# PHYS102 Study Guide

 Site: Saylor Academy Course: PHYS102: Introduction to Electromagnetism Book: PHYS102 Study Guide
 Printed by: Guest user Date: Friday, September 24, 2021, 11:42 PM

## Navigating the 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 the 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. describe the properties of simple harmonic motion and provide examples

1. What kind of force causes the objects to undergo simple harmonic motion?
2. 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:

$x=x_\mathrm{max}\cos(\omega t+\phi)$

Here, $x_\mathrm{max}$ is amplitude, $\omega$ is angular frequency, and $\phi$ is the phase of the oscillations. The period of the oscillations can be calculated as $T=\frac{2\pi}{\omega}$. In general, $x$ does not have to be a position; it could be any variable describing a system.

Mechanical vibrations and waves are similar to electromagnetic waves. Both mechanical and electromagnetic waves begin with an oscillatory system where vibrations propagate in space. The most basic of all oscillatory phenomena is simple harmonic motion. It can occur in a variety of mechanical systems, as discussed in Vibrations.

Harmonic Motion covers the mathematical description and properties of the simple harmonic motion as applied to a system of a mass on a spring.

### 1b. define the following terms related to wave motion: frequency, wavelength, diffraction, and interference

1. What are the properties of wave motion?
2. How is wave motion different from particle motion?
3. What is superposition?
4. What is the relationship between frequency, wavelength, and velocity of a periodic wave?

Waves surrounding us have such a variety of manifestations – ripples on a surface of water, sound, or light – that it is not easy to realize that 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. When more than one wave is present at the same place, they combine. This interference of waves is known as superposition. Waves also demonstrate diffraction, which is the ability to bend around an obstacle. When a wave encounters a boundary between two media, it undergoes reflection (traveling backward) and transmission (also referred to as refraction).

Periodic waves are characterized by their wavelength, which is the distance the wave travels during one period - the time it takes for the oscillation to go through a full cycle. Thus, 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 a reciprocal of period, or the number of cycles the wave goes through in one second: $f=\frac{1}{T}$. Thus, $\lambda = \frac{v}{f}$.

Read Free Waves to review the properties and mathematical description of wave motion.

### 1c. state Hooke's Law

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

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

Springs and Hooke's Law discusses this system in detail.

### 1d. solve problems using simple harmonic motion

1. What are the necessary conditions for a system to undergo simple harmonic motion?
2. What quantities are necessary to calculate the angular frequency, frequency, and period of oscillations?
3. What quantities are necessary to calculate the total energy of oscillations at a given time
4. 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?

Application problems involving simple harmonic motion typically focus on the relationship between the maximum displacement (amplitude) and maximum velocity of a system, and on calculating angular velocity, period, and energy of the oscillations. These are the formulas you should know that pertain to the situations involving SHM:

• Equations of motion: $x=x_\mathrm{max}\cos(\omega t + \phi)$, $v=-x_\mathrm{max}\omega\sin(\omega t + \phi)$. From here, it can be seen that maximum velocity is $v_\mathrm{max}=\omega x_\mathrm{max}$. Typically, the displacement is maximum at $t = 0$. Then, $\phi = 0$ and $x=x_\mathrm{max}\cos(\omega t)$.
• The acceleration at any time is proportional to displacement: $a_x=-\omega^2 x$.
• The angular frequency depends on the system:
For a mass m on the spring with spring constant $k$, $\omega = \sqrt{\frac{k}{m}}$.
For a mathematical pendulum (point mass on the string of length $l$), $\omega = \sqrt{\frac{g}{l}}$.
• The frequency of the oscillations is $f=\frac{\omega}{2\pi}$ and the period is $T=\frac{1}{f}=\frac{2\pi}{\omega}$.
• The energy of oscillations is conserved and equals the sum of potential and kinetic energy. For a mass-and-spring system the energy is $E=\frac{kx^2}{2}+\frac{mv^2}{2}$.

See the solved examples of application problems in Oscillatory Motion.

### Unit 1 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

• Acceleration
• Amplitude
• Angular frequency
• Displacement
• Energy, potential and kinetic
• Equilibrium
• Force constant (same as spring constant)
• Frequency
• Hooke's Law
• Oscillation
• Period
• Phase
• 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

1. 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.
2. What is the expression for Coulomb's force acting on a charge, in vector form?
3. How does the sign of the product of two charges determine the direction of the force?
4. 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, which is the force between charges, is one of the four fundamental forces of nature. Many macroscopic forces that can be easily observed, such as the normal force and friction, are the result of the electric interactions between charged particles on the microscopic level.

The expression for the electrostatic force between two point charges, known as Coulomb's Law, is $\vec{F_{21}}=k\frac{q_1 q_2}{r^2} \widehat{r}$. Here, $\vec{F_{21}}$ is the force on the second charge from the first charge, $k$ is the electrostatic constant that is equal to $8.99\times 10^9\ \mathrm{N\frac{m^2}{C^2}}$, $r$ is the distance between the two charges, and $\widehat{r}$ is the unit vector that points from the first charge toward the second charge. The electrostatic constant can be expressed as $k=\frac{1}{4\pi\varepsilon_0}$, where the electric permittivity of vacuum (free space), $\varepsilon_0$, is $8.85\times 10^{-12}\ \mathrm{\frac{C^2}{Nm^2}}$. The force on the first charge from the second charge would be the same in magnitude but opposite in direction: $\vec{F_{12}}=-k\frac{q_1 q_2}{r^2}\widehat{r}$.

Notice that the sign of the product of the charges determines the direction of the force: when the charges are the same, the product is positive and the force is repulsive; when the charges are different, the product is negative and the force is attractive.

Coulomb's Law is described in Charge and Electric Force (Coulomb's Law).

### 2b. solve problems involving electric forces, electric fields, and electric potentials

1. 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.
2. What information is necessary to calculate the force on a charge in the electric field or electric potential energy of a charge in the electric field?
3. What is the relationship between electric field and electric potential?

These are formulae pertaining to problems involving electric forces, fields, and potential:

• Coulomb's Law: $\vec{F_{21}}=k\frac{q_1 q_2}{r^2}\widehat{r}$. This is a force between two point or two spherical charges. In case of the spherical charges, $r$ is the distance between their centers.
• The electric field of the point charge, or outside of the uniform spherical charge distribution, is $\vec{E}=k\frac{q}{r^2}\widehat{r}$. For a positive charge, it points radially outward; for a negative charge, it points radially inward.
• The superposition principle states that the electric field of several point charges is a vector sum of the fields of each charge: $\vec{E}=\sum_{i=1}^{n}\vec{E_i}$. The force on a point charge placed in this field is $\vec{F}=q\vec{E}=q\sum_{i=1}^{n}\vec{E_i}$.
• Continuous charge distributions are given by charge densities:

Linear charge distribution, like on a charged string or thin rod, is described by linear charge density, $\lambda$. For uniform distributions, the total charge is $Q= \lambda L$, where $L$ is the length of the string or rod. Otherwise, the total charge is the line integral $Q=\int\! \lambda\ \mathrm{d}x$.

Surface charge distribution, like on a charged plane or other surface, is described by surface charge density, $\sigma$. For uniform distributions, the total charge is $Q = \sigma A$, where $A$ is the area of the surface. Otherwise, the total charge is the surface integral $Q=\int\! \sigma\ \mathrm{d}A$.

Volume charge distribution, like for a charged solid object, is described by volume charge density, $\rho$. For uniform distributions, the total charge is $Q = \rho V$, where $V$ is the volume of the object. Otherwise, the total charge is the volume integral $Q=\int\! \rho\ \mathrm{d}V$.
• The field of a continuous charge distribution at a given point is $\vec{E}=\int \! k \frac{\widehat{r}}{r^2}\ \mathrm{d}q$, where $\mathrm{d}q = \lambda\ \mathrm{d}x$, $\mathrm{d}q = \sigma\ \mathrm{d}A$, or $\mathrm{d}q = \rho\ \mathrm{d}V$ for linear, surface, or volume charge distributions, respectively. Here, $r$ is the distance between each point of the distribution and the given point.
• Potential at a point in the field of a point charge is $V=k\frac{q}{r}$. Potential is a scalar, as opposed to an electric field, which is a vector. Potential at a point in the field of several charges is $V=\sum_{i=1}^{n}k\frac{q_i}{r_i}$, and potential at a point in the field of a continuous charge distribution is $V=\int \! k\ \frac{\mathrm{d}q}{r}$.
• Potential energy of one charge in the field of another is $U=k\frac{q_1 q_2}{r}$, where $r$ is the distance between the charges.
• Electric fields and electric potentials are related as $\vec{E}=-\frac{\mathrm{d}V}{\mathrm{d}\vec{r}}$, or $V=\int \! \vec{E}\ \mathrm{d}\vec{r}$. For the case of a uniform one-dimensional field, this means $V=-E_x x$. 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:

• Calculating electric field or force on a charge due to several point charges by using superposition. See the examples Electricity.
• Calculating the electric field of a continuous distribution of charges. Watch Field of Infinite Plane for an example. There is an example of calculating the field of a uniformly charged rod section 22.8 of Light and Matter. This chapter also contains examples of problems involving the relationship between fields and potentials.
• Calculating electric field using Gauss' Law. See learning outcome 2e below.
• Calculating electric potential of a distribution of charges, and electric potential energy of a system of charges. See examples 5.3 and 5.4 in Electric Potential.

### 2c. explain Gauss' law in words

1. How would you illustrate electric field lines, Gaussian surfaces, and electric flux? Draw a few examples, such as field lines of a positive or negative point charge, or field lines of an infinitely large plane.
2. What is the mathematical expression for electric flux as a surface integral? How is it related to the number of field lines?
3. How do the number of field lines entering and leaving a Gaussian surface relate to the charge enclosed by the surface?

Gauss' Law demonstrates that a field is inversely proportional to the square of the distance from its source. It applies to electric fields and others, such as gravitational fields. It states that the electric flux, or number of field lines leaving a closed surface, is proportional to the electric charge enclosed by that surface: $\Phi_E=\oint \! \vec{E}\ \mathrm{d}\vec{A}=\frac{Q_{\mathrm{enc}}}{\varepsilon_0}$. (Here, $\varepsilon_0=8.85\times 10^{-12}\ \mathrm{\frac{C^2}{Nm^2}}$ is the electric permittivity of vacuum.) If the electric field is constant on the surface, the flux equals $\Phi_E=EA\cos(\theta)$, where $\theta$ is the angle between the field lines and the surface area vector (the vector perpendicular to the surface with the magnitude equal to its area). This means that if there is no charge enclosed by a surface, the net electric flux through the surface is zero. Since the electric field lines have to begin or end on the charges, if there is no charge enclosed by the surface, no new lines can be created or terminated inside the surface; all lines that enter the surface have to leave it.

Read more about Gauss' Law in section 22.7 of Light and Matter.

### 2d. compare and contrast the electric potential and the electric field

1. 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?
2. How are field lines and the direction of a field related to the location of its points of equal potential?

The motion of a particle can be described either 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 would be an electric force, and its potential energy is the energy associated with the electric field. A force usually described for a charge at one point in time, whereas electric potential and electric fields are more usually indicated when a charge moves from one location to another.

Review the definition of the electric potential by watching Electric Potential Energy, Electric Potential, and Voltage. The relevant relationships and examples are discussed in Electric Potential.

### 2e. solve problems using Gauss' Law

1. What information must be known about a charge distribution to calculate it using Gauss' Law?
2. What are the guidelines for drawing a Gaussian surface?
3. What information is necessary to calculate the electric flux through an enclosed surface?

Gauss' Law can be used to calculate a field in situations where the charge distribution has symmetry, usually spherical or cylindrical. Do not attempt to use derived formulas, such as for the electric field of a sphere, and adjust them to a different problem; these will not work. Always start with Gauss's Law itself, pick an appropriate Gaussian surface, and find the charge enclosed by this surface. When you need to calculate the flux (not field), all you need to know is how much charge is enclosed by the surface in question; the shape of the surface is irrelevant.

Watch Gauss' Law and Application to Conductors and Insulators to review, and see Gauss' Law for some solved examples.

### 2f. solve problems involving the motion of charged particles in an electric field

1. 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?
2. 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 the 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 $\vec{a}=\frac{q\vec{E}}{m}$, where $\vec{E}$ is the field, $q$ is the charge, and $m$ is the mass of the particle. The trajectory is determined by 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 and be accelerated or decelerated by the electric force. However, if there is an angle between the initial velocity and the field, the trajectory will be parabolic, like the trajectory of a projectile in free fall.

Motion of a Charge in an Electric Field illustrates this effect.

### 2g. define capacitance and describe the factors that determine capacitance

1. When is a parallel-plate capacitor considered ideal (the electric field is uniform between its plates and zero outside)?
2. Consider an ideal parallel-plate capacitor with surface charge density $+\sigma$ on one plate and $-\sigma$ on another. Assume that the area of the plates and the distance between them is known. What is the electrical potential between the plates? How would you use that to find the capacitance of the parallel-plate capacitor?
3. If you have two capacitors, how would you connect them to a battery so that they are connected in 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.
4. If you have two capacitors, how would you connect them to the battery so that they are connected 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 a charge is placed on the surface of a conductor or combination of conductors, the resultant electric potential 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, $C=\frac{\varepsilon_0 A}{d}$, where $\varepsilon_0$ is the 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.

Watch Capacitors and Capacitance and Capacitance, and read Charge Storage, Breakdown, and Capacitance to review these concepts.

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 is determined by the formula $\frac{1}{C_\mathrm{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots$

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 $C_\mathrm{eq}=C_1+C_2+\cdots$

The series and parallel connection of capacitors is discussed in Capacitors in Series and Capacitors in Parallel. Also, see this solved example of determining an equivalent capacitance of a circuit.

### 2h. describe the effect of a dielectric material in a capacitor

1. 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?
2. 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 a dielectric material is placed inside an electric field, the positive and negative charges inside the material experience forces that act in opposite directions. As a result, the molecules inside the dielectric rotate so that the positive charges move toward the direction of the field, while the negative charges move in the direction opposite to the field. The separation of the charges creates the internal field of the dielectric, which is directed opposite to the external field. Therefore, the magnitude of the net field inside the dielectric is smaller than the external field by $K$, 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 $K$.

Dielectrics contains the values of dielectric constants for various materials. See the detailed discussion of how capacitors are affected by dielectrics in Dielectrics in Capacitors and this solved example.

### 2i. define electric potential energy and describe how capacitors can be used to store energy

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

Since work must be performed in order to charge a capacitor, the charged capacitor stores energy equal to that work. The energy stored in a capacitor with the charge $Q$ is $U=\frac{Q^2}{2C}$. Alternatively, since charge is proportional to the voltage as $Q=CV$, the stored energy can be expressed as $U=\frac{1}{2}QV=\frac{CV^2}{2}$.

Refer to Energy of a Capacitor, Energy Stored by Capacitors, and this solved example.

### Unit 2 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

• Capacitance
• Capacitor
• Charge
• Charge density (linear, surface, and volume)
• Coulomb's Law
• Dielectric constant
• Dielectric material
• Electrostatic (Coulomb's) force
• Electric field
• Electric flux
• Electric permittivity
• Electric potential
• Electric potential energy
• Field line
• Gauss' Law
• Gaussian surface
• Parallel connection
• Series connection
• Superposition principle
• Voltage

## Unit 3: Electronic Circuit Theory

### 3a. state Ohm's law in words

1. What happens when a potential difference is applied to the ends of a conductor? What physical quantity measures the rate of the flow of charge?
2. 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 a conductor is placed in an electric field (for example, if its ends are connected to a battery, so that there is a potential difference between each end), the free charges inside the conductor will begin to move. The rate of the flow of charge is called current: $I=\frac{\mathrm{d}q}{\mathrm{d}t}$. The ratio between the applied potential difference and the current is defined as resistance: $R=\frac{\Delta V}{I}$. 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: $\Delta V=IR$.

Resistance and Ohm's Law are discussed in section 21.6 of Light and Matter and summarized in Ohm's Law.

### 3b. apply Ohm's law to simple circuits

1. 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.

Watch Circuits and Ohm's Law to review the application of Ohm's Law to a simple circuit.

### 3c. calculate effective resistance of a network of resistors

1. If you have two resistors, how would you connect them to a battery so that they are connected in series? What will the relationship between the current through each resistor be in this case? What will the relationship of the potential differences between the ends of each resistor and the voltage supplied by the battery be? Use these considerations to determine the equivalent resistance of two resistors connected in series.
2. If you have two resistors, how would you connect them to a battery so that they are connected in parallel? What will the relationship between the current through each resistor be in this case? What will the relationship of the potential differences between the ends of each resistor and the voltage supplied by the battery be? 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 $R_\mathrm{eq}=R_1+R_2+\cdots$

When two or more capacitors 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 $\frac{1}{R_\mathrm{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots$

The series and parallel connection of resistors is discussed in Resistors in Series and in Parallel and can be seen in this solved example of determining an equivalent resistance of a circuit. Also watch Resistors in Series, Resistors in Parallel, and Analyzing a More Complex Resistor Circuit.

### 3d. determine the resistance of a cylindrical wire

1. How do conducting materials resist the flow of a current, at the microscopic level?
2. 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. It 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 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 $R = \rho\frac{l}{A}$.

An explanation of Ohm's Law on the microscopic level is discussed in Resistance and Resistivity. The formula for the resistance of a wire is applied in this solved example. Also watch Resistivity and Conductivity.

### 3e. compare and contrast voltage and current

1. What conditions are necessary for a current to flow?
2. What are some possible ways to create voltage?

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

The ways to establish current and voltage are discussed in sections 21.3 to 21.5 of Light and Matter.

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

1. What is the junction rule? Explain in terms of conservation of charge.
2. 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 of a network of resistors. The current through each resistor in such a circuit can still be determined by using Kirchhoff's Rules, which are the Junction Rule and the Loop Rule. These can be applied 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.

Kirchhoff's Rules are explained and illustrated in Kirchhoff's Rules and in this solved example.

### 3g. explain how a battery works

1. Why is a battery a necessary component of an electrical circuit?
2. 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. The amount of energy per unit of charge supplied by the battery is called electromotive force. 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.

For more about the role of a battery in a circuit, read section 21.5 of Light and Matter, Electric Circuits, and EMF and Internal Resistance.

### 3h. calculate the power in a DC circuit

1. How would you define power using the concepts of work and energy?
2. 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, power is defined the same way as in mechanics: it is a rate at which work is performed, or rate at which energy is dissipated: $P=\frac{\mathrm{d}W}{\mathrm{d}t}$, or $P=\frac{\mathrm{d}E}{\mathrm{d}t}$. 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 $\Delta V = IR$, the power can be calculated as $P=I\Delta V=\frac{\left ( \Delta V \right )^2}{R}$.

To review, read Energy in DC Circuits and this solved example.

### Unit 3 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

• Battery
• Conductivity
• Current
• Electromotive Force (EMF)
• Kirchhoff's Rules: Junction Rule and Loop Rule
• Ohm's Law
• Parallel connection
• Power
• Resistance
• Resistivity
• Resistor
• 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

1. Sketch the magnetic field lines of a bar magnet. Label the north and south poles and the direction of the magnetic field inside and outside of the magnet.
2. 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 a magnetic field.
3. Sketch 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 the current in the loop and the direction of the magnetic field.
4. 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 bar magnet. How 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.

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: $B=\frac{\mu_o I}{2\pi r}$. Here, $I$ is the current, $r$ is the distance to the wire, and $\mu_0=4\pi \times 10^{-7}\ \mathrm{\frac{Tm}{A}}$ is the magnetic permeability of vacuum.

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 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: $B=\mu_0 n I$. Here, $n$ is the number density of the loops, and the number of loops per unit length of the solenoid is $n=\frac{N}{L}$.

Review magnetic fields in section 22.3 of College Physics and Magnetic Field Created by Current in a Wire.

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

1. What quantities does the force on a particle moving in a magnetic field depend on? How can you determine the direction of the force?
2. What quantities does the magnetic force on a current-carrying wire depend on? How can you determine the direction of the force?
3. Describe the interaction between the two long current-carrying wires. When do they repel, and when do they attract?
4. 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 magnetic force on a moving charged particle, sometimes also called the Lorentz force, is determined by the charge and velocity of the particle, and the magnetic field: $\vec{F}=q \vec{v}\times\vec{B}$. 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 $F=qvB\sin(\theta)$, where $\theta$ 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.

To review the properties of vector products of two vectors, watch Cross Product Part I and Part II. The force on a charge moving in a magnetic field is discussed in Magnetic Force on a Charge and in section 22.4 of College Physics.

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. Read The Lorentz force and section 22.7 of College Physics, and watch Magnetic Force on a Current-Carrying Wire.

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 $F=\frac{\mu_o I_1 I_2}{2 \pi d}$, where $d$ is the distance between the wires. Note that like currents (those running in the same direction) attract, while the current in the opposite direction repels. This is in contrast to electrostatic force, which is attractive between unlike charges and repulsive between like charges. To review, read Ampere's Law and watch Magnetic Force between Two Current in the Same Direction and Magnetic Force between Opposing Currents.

The torque due to the magnetic force on a wire loop placed in a magnetic field is a vector product of the magnetic moment of the loop and the field: $\vec{\tau}=\vec{\mu}\times\vec{B}$. The magnetic moment of the loop is a vector perpendicular to the area of the loop, and has a magnitude equal to the product of the current in the loop and the area of the loop: $\mu =IA$. The direction of the magnetic moment is determined by the right-hand rule. The magnitude of the torque is then given by $\tau = IAB\sin(\theta)$, where $\theta$ is the angle between the magnetic moment of the loop and the magnetic field. The torque is zero when $\theta = 0$, when the plane of the loop is perpendicular to the field. The torque is maximum when $\theta = 90^{\circ}$, that is, when the plane of the loop is parallel to the field. As the torque causes the loop to rotate, the interaction between the current and the magnetic field causes the loop to move. The electrical energy of the current then converts into mechanical energy of rotation. This is the basic principle of a motor, as discussed in section 22.8 of College Physics.

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

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

Some objects, particularly ones 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 very weak and cannot be observed 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 the external field is removed, 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.

To review, read sections 22.1 and 22.2 of College Physics and Origins of Permanent Magnetism and watch Magnetism.

### 4d. state Ampere's law

1. Sketch the magnetic field vectors of a current-carrying wire wrapped around a loop centered on the wire.
2. What is the mathematical expression for the circulation of a magnetic field expressed as a line integral?
3. How is the circulation of a magnetic field around an Amperian loop related to the amount of current passing through the surface bounded by the loop?

Ampere's Law for magnetic fields is somewhat analogous to Gauss' Law for electric fields. It states that the circulation of a magnetic field around a closed loop is proportional to the net current entering and leaving the surface bounded by the loop: $\oint \! \vec{B}\ \mathrm{d}\vec{l}=\mu_0 I$. Here, $\mu_0=4\pi \times 10^{-7}\ \mathrm{\frac{Tm}{A}}$ is the magnetic permeability of vacuum.

Ampere's Law is described in Ampere's Circuital Law.

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

1. What factors determine the magnitude and direction of the force on a charged particle entering a magnetic field?
2. 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 a product of the charge and cross-product of the velocity of the particle and the magnetic field: $\vec{F}=q\vec{v}\times\vec{B}$.

From the definition of the vector product, 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 $F=qv_{\perp }B$, where $v_{\perp}=v\sin(\theta)$ 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. The radius of the trajectory can be found from Newton's Second Law, $R=\frac{mv}{qB}$. The angular frequency of the particle's circular motion, also known as its cyclotron frequency, is $\omega = \frac{v}{R}=\frac{qB}{m}$. It does not depend on the speed of the particle, but only on its charge and mass and the strength of the magnetic field. These formulas are derived in Charged Particle in a Magnetic Field. Magnetic Force on a Proton describes the circular motion of a charged particle in a magnetic field in detail. Also, see this solved example.

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. See section 22.5 of College Physics for further discussion of how magnetic fields affect the trajectories of particles in various applications.

### 4f. solve problems using Ampere's law

1. What do you need to know about a current to calculate its magnetic field using Ampere's Law?
2. What are the guidelines for drawing an Amperian loop?
3. What information is necessary to calculate the strength of a magnetic field around a closed loop?

Ampere's Law can be used to calculate a magnetic field in situations when the current distribution is such that the resultant field has radial or cylindrical symmetry. These include the current in a long straight wire, along the surface or volume of a long cylinder, or in a wire coiled to make a solenoid or a toroid. When you need to calculate the circulation, but not the field, all you need to know is how much current passes through the surface bounded by the loop in question; the shapes of the surface or loop are irrelevant.

See the examples of calculating magnetic fields using Ampere's Law in Magnetic Field of a Solenoid and in this solved problem.

### Unit 4 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

• Ampere's Law
• Amperian loop
• Circulation of field
• Diamagnetic material
• Ferromagnetic material
• Magnetic field
• Magnetic flux
• Magnetic force
• Magnetic dipole
• Magnetic moment
• Magnetic permeability
• Magnetization
• Motor
• Paramagnetic material
• Permanent magnet
• Solenoid
• Torque

## Unit 5: Electromagnetic Induction

### 5a. state Faraday's and Lenz's laws

1. Illustrate the concept of magnetic flux through a surface bounded by a current loop of electric flux by drawing an example. What quantities does the magnetic flux depend on?
2. 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.
3. 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 EMF in the loop equal to the rate of change of the flux: $E_\mathrm{ind}=-\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}$. If the loop is made of conducting material, there will be an induced current.

The magnetic flux through a surface, as with flux of any field, is the surface integral of the field: $\Phi_B=\oint_{S} \! \vec{B}\cdot \mathrm{d}\vec{A}$. For a uniform field, this is a cross-product of the field and the surface area vector: $\Phi_B=\vec{B}\cdot \vec{A}=BA \cos({\theta})$, where $A$ is the area of the surface and $\theta$ is the angle between the field and the area vector (the vector perpendicular to the surface, with the magnitude equal to the area of the surface).

There are three ways that the rate of change of magnetic flux might be non-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 the 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,$E_\mathrm{ind}=-\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}$, 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.

To review Faraday's and Lenz's Laws and related concepts, watch Lenz's and Faraday's Laws and read Faraday's Law, Lenz's Law, Magnetic Induction, and Motional EMF.

### 5b. solve problems using Faraday's law

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

Typical problems involving Faraday's Law include:

• "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 $E=Blv$. 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: $F=BIl=\frac{B^2 l^2 v}{R}$. The direction of the current is determined by Lenz's Law: 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. Refer to this solved example.
• 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 the change between the initial and final magnetic flux divided by the time during which the turn took place: $\left | E \right | = \left | \frac{\Phi_f - \Phi_i}{\Delta t} \right |$. If the loop rotates with a constant angular velocity, the flux depends on time as a sine or cosine function ($\Phi = BA\cos(\omega t)$, where $\omega$ is the angular velocity), and the induced EMF is the derivative: $E = BA\omega\sin(\omega t)$. 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. Refer to this solved example.
• 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 have to be calculated. The magnitude of the induced EMF is then found as $\left | E \right |=\left | \frac{\mathrm{d}B}{\mathrm{d}t} \right |A$, 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. Refer to this solved example.

### 5c. define inductance and explain how it affects the change of current in a circuit

1. 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; the tendency of a part of the circuit to provide this resistance is called inductance.

Inductance is the proportionality coefficient between the induced EMF and the rate of change of current: $E_\mathrm{ind}=-L\frac{\mathrm{d}I}{\mathrm{d}t}$. 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 inductance depends on the geometry and magnetic properties of the part of the circuit but is independent of the current. For example, the 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 similar to the way mass, or mechanical inertia, measures resistance to change in velocity.

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 $U_B=\frac{LI^2}{2}$. 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 ($U_E=\frac{Q^2}{2C}$).

To explore the concept of inductance further, watch Lenz's and Faraday's Laws and read Mutual Inductance, Self-Inductance, and Energy Stored in an Inductor.

### 5d. analyze RC, RL, and RCL circuits

1. 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?
2. 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?
3. 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 $Q_\mathrm{max}=CV$, where $V$ is the voltage supplied by the battery: Q=Qmax(1 - e-t/). Here, the constant =RCis called the time constant of the RC circuit. In 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: $Q=Q_\mathrm{max}e^{-t/\tau}$. To review how to analyze RC circuits, refer to The RC Circuit.

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 $I_\mathrm{max}=\frac{V}{R}$, where $V$ is the voltage supplied by the battery: $I=I_\mathrm{max} \left ( 1-e^{-t/\tau} \right )$. 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: $I=I_\mathrm{max}e^{-t/\tau}$. To review how to analyze RL circuits, refer to The RL Circuit.

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 that undergoes. If at $t = 0$ the capacitor is fully charged to charge $Q_\mathrm{max}$, the charge will depend on time with a cosine function: $Q=Q_\mathrm{max}\cos(\omega t)$. The angular frequency of the oscillations (the natural frequency of the system) is $\omega = \frac{1}{LC}$. The energy in this 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 on the resistor, and the amplitudes of charge 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, $\omega = \frac{1}{LC}$. This phenomenon is known as resonance. Read about RLC circuits and their analogy to mechanical systems in sections 25.2 and 25.3 of Light and Matter.

### 5e. compare and contrast electromagnetic generators and motors

1. What are the basic physical principles involved in the operation of a generator?
2. What are the basic physical principles involved in the operation of a motor?

Electromagnetic generators are wire loops rotating 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. The basic principles of the operation of generators are described in The Alternating Current Generator and The Direct Current Generator.

In a motor, the torque of the 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, the 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. Additional modifications are required in order to counteract that effect. The basic principles of operation of motors are described in Professor Fitzpatrick's articles The Alternating Current Motor and The Direct Current Motor. Also, watch Electric Motors.

### Unit 5 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

• AC (alternating) current
• Driving frequency
• Generator
• 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

## Unit 6: Maxwell's Equations

### 6a. state Maxwell's equations and identify each mathematical operator and physical quantity in the equations

1. What is the mathematical expression for the flux of a field? What does the flux of a field represent in the real world?
2. Write the mathematical expression for the circulation of a field. What does the circulation of an electric or magnetic field represent in the real world?
3. According to Maxwell's equations, what are the flux and circulation of an electric field equal to? What are the flux and circulation of a magnetic field equal to?

The flux of a field through a surface is the surface integral of the field vector. For example, the flux of an electric field through the surface $S$ is $\Phi_E=\int_S \vec{E}\ \mathrm{d}\vec{A}$. The flux through a surface indicates the number of field lines passing through the surface. For a closed surface, positive flux indicates the number of the field lines leaving (or coming out of) the surface, while negative flux indicates the number of lines entering the surface.

Maxwell's equations that involve flux are also Gauss' Laws for electricity and magnetism. Gauss' Law for electricity states that the flux of the electric field through a closed surface equals the charge enclosed by the surface divided by electric permittivity of vacuum: $\Phi_E=\oint \vec{E}\ \mathrm{d}\vec{A}=\frac{Q_\mathrm{enc}}{\varepsilon_0}$, where $\varepsilon_0=8.85\times 10^{-12}\ \mathrm{\frac{C^2}{Nm^2}}$.

Gauss' Law for magnetism states that the flux of a magnetic field through any closed surface is always zero: $\Phi_B=\oint \vec{B}\ \mathrm{d}\vec{A}=0$.

The circulation of a field is the line integral of the field vector. For an electric field, it is expressed as $\int_l \vec{E}\ \mathrm{d}\vec{l}$. This is also the expression for the work that the electric field performs on a unit charge. For an electrostatic field, which is conservative, the circulation of the field around a closed loop is zero: $\oint \vec{E}\ \mathrm{d}\vec{l}=0$. However, this changes in the presence of changing magnetic flux. One of Maxwell's equations, Faraday's Law, states that the circulation of electric flux around a closed loop equals the negative rate of change of the magnetic flux through the surface bounded by the loop: $\oint \vec{E}\ \mathrm{d}\vec{l}=\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}$.

For magnetic fields, circulation is expressed as $\int_l \vec{B}\ \mathrm{d}\vec{l}$. This is not equivalent to work, however, because the magnetic force is perpendicular to the field. Ampere-Maxwell's law states that the circulation of a magnetic field around a closed loop equals the magnetic permittivity of vacuum times the sum of the current and the displacement current through the surface bounded by the $\oint \vec{B}\ \mathrm{d}\vec{l}=\mu_0\left ( I+I_d \right )$. Here, $\mu_0=4\pi\times 10^{-7}\ \mathrm{\frac{Tm}{A}}$ is the magnetic permeability of vacuum. The displacement current is not an actual current, but a quantity with the same dimensions as current, defined as the rate of change of electric flux times the electric permittivity of vacuum: $I_d=\varepsilon_0\frac{\mathrm{d}\Phi_E}{\mathrm{d}t}$.

Note that each of these statements about Maxwell's equations is for fields in a vacuum, as opposed to media that have different electric permittivity and magnetic permeability. Read about Maxwell's equations in the introduction to A Dash of Maxwell's, which summarizes how all that is known about electric and magnetic fields is expressed in these four equations.

### 6b. state in words the information contained in each of Maxwell's equations

1. According to Gauss' Law for electric fields and Faraday's Law, what are the possible sources of electric fields? How do electric fields differ when created by different kinds of sources?
2. According to Ampere-Maxwell's Law and Gauss' Law for magnetic fields, what are the possible sources of magnetic fields? What features are common to all magnetic fields?

There are four Maxwell's equations, each of which describes possible sources for electric and magnetic fields:

Gauss' Law for electricity states that the electric flux through a closed surface is proportional to the net charge enclosed by the surface. This means that an electric field is created by the presence of an electric charge. Electric field lines begin at positive charges and terminate at negative charges.

Ampere-Maxwell's Law states that the circulation of a magnetic field around a closed loop is proportional to the sum of currents through the surface bounded by the loop plus the displacement current through that surface. The displacement current is a quantity proportional to the rate of change of the electric flux through the surface. This means that a magnetic field can be created by a current or a moving charge, and by the presence of changing electric flux.

Faraday's Law states that the circulation of an electric field around a closed loop equals the negative rate of change of the magnetic flux through that loop. This means that an electric field can be created by the presence of changing magnetic flux. The lines of the electric field created in this way form closed loops. Unlike the field created by stationary electric charges, this field not conservative.

Gauss' Law for magnetism is analogous to Gauss' Law for electricity. It states that the magnetic flux through any closed surface is zero. This means that there is no such a thing as a magnetic charge: if there was a magnetic charge similar to an electric charge, it would be possible to surround it with a surface, and the magnetic flux through that surface would be non-zero. This is not the case, however: all sources of magnetic fields, such as bar magnets and currents, contain north and south poles. Separating them and creating a magnetic monopole, or a 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.

Watch Maxwell's Equations and Electromagnetic Waves I to review each of Maxwell's equations. You can skip the first twenty minutes, which are devoted to the derivation of the mechanical wave equation.

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

1. What mathematical form does the solution of Maxwell's equations take that indicates that electric and magnetic fields can propagate?
2. 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.

The solution of Maxwell's equations, which show that electric and magnetic field are a function of time and space, take the form of $\vec{E}(x,t)=\vec{E_0}\cos{(kx-\omega t)}$ and $\vec{B}(x,t)=\vec{B_0}\cos{(kx-\omega t)}$. Alternatively, these can be written as $\vec{E}(x,t)=\vec{E_0}\cos[{k(x-vt)}]$ and $\vec{B}(x,t)=\vec{B_0}\cos[{k(x-vt)}]$. This kind of function of the argument $x-vt$ describes a wave propagating with a velocity $v$. Here, the angular velocity $\omega$, wave number $k$, and speed of propagation $v$ are related as $v=\frac{\omega}{k}$. The speed of propagation $v$ from Maxwell's equations is $v=\frac{1}{\sqrt{\varepsilon_0 \mu_0}}$. Plugging in the values for $\varepsilon_0$ and $\mu_0$ results in $v=3\times 10^{8}\ \mathrm{m/s}$.

At the time when the theory of electromagnetism was being developed by Maxwell and others, 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. This was confirmed experimentally by Hertz, who showed that light has the properties that electromagnetic waves are expected to have.

For a detailed explanation of how Maxwell's equations result in the prediction of the existence of electromagnetic waves, watch Maxwell's Equations and Electromagnetic Waves II and read Why Things Radiate.

### 6d. list several examples of electromagnetic phenomena

1. 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?
2. What wavelength and frequency do microwaves exist within? How are microwaves used? How are microwaves produced?
3. 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?
4. What wavelength and frequency do x-rays exist within? What are some applications of x-rays? How are x-rays produced?
5. 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 with the speed of light $c=3\times 10^8\ \mathrm{m/s}$, 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 1010 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. Radio waves are also naturally produced in space by astronomical events.

Microwaves overlap with radio waves but have shorter wavelengths and higher frequencies. Their wavelength is between several millimeters to several centimeters, which correspond to frequencies from 108 to 1010 Hz. This is the highest possible frequency of electromagnetic waves that can be produced by electronic circuits. Microwaves can also be produced and absorbed by molecules, 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. This property is widely used in the household appliance of the same name.

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 × 10-7 m) and frequency of about 4 × 1014 Hz, to violet light, with a wavelength of about 400 nm (4 × 10-7 m) and frequency of about 7 × 1014 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 × 1014 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 1015 to 1018 Hz). UV light is also produced by electronic transitions, and it is known to have an effect on vision and can damage the skin.

X-rays have a wavelength of 10-12 to 10-10 m and a frequency of 1018 to 1020 Hz. They overlap with high-frequency UV waves and low-frequency gamma rays. X-rays are produced by high-energy electronic transitions and 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 1023 Hz. Nuclear medicine is based on the properties of gamma rays, and they are used in many of the same applications as x-rays.

See the detailed description of various electromagnetic phenomena and their applications in section 24.3 of College Physics.

### 6e. solve problems involving properties of electromagnetic waves

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

Typically, problems involving the properties of electromagnetic wave include:

• Calculating wavelength $\lambda$, frequency $f$, angular frequency $\omega$, or wave number $k$ of the wave. The appropriate formulas are:

$\lambda=\frac{c}{f}$ ($c =3\times10^8 \mathrm{m/s}$ is the speed of light in vacuum)

$\omega=2\pi f$

$k=\frac{2\pi}{\lambda}=\frac{\omega}{c}$
• 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 and is the same as the direction of the cross-product of $\vec{E}$ and $\vec{B}$. 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: $E=cB$.
• The intensity of a radiation is measured is power per unit area: $I=\frac{P}{A}$. For a source of spherical radiation, spreading 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 of propagating electromagnetic wave is proportional to the product of the amplitudes of the electric or magnetic field: $I_\mathrm{avg}=\frac{E_\mathrm{max} B_\mathrm{max}}{2\mu_0}$. This can be expressed in terms of the amplitude of electric or magnetic field alone, by using $E=cB$ and $c=\frac{1}{\sqrt{\varepsilon_0 \mu_0}}$: $I_\mathrm{avg}=\frac{cB^2_\mathrm{max}}{2\mu_0}=\frac{c\varepsilon_0E^2_\mathrm{max}}{2}$.

Refer to sections 24.2 and 24.4 of College Physics to see applications of these formulas.

### Unit 6 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

• Ampere-Maxwell's Law
• Circulation
• Displacement current
• Flux
• Frequency
• Gamma ray
• Gauss' Law
• Intensity
• Light
• Magnetic monopole
• Maxwell's Equations
• Microwave
• Visible light
• Wave
• Wave number
• Wavelength
• X-ray

## Unit 7: Optics

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

1. How are images formed by mirrors?
2. What is the primary difference between concave and convex mirrors?
3. 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?
4. How are the focal length and curvature of a mirror related?
5. How are the locations of an object and its image related by the mirror equation?
6. How are images formed by lenses?
7. What is the primary difference between converging and diverging lenses?
8. How is the type of an image formed by a lens determined by its location relative to the lens?
9. How are the locations of an object and its image related by the lens equation?
10. 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 generated by an object bounce off the reflective surface of the mirror, they intersect and form an image. The angle of reflection 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. A parabolic surface 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 towards 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 the camera film.

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

Positions in front of the mirror are considered positive. The distance between an object and the mirror is denoted by $p$ and is always positive, as the object is always located in front of the mirror. The location of an image is denoted by $q$, and is positive when the image is in front of the mirror and negative when the image is behind the mirror. Thus, $q$ is positive for real images and negative for virtual images.

Magnification is defined 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 it is considered negative, which means $M$ is also negative. From geometric considerations, it can be shown that $M=-\frac{q}{p}$. From here, it follows that $M$ is positive when $q$ is negative, so a virtual image is always upright; and that $M$ is negative when $q$ is positive, so a real image is always inverted.

There are three rays that can be used to draw the images formed by mirrors:

• The reflection of the rays parallel to the axis of symmetry pass through the focal point
• The reflection of the rays passing through the focal point are parallel to the axis of symmetry
• The rays perpendicular to the surface (or passing through the center of curvature, for spherical surfaces) 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: $p = -q$. This image is virtual, upright, and has the same size as the object: $h_i = h_o$.

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 the location $p$ and size $h_o$ of the object are known, the location of the image can be found from the mirror equation: $\frac{1}{p}+\frac{1}{q}=\frac{1}{f}$. The size of the image can be found using the magnification formula: $M=-\frac{q}{p}=\frac{h_i}{h_o}$.

To review how to draw images formed by mirrors and determine the location, size, and nature of an image, review Spherical Mirrors and Image Formation by Concave Mirrors, and explore this solved example. Also, watch Image Formation by Convex Mirrors and see this solved example, as well as Image Formation by Plane Mirrors. Finally, see each video in the Mirrors and Lenses series.

Lenses form images of objects by refraction. As rays of light change direction when passing through 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 the 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 $p$, 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, $q$, 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: $M=\frac{h_i}{h_o}$. 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 it is considered negative, which means $M$ is also negative. From geometric considerations, it can be shown that $M=-\frac{q}{p}$. From here, it follows that $M$ is positive when $q$ is negative, so the virtual image is always upright, and that $M$ is negative when $q$ 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:

• 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
• The incident rays passing through the focal point refract so that they are parallel to the axis of symmetry
• 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 p and size ho of the object are known, the location of the image can be found from the lens equation: $\frac{1}{p}+\frac{1}{q}=\frac{1}{f}$. Then, the size of the image can be found using the magnification formula: $M=-\frac{q}{p}=\frac{h_i}{h_o}$.

To review how to draw images formed by lenses and determine the location, size, and nature of an image, read Thin Lenses and Image Formation by Thin Lenses and see these examples of converging and diverging lenses. Also, watch the Mirrors and Lenses series.

### 7b. solve problems using the law of refraction

1. How are the angle of incidence and angle of refraction related?
2. 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 $c=3\times 10^8\ \mathrm{m/s}$. In other media, it is decreased by a factor of $n$, the index of refraction of the medium: $v_\mathrm{light} = \frac{c}{n}$. 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 critical angle can be found by setting the angle of refraction to 90°, which means $\sin{\theta_r}=1$. Then, $\sin{\theta_c}=\frac{n_r}{n_i}$.

To review the laws of reflection and refraction, watch Reflection and Refraction and read Law of Geometric Propagation, Law of Reflection, Law of Refraction, and Total Internal Reflection. Also, review this solved example.

### 7c. describe the interference pattern in a double-slit experiment and explain the experiment's results

1. 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?
2. 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. This pattern can be explained, and the location of the bright and dark bands predicted, 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 recombine and 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 the distance 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, a bright band is observed. However, if the path difference is an odd multiple of half wavelengths: $\delta=\left ( m+\frac{1}{2} \right )\lambda$, the waves will undergo destructive interference and cancel each other out. In this case, a dark band is observed.

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 $y_\mathrm{bright}=\frac{m\lambda L}{a}$ and $y_\mathrm{dark}=\frac{\left (m+\frac{1}{2} \right )\lambda L}{a}$. 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 the details of the double-slit experiment by watching Interference of Light Waves and reading Young's Double-Slit Experiment. Also, review this solved example.

### 7d. explain how rainbows are produced

1. Why do light waves of different frequencies refract at different angles?
2. 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 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.

The most familiar natural phenomenon that demonstrates dispersion is a rainbow. 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. This is perceived by observers as a rainbow.

Watch the last video in the Reflection and Refraction series. Also, review Unweaving the Rainbow, which illustrates why rainbows appear as arcs.

### Unit 7 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

• Concave mirror
• Converging lens
• Convex mirror
• Diffraction
• Dispersion
• Diverging lens
• Focal length
• Focal point
• Interference, constructive and destructive
• Magnification
• Mirror equation
• Lens equation
• Path difference
• Plane mirror
• Real Image
• Reflection, specular and diffuse
• Refraction
• Snell's Law
• Total internal reflection
• Virtual Image

## Unit 8: Special Relativity

### 8a. identify the postulates upon which the Special Theory of Relativity is based

1. What is an inertial reference frame?
2. According to the Special Theory of Relativity, how do physical laws vary when observed from different inertial reference frames?
3. 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 the 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 would look exactly the same in all inertial frames.

The second postulate of the Special Theory of Relativity is that the speed of light in a vacuum is a constant, and is approximately 3 × 108 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.

Introduction to Relativity gives the historical context for the Special Theory and introduces its two postulates.

### 8b. solve problems involving time dilation and length contraction

1. What is the Lorentz factor, and how does it depend on the velocity of a traveling object?
2. How does time flow differently in reference frames traveling at speeds near the speed of light?
3. 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 that 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 transformations of positions measured in different inertial frames are called Lorentz transformations, which 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 can be seen algebraically that this factor is always greater than 1.

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 $\gamma$ 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.

Work through this solved example, which illustrates the consistency of the time dilation and length contraction expressions in the Special Theory of Relativity.

### 8c. explain the principle of equivalence as introduced in the General Theory of Relativity

1. The Special Theory of Relativity deals only with inertial frames of reference when the reference frames move at a constant velocity relative to one another. How does the General Theory include non-inertial frames of reference by establishing the principle of equivalence?

Non-inertial frames of reference, by definition, are reference frames that move under acceleration as observed from an inertial frame of reference. The principle of equivalence states that accelerated motion is indistinguishable from motion under the influence of gravity. This means that the acceleration of an object is equivalent to the gravitational pull on it from another object of the appropriate mass. In order for an observer in a non-inertial frame to provide a picture of an event consistent with that of an observer in an inertial frame, gravity has to be included. This could be expressed by comparing the inertial mass and gravitational mass. Newton's Second Law states that $F = ma$, where $m$ is inertial mass; it indicates how much the velocity of an object changes under the influence of a force $F$. If $F$ is the force of gravity exerted by the Earth on the object, then $a$ must be $g$, gravitational acceleration, so that $F_\mathrm{gravity}=mg$. This also means that $m$ must be gravitational mass, which is the gravitational attraction between the object and the Earth. The principle of equivalence postulates that the two masses are exactly the same.

Read more about the principle of equivalence in General Relativity and Black Holes.

### 8d. compare and contrast the special and general theories of relativity

1. Which postulate is part of both the special and general theories of relativity?
2. What is the primary difference between the special and general theories?

The common basis of both the special and general theories of relativity is the relativity principle. It states that all physical laws are the same in all reference frames. For the Special Theory of Relativity, this means all inertial reference frames. The General Theory of Relativity includes both inertial and non-inertial reference frames.

The primary difference between the two theories is that the Special Theory of Relativity is limited in scope: it applies only to events observed from inertial frames of reference. The General Theory of Relativity includes both inertial and non-inertial frames. Most importantly, it also incorporates gravity. The general theory is mainly a theory of gravity, while the special theory primarily describes events in different inertial frames of reference.

To compare and contrast the two theories, read Special Relativity and The General Theory of Relativity.

### 8e. list the most significant consequences of Einstein's special and general theories of relativity

1. What are three major consequences of the Special Theory of Relativity? Consider our understanding of space, time, and energy.
2. What are two major consequences of the General Theory of Relativity? Consider the effect gravity has on light.

The consequences of special relativity include:

• Measurements of length and time are relative and depend on the observer who measures them. Time flows more slowly if measured in a reference frame moving at a speed close to the speed of light, which is known as time dilation. Lengths appear shorter when measured at a speed close to the speed of light, which is known as length contraction.
• No massive object can move at a speed greater than the speed of light in a vacuum relative to any observer. No information can propagate at a speed greater than the speed of light in a vacuum.
• All massive objects possess energy proportional to their mass at rest ("rest mass"), $E=mc^2$. This energy can convert to other forms of energy. In nuclear reactions, a particle can become a particle of a different mass, with a corresponding release or intake of energy.

See the list of the consequences of the theory in Special Relativity.

The consequences of general relativity include:

• Light can be deflected by gravity in a phenomenon known as gravitational lensing. In the General Theory of Relativity, the presence of a massive body is equivalent to a curvature in space-time. Since light travels along the shortest path between two points, it will not take a straight line to get from one point to another on a curved surface, since that would not be the shortest path. Instead, it would move along a curve.
• Time flows more slowly in a strong gravitational field. While this sounds similar to the time dilation arising as a consequence of special relativity, gravitational time dilation is a distinct effect.

The experimental confirmation of these consequences, as well as some others, is found in General Relativity and Black Holes and The General Theory of Relativity.

### 8f. explain the results of the Michelson-Morley experiment using the Special Theory of Relativity

1. What was the goal of the Michelson-Morley experiment?
2. Did the experiment produce the expected result?
3. 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 since light was a wave, it would require a medium in order to propagate, much like sound requires air. The setup of the experiment involved producing an interference pattern between two beams of light, one parallel and another perpendicular to the surface of the Earth. The interference pattern would depend on the orientation of the interferometer, the time of the day, and the time of the year; the changes in the pattern would yield measurements of the speed of the Earth relative to the ether. However, no such changes were ever detected. 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.

In 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.

Read a short summary of the Michelson-Morley experiment in Relativity.

### Unit 8 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

• Frame of reference (inertial and non-inertial)
• General Theory of Relativity
• Gravitational Lensing
• Length contraction
• Lorentz factor
• Mass, inertial and gravitational
• Principle of Equivalence
• Principle of Relativity
• Rest energy
• Rest mass
• Special Theory of Relativity
• Time Dilation