10.3 Electric Fields

An electric field is the force per unit charge that a charged object creates in the space around it, written as $\vec{E} = \frac{\vec{F}_E}{q}$. Fields point away from positive charges and toward negative charges, and you find the total field from several charges by adding the individual field vectors.

Why This Matters for the AP Physics 2 Exam

Electric fields are the bridge between charge and force, so this topic shows up across multiple-choice and free-response work in electricity. You will be asked to describe and sketch fields, build field vector maps and field line diagrams, and translate between a picture of charges and a symbolic expression for the field. That kind of representation-to-representation thinking is exactly what the second free-response question rewards, where you might sketch field or equipotential patterns around a charged sphere and then explain how your representations agree. Getting comfortable with field direction, superposition, and conductor versus insulator behavior sets you up for electric potential, capacitors, and energy in the rest of the unit.

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Key Takeaways

• The electric field at a point is the electric force on a positive test charge divided by that charge: $\vec{E} = \frac{\vec{F}_E}{q}$, with units of N/C.
• A test charge is small enough that it does not noticeably change the field it is measuring.
• Field lines point away from positive charges and toward negative charges; closer lines mean a stronger field.
• The net field from several charges is the vector sum of each charge's field (superposition).
• Inside a conductor in electrostatic equilibrium the field is zero, charge sits on the surface, and the surface field is perpendicular to the surface.
• Inside a charged insulator the field can be nonzero, because excess charge stays spread through the material instead of moving to the surface.

Electric Field Fundamentals

An electric field describes the force a charged object would exert on other charges in the space around it. When you place a charged particle in a field, it feels a force proportional to its charge.

The electric field at a point is defined as the electric force that would act on a positive test charge placed there, divided by the magnitude of that test charge:

$\vec{E} = \frac{\vec{F}_{E}}{q}$

A test charge is a point charge small enough that its presence does not significantly affect the field it is measuring. For a positive test charge, the electric force points in the same direction as the field. For a negative charge, the force points opposite the field.

• Electric fields point away from isolated positive charges.
• Electric fields point toward isolated negative charges.
• Field strength decreases with distance following the inverse square law.

For a single point charge, the field magnitude is:

$E = k\frac{|q|}{r^2}$

The direction is radially outward from a positive source charge and radially inward toward a negative source charge. In vector form:

$\vec{E} = k\frac{q}{r^2}\hat{r}$

where $\hat{r}$ points radially outward from the source charge and $k$ is Coulomb's constant, $k = 8.99 \times 10^9 \text{ N} \cdot \text{m}^2/\text{C}^2$.

Electric Field Visualization

Electric fields can be represented with vector field maps, which use arrows at many locations to show both the direction and magnitude of the field. In these maps:

• Arrow direction shows the direction of the field.
• Arrow length or color intensity shows the field strength.

Electric field line diagrams are simplified models of these vector maps. They show the relative magnitude and direction of the field at any position. In a field line diagram:

• The field at any point is tangent to the field line there.
• Where lines are closer together, the field is stronger.
• Lines point away from positive charges and toward negative charges.

When more than one charge is present, the total field at any point is the vector sum of the individual fields from each charge. This is the superposition principle, and it is how you handle configurations of up to four point charges.

Electric Fields in Conductors

When a conductor reaches electrostatic equilibrium:

• Excess charge distributes entirely on the surface of the conductor.
• The electric field inside the conductor is zero.
• At the surface, the field is perpendicular to the surface.

For an isolated sphere with a spherically symmetric charge distribution, the field outside the sphere is the same as the field of a point charge with the same net charge located at the center.

Electric Fields in Insulators

Insulators behave differently from conductors when charged:

• Excess charge can be distributed throughout the volume of the material as well as on the surface.
• The electric field inside an insulator may have a nonzero value.

This contrast matters: in a conductor at electrostatic equilibrium, the interior field is always zero, but in an insulator the interior field can be nonzero because the excess charge stays spread through the material. On the AP Physics 2 exam you only need to describe this behavior qualitatively, usually by comparing it to the conductor case.

How to Use This on the AP Physics 2 Exam

Problem Solving

• Use $\vec{E} = \frac{\vec{F}_E}{q}$ when a problem gives you a force on a known charge and asks for the field, or gives a field and asks for the force.
• Use $E = k\frac{|q|}{r^2}$ for the magnitude of a point-charge field, and track distance in meters and charge in coulombs so your answer comes out in N/C.
• For multiple charges, find each field vector separately, break them into components, then add component by component before recombining.

Free Response

• Be ready to sketch field vectors or field line diagrams around one or more charges. Show direction with arrows and relative strength with spacing or length.
• When a question asks you to translate between a diagram and an equation, state clearly how field direction and magnitude in your sketch match the symbolic expression.
• For a charged conductor question, state that the interior field is zero and that surface field lines leave perpendicular to the surface, then justify with electrostatic equilibrium.

Common Trap

• Field direction is defined by a positive test charge. If the test charge is negative, the force flips relative to the field, but the field itself does not change.

Practice Problem 1: Electric Field of a Point Charge

Solution

For a point charge, use: $E = k\frac{|q|}{r^2}$

Where:

• $k = 8.99 \times 10^9 \text{ N} \cdot \text{m}^2/\text{C}^2$
• $q = 5.0 \times 10^{-6} \text{ C}$
• $r = 0.3 \text{ m}$

Substituting: $E = (8.99 \times 10^9 \text{ N} \cdot \text{m}^2/\text{C}^2) \times \frac{5.0 \times 10^{-6} \text{ C}}{(0.3 \text{ m})^2}$

$E = (8.99 \times 10^9) \times \frac{5.0 \times 10^{-6}}{0.09} \text{ N/C}$

$E = 4.995 \times 10^5 \text{ N/C} \approx 5.0 \times 10^5 \text{ N/C}$

The field points radially outward from the positive charge, directly away from it in all directions.

Practice Problem 2: Electric Field Inside a Conductor

Solution

For a conductor in electrostatic equilibrium:

1. At the center of the sphere: The field inside any conductor in electrostatic equilibrium is zero, because the free electrons rearrange until there is no net interior field. So the field at the center is 0 N/C.

2. Halfway from the center to the surface: This point is still inside the conductor, so the same reasoning applies. The field here is also 0 N/C.

The charge sits only on the surface, and the field is nonzero only outside the conductor.

Common Misconceptions

• The field is created by the test charge. The test charge only measures the field. The field comes from the source charges, and a true test charge is small enough not to disturb it.
• A negative charge changes the field's direction. Field direction is set by the source charges and the positive-test-charge convention. A negative charge just feels a force opposite to the field.
• The field is zero everywhere for a charged conductor. The field is zero only inside the conductor. Outside, it can be strong, and at the surface it points perpendicular to the surface.
• Conductors and insulators behave the same inside. Inside a conductor at equilibrium the field is zero, but inside a charged insulator the field can be nonzero because charge stays spread through the material.
• Field lines and equipotential lines are the same thing. They are different representations. Field lines show field direction; you handle equipotential lines in the electric potential topic.
• Field strength drops off linearly with distance. A point-charge field follows an inverse square law, so doubling the distance cuts the field to one quarter, not one half.

Related AP Physics 2 Guides

• 10.1 Electric Charge and Electric Force
• 10.2 The Process of Charging
• 10.5 Electric Potential
• 10.6 Capacitors
• 10.7 Conservation of Electric Energy
• 10.4 Electric Potential Energy