Electric Potential | AP Physics 2 Unit 2 Study Guide

Getting Started

We will investigate the energy landscape created by static electric charges. This chapter moves beyond the concept of electric force to describe the properties of the space itself. Our physical system consists of one or more fixed "source" charges and the electric field they produce, and we will ask: How can we map the energy associated with any point in this field, independent of the specific charge we might place there?

What You Should Be Able to Do

After studying this chapter, you will be able to:

Calculate the total electric potential at a point in space due to a collection of point charges.
Describe electric potential as the potential energy per unit charge.
Relate the work done on a charge by an electric field to the change in the charge's potential energy and the potential difference it moves through.
Interpret equipotential line diagrams to determine the direction and relative strength of the electric field.
Use the relationship between electric field and electric potential to solve problems.

Key Concepts & Mechanisms

This topic is best understood through the lens of Interactions and Conservation. The core idea is that a configuration of charges alters the energy of the space around it. We can then analyze how a new charge interacts with this "energy landscape" and how its total energy is conserved as it moves.

System & Preconditions

System: Our system consists of a collection of static (non-moving) source charges that create an electric field. We analyze this field by considering the effect it would have on a hypothetical positive "test charge."
Idealizations: We assume the source charges are fixed in place and that all interactions occur in a vacuum, so we can use the vacuum permittivity constant. We also treat charges as point-like, meaning their physical size is negligible. The zero point of potential is typically defined as being infinitely far away from the source charges.

Key Steps / Relations

From Work to Potential Energy: Recall that work must be done by an external force to move a charge against the electric force exerted by a field. This work is stored as electric potential energy ( $U_{E}$ ), measured in joules (J). The work done by the electric field ( $W_{f i e l d}$ ) is equal to the negative change in potential energy: $Δ U_{E} = - W_{f i e l d}$ . This is a direct consequence of the conservation of energy.
Defining Electric Potential: To create a property of the field itself, independent of the test charge, we define electric potential ( $V$ ) as the electric potential energy per unit charge.
$V = \frac{U _{E}}{q}$
Electric potential is a scalar quantity measured in volts (V), where 1 Volt = 1 Joule/Coulomb. It describes the "energy level" of a location in space. A potential difference ( $Δ V$ ), often called voltage, is the change in electric potential between two points.
Potential from a Point Charge: The electric potential created by a single point charge Q at a distance r from its center is given by:
$V = \frac{1}{4 π ϵ _{0}} \frac{Q}{r}$
where $\frac{1}{4 π ϵ _{0}} = k \approx 9.0 \times 1 0^{9} N \cdot m^{2} / C^{2}$ . Note that potential is a scalar: it is positive for a positive source charge and negative for a negative source charge. It decreases with distance for a positive charge.
Superposition of Potentials: The true power of potential lies in its scalar nature. To find the total potential at a point due to multiple source charges, we simply perform an algebraic sum of the potentials from each charge. This is the principle of superposition for potential.
$V_{t o t a l} = \sum_{i} V_{i} = \frac{1}{4 π ϵ _{0}} \sum_{i} \frac{q _{i}}{r _{i}}$
This is significantly simpler than the vector summation required to find the net electric field.
Relating Electric Field and Potential: The electric field and potential are two descriptions of the same phenomenon. The electric field is related to how rapidly the potential changes from point to point. A strong electric field exists in regions where the potential changes quickly, and a weak field exists where the potential changes slowly. For a uniform electric field, or as an average over a small distance $Δ r$ , this relationship is:
$∣ E ∣ = ∣ \frac{Δ V}{Δ r} ∣$
This equation shows that the units for the electric field, N/C, are equivalent to V/m. The electric field vector $E$ always points in the direction of the most rapid decrease in potential.

Outputs & Effects

A configuration of charges establishes a scalar potential field in the surrounding space.
A charged particle placed in this field will possess electric potential energy ( $U_{E} = q V$ ).
If released, the particle will accelerate in a direction that lowers its potential energy. A positive charge accelerates from high potential to low potential. A negative charge accelerates from low potential to high potential.
In the absence of non-conservative forces, the total mechanical energy ( $K + U_{E}$ ) of the particle is conserved as it moves through the field.

Regulation & Limits

The equation $∣ E ∣ = ∣ \frac{Δ V}{Δ r} ∣$ is an approximation for the average field magnitude over the displacement $Δ r$ . It is only exact for a uniform electric field where the potential changes linearly with position.
The formula for the potential of a point charge assumes that the potential is zero at an infinite distance away. This is a convention, and only potential differences are physically meaningful for predicting motion.

Key Models & Diagrams

Visualizing the energy landscape is crucial. We use electric field lines and equipotential lines to do this. The table below links these representations to their mathematical and physical meaning.

Representation	What It Encodes	How to Read/Use It	Physical Interpretation
Electric Field Lines	The direction and relative strength of the electric field vector ( $E$ ).	Lines point away from positive charges and toward negative charges. Line density indicates field strength.	Shows the path a positive test charge would initially take. The force on a charge `q` is `F = qE`.
Equipotential Lines	Locations in space that have the same electric potential ( $V$ ).	These are contour lines of the "voltage map." No work is done moving a charge along an equipotential line.	Represents lines of constant potential energy for a given charge. The change in potential energy is `ΔU = qΔV`.
Combined Map	The relationship between the force field ( $E$ ) and the energy field ( $V$ ).	Electric field lines are always perpendicular to equipotential lines. The field points from higher potential to lower potential.	Where equipotential lines are close together, the potential is changing rapidly, so the electric field is strong.

Key Components & Evidence

Electric Potential ( $V$ ): A scalar property of a location in an electric field, representing the electric potential energy per unit charge. Its SI unit is the volt (V).
Electric Potential Energy ( $U_{E}$ ): The energy a charge possesses due to its position in an electric field. It is a property of the system (charge + field). Its SI unit is the joule (J).
Potential Difference ( $Δ V$ ): The difference in electric potential between two points, also known as voltage. It determines the work done per unit charge to move between the points.
Electric Field ( $E$ ): A vector field representing the force per unit charge at any point in space. Its SI units are newtons per coulomb (N/C) or volts per meter (V/m).
Equipotential Surface: A three-dimensional surface on which the electric potential is the same everywhere. On a 2D diagram, these are represented by equipotential lines.
Superposition Principle: A fundamental principle stating that the total potential at a point is the algebraic sum of the potentials created by each individual source charge.
Conservation of Energy: The total energy of a charged particle (kinetic + potential) remains constant as it moves under the influence of a conservative electric field.
Point Charge: An idealized model of a charged object where its size is considered negligible compared to the distances involved.

Skill Snapshots

Causation

A configuration of source charges causes a scalar electric potential field to exist in the surrounding space.
A potential difference ( $Δ V$ ) between two points causes an average electric field ( $∣ E ∣ \approx ∣Δ V /Δ r ∣$ ) to exist in the region between them.
Placing a charge in a region with a potential difference causes it to experience a change in potential energy, which can be converted to kinetic energy.

Comparison

Electric field is a vector quantity that describes the force on a charge, while electric potential is a scalar quantity that describes the energy of a charge.
Electric field lines point in the direction of force on a positive charge, while equipotential lines are surfaces of constant potential and are always perpendicular to field lines.
For a single point charge, the electric field magnitude falls off with distance as 1/r², whereas the electric potential falls off as 1/r.

Change Over Time

Baseline: A charged particle in a region of uniform electric potential (an equipotential region) experiences zero electric force and will not accelerate.
Change: As a positive charge moves in the direction of the electric field, its electric potential decreases, and its kinetic energy increases (if starting from rest).
Change: As a negative charge moves opposite to the direction of the electric field, its electric potential increases, and its kinetic energy increases (if starting from rest).
Continuity: For a charge moving under the influence of the electric field alone, the sum of its kinetic and electric potential energy remains constant.

Common Misconceptions & Clarifications

Misconception: A point of zero electric potential must also have zero electric field.
- Clarification: This is false. The electric field is related to the change in potential (the "slope"), not its absolute value. For example, at the point exactly midway between two equal and opposite charges, the potential is zero ( $V = k (+ q) / r + k (- q) / r = 0$ ), but the electric field is strong and non-zero because both charges create fields that point in the same direction there.
Misconception: All charges naturally move to regions of lower potential.
- Clarification: All charges naturally move to regions of lower potential energy. For a positive charge ( $q > 0$ ), lower potential energy ( $U_{E} = q V$ ) means lower potential ( $V$ ). For a negative charge ( $q < 0$ ), lower potential energy means higher potential ( $V$ ).
Misconception: Electric potential and electric potential energy are the same thing.
- Clarification: Electric potential ( $V$ , in volts) is a property of a location in space. Electric potential energy ( $U_{E}$ , in joules) is a property of a charge placed at that location. They are related by $U_{E} = q V$ .
Misconception: Electric field lines point towards higher potential.
- Clarification: Electric field lines always point in the direction of the force on a positive charge, which is from a region of higher potential to a region of lower potential. Think of them as pointing "downhill" on a potential map.

One-Paragraph Summary

Electric potential is a powerful scalar concept that describes the energy landscape created by a configuration of charges. Defined as the electric potential energy per unit charge, it allows for the calculation of the total potential at any point in space through simple algebraic summation, a key advantage over vector-based field calculations. The potential difference, or voltage, between two points is directly related to the work required to move a charge between them. This energy map is visualized using equipotential lines, which are always perpendicular to electric field lines. The spacing of these lines reveals the strength of the electric field, as the field is strongest where the potential changes most rapidly. Ultimately, understanding electric potential allows us to predict how charged objects will move and how their energy will transform within an electric field.

Electric Potential - AP Physics 2: Algebra-Based Study Guide