PHYS102: Introduction to Electromagnetism

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

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

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

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

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

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

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

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

Review this material in Coulomb's Law.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

When two or more capacitors are connected in series:

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

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

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

When two or more capacitors are connected in parallel:

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

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

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

Review these explanations in Capacitors in Series and Parallel.

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

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

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

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

Review this material in Capacitors and Dielectrics.

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

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

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

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

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

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

Unit 2 Vocabulary

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

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