensity, and power help us characterize electromagnetic waves in Energy in Electromagnetic Waves.

 

Unit 6 Vocabulary

You should be familiar with these terms to complete the final exam.

  • Ampere-Maxwell's Law
  • antenna
  • displacement current
  • Faraday's Law
  • flux
  • frequency
  • gamma ray
  • Gauss' Law
  • infrared radiation
  • intensity
  • light
  • magnetic monopole
  • Maxwell's Equations
  • microwave
  • radiation
  • radio wave
  • ultraviolet radiation
  • visible light
  • wave
  • wavelength
  • x-ray

Unit 7: Optics

 

7a. Determine the size, location, and nature of images by using the mirror and lens equations

  • How do mirrors form images?
  • What is the primary difference between concave and convex mirrors?
  • What are the two possible types of images? How is the type of image formed by a mirror determined by its location relative to the mirror?
  • How are focal length and curvature of a mirror related?
  • How are the locations of an object and its image related by the mirror equation?
  • How do lenses form images?
  • What is the primary difference between converging and diverging lenses?
  • How is the type of an image formed by a lens determined by its location relative to the lens?
  • How are the locations of an object and its image related by the lens equation?
  • What is magnification, and what does its sign (positive/negative) indicate about the image?

Mirrors form images of objects by reflection. As the rays of light that an object generates bounce off the reflective surface of a mirror, they intersect and form an image. In specular reflection, the angle between the reflected ray and the normal of the surface is equal to the angle of incidence between the incident ray and the normal of the surface. When the surface is rough or dirty, light rays do not get reflected in this simple manner, and the surface appears dull because the rays do not reproduce an image. This is called diffuse reflection.

A concave parabolic surface (a concave mirror) has the property of reflecting all rays parallel to its axis of symmetry in such a way that they all intersect at one point: the focal point of the parabola.

Spherical surfaces serve as a good replacement for parabolic ones; as long as the incident rays fall near the axis of symmetry, their reflections also intersect at one focal point that is located on the axis of symmetry halfway between the center of curvature and the mirror. Thus, the focal length f of the spherical mirror and its radius of curvature R are related as f=\frac{R}{2}.

Images formed by the intersection of rays reflected back toward the object are real images. If the reflected rays do not intersect, then their extensions intersect behind the mirror, forming a virtual image. A virtual image can be seen, but cannot be captured with a camera film or detector.

When describing the locations of objects and images formed by mirrors, the following conventions are used:

Positions in front of a mirror are considered positive. The distance between an object and the mirror is denoted by d_{o} and is always positive, as the object is always located in front of the mirror. The location of an image is denoted by d_{i}, and is positive when the image is in front of the mirror and negative when the image is behind the mirror. Thus, d_{i} is positive for real images and negative for virtual images.

We define magnification as the ratio of the sizes of the image and the object: M=\frac{h_{i}}{h_{o}}. Here, h_{i} is the height of the image, and h_{o} is the height of the object. If the image is upside down (or inverted) then h_{i} is considered negative, which means M is also negative. A virtual image is always upright; a real image is always inverted.

We can use three rays to draw the images formed by mirrors:

  1. The reflection of the rays parallel to the axis of symmetry pass through the focal point

  2. The reflection of the rays passing through the focal point are parallel to the axis of symmetry

  3. The rays perpendicular to the surface (or passing through the center of curvature, for spherical surfaces) are reflected along the same line

Plane mirrors have an infinite radius of curvature, and thus no focal point. The image of an object located in front of a mirror is located at the same distance behind the mirror. This image is virtual, upright, and has the same size as the object.

Concave mirrors have a reflective surface on the inside of a spherical surface, so their focal point is in front of the mirror and on the same side as an object. Their focal length is positive: f > 0. Concave mirrors can form either real or virtual images, depending on the location of the object.

Convex mirrors have a reflective surface on the outside of a spherical surface, so their focal point is behind the mirror. Their focal length is negative: f < 0. Concave mirrors always form virtual images.

If we know the location p and size h_{o} of the object, we can use the mirror equation: \frac{1}{d_{o}}+\frac{1}{d_{i}}=\frac{1}{f} to find the location of the image. We can use the magnification formula M=-\frac{d_{i}}{d_{o}}=\frac{h_{i}}{h_{o}} to find the size of the image.

Lenses form images of objects by refraction. As rays of light change direction when crossing into or out of the material of the lens (typically glass or plastic), they intersect and form an image. The lens has to be very thin compared to the distance between the lens and object. Lenses are made in a variety of shapes, and they have focal points where the refracted rays converge if the incident rays are parallel to the lens' axis of symmetry. Lenses can be made to be either converging or diverging.

For lenses, the object and image locations are similar to that of mirrors. The distance between an object and the lens is denoted by d_{o}, which is always positive. However, real images are located on the opposite side (behind the lens), so the distance between the image and the lens, d_{i}, is positive is when the image is behind the lens (a real image) and negative when the image is in front of the lens (a virtual image)

For lenses, magnification is defined as with mirrors, as the ratio of the sizes of the image and the object: M=\frac{h_{i}}{h_{0}}. Here, h_{i} is the height of the image and h_{0} is the height of the object.

If the image is upside down (or inverted) then h_{i} is considered negative, which means M is also negative. From geometric considerations, it can be shown that M=-\frac{d_{i}}{d_{0}}. From here, it follows that M is positive when d_{i} is negative, so the virtual image is always upright, and that M is negative when d_{i} is positive, so the real image is always inverted.

Converging lenses refract incident rays so that they bend toward the axis of symmetry, while diverging lenses bend incident rays away from the axis of symmetry. There are three rays that can be used to draw the images formed by the lenses:

  1. The incident rays parallel to the axis of symmetry refract so that the refracted rays pass through the focal point behind the converging lens, or so that their extensions pass through the focal point in front of the diverging lens

  2. The incident rays passing through the focal point refract so that they are parallel to the axis of symmetry

  3. The rays passing through the center of the lens are not refracted

Converging lenses have a positive focal length and can form real and virtual images, depending on the location of the object. Diverging lenses have a negative focal length and always form virtual images.

If the location d_{0} and size h_{0} of the object are known, the location of the image can be found from the lens equation: \frac{1}{d_{0}}+\frac{1}{d_{i}}=\frac{1}{f}. Then, the size of the image can be found using the magnification formula: M=-\frac{d_{i}}{d_{0}}=\frac{h_{i}}{h_{0}}.

Review The Ray Aspect of Light, Image Formation by Mirrors, and the corresponding rules for lenses in Image Formation by Lenses.

 

7b. Solve problems using the law of refraction

  • How are the angle of incidence and angle of refraction related?
  • When does total internal reflection occur? What is a critical angle?

Refraction is the change of direction, or bending, of a light ray when it crosses a boundary between two media. It occurs because the light propagates at different speeds in media with different optical properties, which are determined by the electric permittivity and the magnetic permeability of the media. The speed of light in vacuum is c=3\times 10^{8} m/s. In other media, it is decreased by a factor of n, the index of refraction of the medium: v_{light}=\frac{c}{n}. In air, the index of refraction is very close to 1, so the speed of light in air is considered to be equal to that of vacuum. Other transparent media have indices of refraction greater than 1.

As a ray of light crosses the boundary between two media, its frequency remains unchanged (since frequency depends only on the source of the light), but its speed of propagation, and therefore wavelength, changes. This results in a change of direction of the ray. The direction is determined by the angle the ray makes with the normal of the boundary between the media. The law relating the angle of incidence \theta_{i} to the angle of refraction \theta _{r} is called Snell's Law: \frac{\sin \theta _{i}}{\sin \theta _{r}}=\frac{n_{r}}{n_{i}}. Alternatively, it can be written as n_{i}\sin \theta _{i}=n_{r}\sin \theta _{r}.

Notice that if \theta _{i}=0, that is, the incident light is perpendicular to the boundary, then \theta _{r}=0 as well, so the light will not be refracted.

From Snell's Law, it follows that when light crosses the boundary to the medium with the greater index of refraction (for example, from air to water), the refracted ray will be closer to the normal than the incident ray. Also, the expression for the angle of refraction \sin \theta _{r}=\frac{n_{i}\sin \theta_{i}}{n_{r}} will always have solutions, since the right-hand side of the equation will be less than 1 for any incident angle. However, when light goes from a medium with a greater index of refraction to a medium with a smaller one (as with from water to air), the refracted ray will be further away from the normal than the incident ray.


It can make a 90° angle with the normal, but it cannot go further, since it would then longer be in the refractive medium and would reflect back to the incident medium. This phenomenon is known as total internal reflection. When the incident angle is larger than a certain value called the critical angle, light will not refract, just reflect. The value of the critical angle can be found by setting the angle of refraction to 90°, which means \sin \theta _{r}=1. Then, \sin \theta _{c}=\frac{n_{r}}{n_{i}}.

Review the physics of why lenses work in The Law of Refraction. Refraction and reflection usually come together; their interplay is especially visible in Total Internal Reflection.

 

7c. Explain the interference pattern in a double-slit experiment and what these results mean

  • In the double-slit experiment, light is emitted onto a sheet with two small openings, and a pattern is observed on a screen some distance away. If the light is modeled as a beam of particles, what pattern should be observed on the screen?
  • If light is instead modeled as a wave, what pattern should be observed? What wave property explains this pattern?

The double-slit experiment conclusively demonstrated the wave nature of light. If light was a beam of particles, the pattern on the screen in the double-slit experiment would have consisted of two bright spots in front of the slits. Instead, the double-slit experiment yielded a pattern of alternating bright and dark bands. We can explain this pattern, and predict the location of the bright and dark bands, by explaining light as a wave.

Two light waves with the same frequency and same initial phase leave the slits. When the waves reach the screen, they interfere with one another. By the time they reach the screen, they have traveled different distances and are no longer in phase. The difference between these distances each wave travels is called path difference and is denoted by \delta.

If the path difference equals an integer number m of wavelengths: \delta=m \lambda, then the waves undergo constructive interference, and their amplitudes add up. In this case, we observe a bright band. However, if the path difference is an odd multiple of half wavelengths: \delta =(m+\frac{1}{2})\lambda, the waves will undergo destructive interference and cancel each other out. In this case, we observe a dark band.

From geometric considerations and the assumption that the screen is far away from the sheet with the slits, the locations of bright and dark bands can be found as y_{bright}=\frac{m\lambda L}{a} and y_{dark}=\frac{(m+\frac{1}{2})\lambda L}{a}. Here, L is the distance between the sheet and the screen and a is the distance between the slits. Note that the zeroth bright band occurs at y=0, in the center of the screen, opposite the midpoint between the slits. Note also that the dark and bright bands will be equidistant from one another as long as they are close enough to the center of the screen. Further away from the center of the screen, the brightness of the bright bands becomes less intense, and the bright bands spread further away from one another.

Review Figure 27.13 in Young's Double Slit Experiment.

 

7d. Explain how rainbows are produced

  • Why do light waves of different frequencies refract at different angles?
  • How do these frequencies and angles correspond to the different colors of visible light?

Snell's Law describes how the angle of refraction depends on the ratio of the indices of refraction of the incident and refractive media. For many media (including water and glass), the index of refraction depends slightly on the frequency or wavelength of incident light. This dependency is called dispersion, and a medium for which n depends on f (or \lambda) is called dispersive.

A rainbow is the most familiar natural phenomenon that demonstrates dispersion. The light from the sun contains waves with all frequencies of the visible spectrum. The combination of all these waves is perceived as white light. If there are water droplets in the air, the light refracts when it enters the droplets, and then refracts again as it leaves the droplets. Since the angle of refraction is different for light of different frequencies, waves of different colors separate. Observers see this as a rainbow.

Review how the dependence of refractive index on the color of light can cause light rays of different colors to take different paths in Dispersion: The Rainbow and Prisms.

 

7e. Explain how the Huygens principle leads to diffraction at a single slit

  • Draw a sketch of the light rays sent out by a star. Then add lines representing the wavefronts to this drawing. How is this similar to the ripples in a pond when a pebble is thrown in?
  • Which general principle of wave physics makes it possible to construct a straight wavefront from many circular wavefronts?
  • Why does diffraction prove that light is a wave and not a stream of particles?

As Maxwell predicted, you do not need to know that light is a wave to understand most optical instruments. It is enough to describe light by rays that follow the geometric laws of refraction and reflection. This is because the wavelength of visible light is much shorter than the wavelength of more familiar waves, such as water waves. But the double-slit experiment shows that light is a wave since it forms an interference pattern between two overlapping waves coming from different slits. Diffraction answers the question how these two waves were able to overlap in the first place, after they came through the slits: they can go around corners.

To explain diffraction, think of every point in a slit as the source of a spherical wave, also called a wavelet. The wavefronts coming from this point source look like concentric circles, whereas the rays for the same point source form a star-shaped pattern. This means that rays and wavefronts intersect each other at right angles. When we put many of these point sources next to each other, their wavefronts combine to form a single, new, wavefront. The superposition principle that applies to all harmonic waves makes this possible.

Huygens' principle uses this idea to explain how wavefronts propagate, even around obstacles. It does not involve drawing any rays at all, only wavefronts. The principle states: Every point on a wavefront is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wavefront is a line tangent to all of the wavelets.

You can use this to construct the wavefronts of a light wave with straight wavefronts passing through a slit. The result is that the wavelets at the corners are tangent to a new, curved, wavefront. This curved wavefront propagates outward from the slit in a fan shape, so that light coming from two neighboring slits can overlap and interfere. When a single slit gets very narrow, Huygens' principle says it will act more and more like a single point source of light, with wavefronts that become almost semicircles behind the slit, allowing light to spread out in all directions.

For a single slit that is not too narrow, all of the points inside the slit create their individual wavelets, and these may have path-length differences between them that lead to destructive interference in certain directions, just as we have discussed for the two-slit interference. This means that even a single slit will show an interference pattern if one looks closely at the light reaching a screen behind the slit.

These ideas help explain most of the wavelength-dependent effects you can observe with light. Review Huygens' Principle: Diffraction and see an example of a single slit in Single Slit Diffraction.

 

7f. Use the Rayleigh criterion for the resolution limit at different wavelengths

  • Astronomers cannot use binoculars to see the moons of Jupiter. Why do they need to use a telescope with a large diameter instead?
  • Draw two dots next to each other on the wall, then back away from the wall. At some point, you will not be able to see the dots as distinct anymore. They will blur into one. Why does this happen?
  • Which two quantities can be changed to reduce the angle at which two light sources can still be resolved as two separate spots by an optical instrument?

For a single slit with the width D, the light collected on a screen at angle \theta relative to the forward direction will show destructive interference, leading to dark regions, at angles that satisfy the condition:

D\sin \theta =m\lambda, for m = 1, –1, 2, –2, 3,… .

Here, \lambda is the wavelength of the light. The brightest direction behind the slit is the forward direction, corresponding to \theta =0. But this central bright line (roughly in the shape of the slit itself) is surrounded by darkness at the angle of the first destructive interference, where m=1 or -1. The width of the bright line can be estimated from this angle range. The narrower the slit is, the broader the bright region becomes, because diffraction is more pronounced.

We can do the same if we replace the slit with a circular hole. On a screen behind the hole, we will see a bright spot surrounded by a dark circle of destructive interference. The smaller the hole, the larger the bright spot will be. By calculating the diffraction and interference pattern for this hole, one can find the formula that describes the size of the bright spot. It is usually given in terms of the angle \theta between the forward direction and a ray that goes from the hole to the edge of the bright spot. Aperture is the diameter D of the hole. The angle characterizing the bright spot is given by:

\theta = 1.22 \frac{\lambda}{D}.

The shorter the wavelength \lambda, the smaller the bright spot will be. In this formula, it is assumed that angles are measured in radians, not in degrees. This is not important in the earlier formulas involving \sin \theta, but you will get the wrong result if you do not use radians here.

This case of a circular aperture can be used to estimate how close together two small light sources can be brought, before the bright spots they create behind that same aperture overlap and merge into a single spot. The angle at which this happens is called the resolution limit. In terms of the angle between the light sources, the Rayleigh criterion says that the smallest allowable angle is just equal to the angular size of a single bright spot as given above: \theta = 1.22\ \lambda/ D. This means that we must either use a large aperture D or a short wavelength \lambda to resolve two light sources that are at a small angle to each other.

In astronomy, telescopes with a large aperture are always better. The reason is that astronomers try to resolve celestial objects that may not only be far away (and therefore may have low intensity) but also close together.

For example, to see the moons orbiting a planet such as Jupiter, or to see Saturn's rings, we have to think of each of these as separate sources that send their light through the aperture of our telescope. The angle between Jupiter and its moon Europa, as seen from our telescope, is very small. So we want a large aperture so our telescope can create two clear, distinct bright spots for Jupiter and Europa on the detector, camera film, or on the retina of the eye that looks through that aperture from the other end.

If you cannot change the aperture or wavelength, you can still affect the resolution your eye can achieve. Just move closer to the object you want to resolve. This can also be explained by the Rayleigh criterion.

The reason is that the distance between you and the object determines the angle at which different points of the object will appear from your point of view. The closer you are, the larger the angles get. In this case, the aperture is the diameter of your eye's pupil, and it stays roughly the same. So the resolution limit is the same, but by getting closer you make the object's angular size larger than the resolution limit.

Review the drawing that illustrates how we measure angles with respect to the aperture in Limits of Resolution: The Rayleigh Criterion.

 

7g. Explain how polarized light is created and detected

  • When you take two pairs of 3D glasses in a movie theater and hold them against the light, you can change the brightness by looking through both combined and rotating one of them. Why?
  • Why do polarized sunglasses reduce glare from water surfaces?
  • When you look at a laptop screen with polarized sunglasses, you may see the brightness change if you tilt your head. Why?

Polarized light is created when the electromagnetic wave that the light is made of shows a particularly regular oscillation. This manifests itself in the fact that the electric field at a given point in the wave oscillates back and forth only along a single line: for example up and down, or left and right. The former would be called vertical polarization, the latter horizontal polarization. We learned earlier that the electric and magnetic fields in an electromagnetic wave are perpendicular to each other, and also perpendicular to the direction in which the wave is traveling.

For light created by most sources, such as stars or light bulbs, the light waves that propagate in any given direction are not polarized, because they consist of wave trains (or wave packets) that randomly oscillate with the electric fields pointing in arbitrary directions perpendicular to the general forward direction. But one can create polarized light by sending such unpolarized light waves through a polarization filter.

A special type of adjustable polarization filter can be made using liquid crystals. They change their orientation in response to electric fields, and this orientation determines whether they let certain polarizations pass through. This effect is used in computer and phone displays. It is described by the following formula, which assumes two polarization filters whose polarization axes are tilted relative to each other by an angle \theta. Then the transmitted light intensity I is related to the intensity I_{o} of the original light beam by:

I=I_{o}\cos^{2} \theta.

The square of the cosine means that you get perfect transmission (I=I_{o}), for two angles: \theta equal to 0° and \theta equal to 180°.

Another way to create polarized light is by reflecting unpolarized light from a semi-transparent surface, such as water. The amount of light that is reflected by the surface, compared to the amount transmitted into the material, depends on the polarization of the electromagnetic wave. In particular, if the electric field is in the same plane as the lines that follow the propagation directions of the incident and reflected light, there is a special angle of incidence at which no light is reflected at all. This is called the Brewster angle, \theta_{b}. It depends on the index of refraction of the materials.

For example, if the light comes in from the air and hits a water surface, the Brewster angle is given by \theta _{b}=\arctan(1/n), where n=1.33 is the index of refraction for water.

Review Polarization for examples of polarization and how to create it.

 

Unit 7 Vocabulary

You should be familiar with these terms to complete the final exam.

  • aperture
  • Brewster angle
  • concave mirror
  • converging lens
  • convex mirror
  • diffraction
  • dispersion
  • diverging lens
  • focal length
  • focal point
  • image
  • index of refraction
  • interference (constructive and destructive)
  • lens equation
  • magnification
  • mirror equation
  • object
  • path difference
  • plane mirror
  • polarization
  • polarization filter
  • Rayleigh criterion
  • real Image
  • reflection (specular and diffuse)
  • refraction
  • resolution limit
  • Snell's Law
  • total internal reflection
  • virtual Image

Unit 8: Special Relativity

 

8a. Identify the postulates that form the basis for the Special Theory of Relativity

  • What is an inertial reference frame?
  • According to the Special Theory of Relativity, how do physical laws vary when observed from different inertial reference frames?
  • According to the Special Theory of Relativity, what is the difference between the speed of light and other materials, such as physical objects or mechanical waves?

The Special Theory of Relativity describes how observations of events change when conducted in different inertial reference frames. Inertial reference frames all move with constant velocities relative to one another; there is no preferred inertial frame.

The first postulate of the Special Theory of Relativity is that all physical laws are the same in all inertial frames of reference. In other words, if you try to perform an experiment to determine whether your reference frame is in motion, you would not be successful. All experiments will look exactly the same in all inertial frames.

The second postulate of the Special Theory of Relativity states that the speed of light in a vacuum is a constant, and is approximately 3\times 10^{8} m/s. This means that measurements of the speed of light performed by observers in inertial reference frames traveling at different velocities will all yield the same result. The speed of light in a vacuum is also independent of the source.

Review the origins of Einstein's ideas in Einstein's Postulates.

 

8b. Solve problems involving time dilation and length contraction

  • What is the Lorentz factor, and how does it depend on the velocity of a traveling object?
  • How does time flow differently in reference frames traveling at speeds near the speed of light?
  • How are the measurements of length different in reference frames traveling at speeds near the speed of light?

One consequence of the Special Theory of Relativity (which follows directly from its two postulates) is that measurements of time and length are not the same in different inertial reference frames that move at different relative velocities. When you measure the length of a stick when it is on the ground and again when it is on a moving train, the result will be the same. Also, when you measure the duration of an event occurring on the ground and again on a moving train, you expect the results to be the same. However, according to the Special Theory of Relativity, this is not the case. The difference between the two results is not significant in real life, since a train moves very slowly. However, when that train moves at a speed close to the speed of light, the difference becomes significant.

The changes in lengths and time intervals measured in different inertial frames involve a factor \gamma that depends on the speed v of the relative motion of the frames.

This is called the Lorentz factor: \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}. It turns out that no object can move at speeds larger than the speed of light, c. This means that \gamma is larger than 1 (or equal to 1, for an object at rest).

The time interval t measured by an observer in a frame moving at speed v will be measured as t'=\gamma t=\frac{t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}. Since \gamma is greater than 1, t' is greater than t. Time slows down when measured in a moving reference frame; this phenomenon is known as time dilation.

The length L measured by an observer in a frame moving at speed v will be measured as

L'=\frac{L}{\gamma}=L\sqrt{1-\frac{v^{2}}{c^{2}}}. Since is greater than 1, L' is less than L. Lengths decrease when measured in a moving reference frame; this phenomenon is known as length contraction.

Review time dilation and length contraction in Simultaneity and Time Dilation and Length Contraction.

 

8c. State the law of velocity addition in special relativity

  • What is the maximum speed any object can have?
  • How does the law of velocity addition get modified when objects move at high speeds? Does the end result get larger or smaller than expected from the classical laws of Galileo and Newton?
  • Can the relativistic law of velocity addition be applied to objects that are moving in opposite directions?

Imagine you are running on a moving train, from the back to the front of the compartment. To someone on the ground, it looks as if you are running faster than the train itself because the velocity of the train adds to the velocity with which you are running. But Einstein realized that this cannot actually be completely accurate, because in the hypothetical case that all the velocities are very large they could add up to more than the speed of light. The speed of light, c, is 300,000 km/s, which is the "universal speed limit" for all physical objects. The law of velocity addition as just described must break down when speeds approach c.

To write down the formula that fixes this issue, take the example of a person running on a moving train. Call the velocity of the train v, and the velocity of the running person relative to the train u'. Then the velocity of the person relative to the ground is

u=(v+u')/(1+vu'/c^{2})

The term "velocity addition" does not quite describe what you have to do here, because after adding the velocities, you have to divide the total by the extra factor (1+vu'/c^{2}). The effect is that the result ends up being less than the sum v+u', just as Einstein knew it should, so that the combined velocity u can never exceed the speed of light.

Velocities are vectors, so they have a direction, too. The above formula works for any two velocities that point either in the same or in opposite directions. If the velocities point in opposite directions, you have to call one direction positive and the other direction negative before combining them. But the formula does not work if the velocities are not aligned with each other, as in a boat crossing a river from north to south while the river flows from east to west.

Review Einstein's Law for Velocity Addition with examples in Relativistic Addition of Velocities.

 

8d. Explain the results of the Michelson-Morley experiment using the Special Theory of Relativity

  • What was the goal of the Michelson-Morley experiment?
  • Did the experiment produce the expected result?
  • What postulate of special relativity explains the results of the Michelson-Morley experiment?

The goal of the Michelson-Morley experiment was to measure the speed of Earth relative to the ether, the hypothetical medium where light propagated. At the time, scientists assumed that light would require a medium to propagate because it is a wave, much like sound requires air.

The setup for their experiment involved producing an interference pattern between two beams of light, traveling back and forth at right angles to each other. The interference pattern would depend on the orientation of the interferometer, the time of the day, and the time of year; the changes in the pattern would yield measurements of the speed of the Earth relative to the ether. However, they never detected these types of changes. None of the proposed explanations were able to reconcile this result (or lack of result) with what was known about electricity, magnetism, and wave propagation.

Within the framework of the Special Theory of Relativity, the results of the Michelson-Morley experiment make sense. One of the postulates of the theory is that the speed of light is constant in all reference frames moving at any speed. Since light always propagates at the same speed, there is no need for a medium of propagation relative to which the speed of light should be measured. Thus, there is no need for ether. The results of the Michelson-Morley experiment indicate that ether does not exist. They also confirm that the speed of light is the same in all inertial frames of reference.

Review this brief description in Einstein's Postulates.

 

8e. Explain how the special theory of relativity relates to mass and energy

  • What happens to the energy of an object if its speed approaches the speed of light?
  • How is the energy of an object related to its mass?

An object's kinetic energy is the most directly observable form of energy because it is related to the speed at which the object is moving. But Einstein knew that no object can go faster than the speed of light, so it must become impossible to increase the energy E of a fast-moving object when it is already going at this maximum speed. The formula that contains this insight is relativistic energy:

E=\gamma \ mc^{2}.

m is the rest mass of the object (it is the same mass that appears in Newton's Second Law, so the qualifier "rest" is not really needed). \gamma is the Lorentz factor, which contains the speed: \gamma =\frac{1}{\sqrt{1-v^{2}/c^{2}}}.

When the speed approaches the speed of light c, the Lorentz factor and the energy E blow up to infinity. You cannot get larger than infinity, and this explains why the object cannot get any faster when it reaches this limit.

Review Relativistic Energy to practice applications of these relations.

 

8f. Calculate the rest energy from the mass of an object

  • What happens to the energy of an object if its speed is zero?
  • If a star emits energy in the form of light, how does this affect the mass of the star?

For slow-moving objects, the formula E=\gamma \ mc^{2} agrees with the kinetic-energy formula known from Newton's classical physics, except for a constant term that can be considered just part of the potential energy of the object. This constant term is obtained by setting \gamma =1, corresponding to zero speed, in the formula for the energy:

E=mc^{2}.

This is also called the rest energy of the object. Therefore, Einstein's special relativity predicts that the rest energy and mass of an object are proportional to each other.

Around the same time that Einstein discovered the energy-mass relation above, other physicists also discovered the phenomenon of radioactive decay, where the nucleus of an atom can change its mass while emitting new forms of radiation. The energy \Delta E contained in that radiation is related to the change in mass \Delta m of the nucleus, and Einstein's formula predicts:

\Delta E=(\Delta m)c^{2}

See example 28.7 in Relativistic Energy.

 

Unit 8 Vocabulary

You should be familiar with these terms to complete the final exam.

  • inertial frame of reference
  • length contraction
  • Lorentz factor
  • Michelson-Morley experiment
  • principle of relativity
  • relativistic energy
  • rest energy
  • rest mass
  • Special Theory of Relativity
  • speed of light
  • time dilation
  • velocity addition law