10.4 Electric Potential Energy

Electric potential energy is the energy stored in a system of charged objects, equal to the work needed to bring those charges from infinitely far apart to their current positions. For two point charges, use $U_E = k \frac{q_1 q_2}{r}$, and for systems of several charges, add up the potential energy of every pair.

Why This Matters for the AP Physics 2 Exam

Electric potential energy connects forces, fields, and energy conservation, so it shows up in both multiple-choice and free-response reasoning. The second free-response question, the Translation Between Representations question, often asks you to move between pictures, energy bar charts, equipotential diagrams, and equations. A point charge released from rest near a charged object is a classic setup, and explaining how its energy changes relies on understanding electric potential energy. Being able to calculate $U_E$, track its sign, and connect it to kinetic energy gives you the tools to support claims about charged systems with both math and words.

more resources to help you study

practice multiple choiceFRQ practice & scoringcheatsheetsscore calculator

Key Takeaways

• Electric potential energy equals the work an external force must do to assemble charges from infinity, with infinity set as the zero reference.
• Use $U_{E}=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r}=k \frac{q_{1} q_{2}}{r}$ for two point charges, keeping the signs of the charges in the calculation.
• Positive $U_E$ means a repulsive interaction (like charges); negative $U_E$ means an attractive interaction (unlike charges).
• For multiple charges, add the potential energy of each unique pair. Three charges give three pairs; four charges give six pairs.
• $U_E$ is a scalar measured in joules, so you add values with their signs, not as vectors.
• Energy is conserved: as charges move, $U_E$ and kinetic energy trade off so the total stays constant in an isolated system.

Describing Electric Potential Energy

Electric potential energy measures the work needed to assemble a system of charged particles. When charges are brought together from infinity, energy is either stored or released depending on whether the charges attract or repel each other.

• Like charges (both positive or both negative) have positive potential energy because work must be done against repulsive forces to bring them together.
• Unlike charges (one positive, one negative) have negative potential energy because they naturally attract, releasing energy as they come together.
• The reference point is at infinity, where the potential energy is defined as zero.

The general equation for electric potential energy between two charged objects is:

$U_{E}=\frac{1}{4 \pi \varepsilon_{0}} \frac{q_{1} q_{2}}{r}=k \frac{q_{1} q_{2}}{r}$

Where:

• $U_E$ = electric potential energy (J)
• $q_1, q_2$ = charges of the two objects (C)
• $r$ = distance between the charges (m)
• $\varepsilon_0$ = permittivity of free space constant ($8.85 \times 10^{-12}$ $\frac{C^2}{N \cdot m^2}$)
• $k$ = Coulomb's constant ($8.99 \times 10^9$ $\frac{N \cdot m^2}{C^2}$)

Notice that $U_E$ depends on $r$, not $r^2$. That is different from the force equations from earlier in the unit, so keep the two straight: force uses $r^2$, energy uses $r$.

Multiple Charge Systems

For systems with multiple charges, you consider every possible pair of interactions. The total electric potential energy is the sum of the potential energies for each unique pair of charges.

• For three charges ($q_1$, $q_2$, and $q_3$), calculate:

  1. Potential energy between $q_1$ and $q_2$
  2. Potential energy between $q_1$ and $q_3$
  3. Potential energy between $q_2$ and $q_3$

• The total potential energy is the sum of these three interactions: $U_{total} = k\frac{q_1q_2}{r_{12}} + k\frac{q_1q_3}{r_{13}} + k\frac{q_2q_3}{r_{23}}$

• For four charges, you would have six unique pairs to consider.

Conservation of Energy in Electric Systems

Electric potential energy is a form of potential energy that can be converted to other forms of energy:

• When opposite charges move closer together, electric potential energy decreases and may convert to kinetic energy.
• When like charges move closer together, work must be done against the repulsive force, increasing the electric potential energy.
• The total energy (kinetic + potential) remains constant in isolated systems.

How to Use This on the AP Physics 2 Exam

Problem Solving

• Plug charges in with their signs. The product $q_1 q_2$ sets the sign of $U_E$ automatically, so do not force the answer to be positive.
• Use $r$ (the separation), not $r^2$, in the energy equation.
• Convert microcoulombs to coulombs ($1 \mu C = 10^{-6}$ C) before substituting.
• For multiple charges, list every pair first, find each distance, then add the pair energies with their signs.

Free Response

• For the Translation Between Representations question, be ready to connect an energy bar chart to the math. A charge released from rest starts with all energy as $U_E$ and converts it to kinetic energy as it moves.
• When the system has only attractive or repulsive interactions, predict the sign of $U_E$ before calculating, then check that your answer matches.
• Justify motion with energy conservation: if $U_E$ drops, kinetic energy rises by the same amount in an isolated system.

Common Trap

• A negative $U_E$ does not mean negative energy in a broken sense. It means the system is more bound than it would be at infinite separation, so you would need to add energy to pull the charges apart.

Practice Problem 1: Two Point Charges

Solution: First, calculate the electric potential energy of the system using the equation: $U_E = k\frac{q_1q_2}{r}$

Given:

• $q_1 = +2.0 \times 10^{-6}$ C
• $q_2 = -5.0 \times 10^{-6}$ C
• $r = 3.0$ m
• $k = 8.99 \times 10^9$ $\frac{N \cdot m^2}{C^2}$

Substituting these values: $U_E = (8.99 \times 10^9)\frac{(2.0 \times 10^{-6})(-5.0 \times 10^{-6})}{3.0}$ $U_E = (8.99 \times 10^9)\frac{-10.0 \times 10^{-12}}{3.0}$ $U_E = -29.97 \times 10^{-3}$ J $U_E = -0.03$ J

The negative sign indicates that the charges attract each other. To separate these charges to infinity would require doing work against this attractive force. The work required equals the negative of the potential energy: Work required = $-U_E = -(-0.03$ J$) = 0.03$ J

Practice Problem 2: Three-Charge System

Solution: Find the potential energy for each pair of charges, then add them together.

First, calculate the distances between each pair:

• Distance between $q_1$ and $q_2$: $r_{12} = 4.0$ m
• Distance between $q_1$ and $q_3$: $r_{13} = 3.0$ m
• Distance between $q_2$ and $q_3$: $r_{23} = \sqrt{4.0^2 + 3.0^2} = \sqrt{16 + 9} = \sqrt{25} = 5.0$ m

Now calculate the potential energy for each pair:

1. For $q_1$ and $q_2$: $U_{12} = k\frac{q_1q_2}{r_{12}} = (8.99 \times 10^9)\frac{(3.0 \times 10^{-6})(-2.0 \times 10^{-6})}{4.0}$ $U_{12} = -13.49 \times 10^{-3}$ J

2. For $q_1$ and $q_3$: $U_{13} = k\frac{q_1q_3}{r_{13}} = (8.99 \times 10^9)\frac{(3.0 \times 10^{-6})(1.0 \times 10^{-6})}{3.0}$ $U_{13} = 9.0 \times 10^{-3}$ J

3. For $q_2$ and $q_3$: $U_{23} = k\frac{q_2q_3}{r_{23}} = (8.99 \times 10^9)\frac{(-2.0 \times 10^{-6})(1.0 \times 10^{-6})}{5.0}$ $U_{23} = -3.6 \times 10^{-3}$ J

Total electric potential energy: $U_{total} = U_{12} + U_{13} + U_{23}$ $U_{total} = -13.49 \times 10^{-3} + 9.0 \times 10^{-3} + (-3.6 \times 10^{-3})$ $U_{total} = -8.09 \times 10^{-3}$ J $U_{total} = -0.00809$ J

Common Misconceptions

• Electric potential energy is not the same as electric potential. Potential energy ($U_E$, in joules) belongs to the whole system of charges, while electric potential ($V$, in volts) is energy per unit charge at a point. You will sort out potential in the next topic.
• The energy equation uses $r$, not $r^2$. Students often copy the force formula's $r^2$ into the energy formula by mistake.
• A negative $U_E$ is not an error. It just means the charges attract and the system is bound relative to infinite separation.
• Electric potential energy is a scalar, so you add pair values with their signs. Do not try to add them like vectors with angles.
• Do not drop the charge signs to "make it positive." The sign of the product $q_1 q_2$ carries real physical meaning about attraction versus repulsion.
• Bringing like charges closer together increases $U_E$; bringing unlike charges closer together decreases it. Plenty of students reverse this.

Related AP Physics 2 Guides

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