5.2.1. Electric Potential Energy

Table of Contents

Energy in an Electric Interaction

Electric potential energy is the energy associated with the positions of electric charges relative to one another. When charges interact, the electric force can make them move, and during that motion energy can be stored or released. Electric potential energy plays the same role in electricity that gravitational potential energy plays in gravity.

If two charges are brought close together or moved farther apart, work may be required, or work may be produced by the electric force itself. That change is described by electric potential energy, usually written as $U$.

Why Electric Potential Energy Matters

Suppose you have two positive charges. They repel each other. If you try to push them closer together, you must do work against the electric force. That work becomes stored electric potential energy. If you release them, they move apart and that stored energy can turn into kinetic energy.

Now consider a positive charge and a negative charge. They attract each other. If they are far apart and move closer together, the electric force does work and the electric potential energy decreases.

This idea gives a useful rule. Systems tend to move toward lower potential energy when they are free to move under the electric force.

A change in electric potential energy is related to work done by the electric force:
$$\Delta U = -W_{\text{electric}}$$
If the electric force does positive work, the potential energy decreases.

Electric Potential Energy of Two Point Charges

For two point charges $q_1$ and $q_2$ separated by distance $r$, the electric potential energy is

$$U = k \frac{q_1 q_2}{r}$$

where $k$ is Coulomb's constant.

This formula shows several important things. The energy depends on both charges and on how far apart they are. It is positive if the charges have the same sign, and negative if they have opposite signs.

Charge combination	Sign of $q_1 q_2$	Sign of $U$	Meaning
Positive and positive	Positive	Positive	Repulsive configuration
Negative and negative	Positive	Positive	Repulsive configuration
Positive and negative	Negative	Negative	Attractive configuration

A positive value of $U$ means energy had to be added to create that arrangement from very far away. A negative value means the arrangement is more bound than charges infinitely far apart.

For two point charges,
$$U = k \frac{q_1 q_2}{r}$$
Same sign charges give $U > 0$.
Opposite sign charges give $U < 0$.

Reference Point at Infinity

Potential energy is always defined relative to a reference. For electric interactions between point charges, the usual choice is

$$U = 0 \quad \text{when} \quad r \to \infty$$

This means that when the charges are infinitely far apart, we take the potential energy to be zero. Then any finite separation gives a value of $U$ measured relative to that zero level.

Because of this choice, opposite charges have negative potential energy at finite distance, and like charges have positive potential energy.

Change in Electric Potential Energy

Often we care more about the change in energy than the absolute value. If two charges move from separation $r_i$ to separation $r_f$, then

$$\Delta U = U_f - U_i = k q_1 q_2 \left( \frac{1}{r_f} - \frac{1}{r_i} \right)$$

This formula tells us whether energy is gained or lost in the process.

If like charges are pushed closer together, $r_f < r_i$, so $\frac{1}{r_f} > \frac{1}{r_i}$ and $\Delta U > 0$. The potential energy increases.

If opposite charges move closer together, then $q_1 q_2 < 0$, so $\Delta U < 0$. The potential energy decreases.

The change in electric potential energy between two separations is
$$\Delta U = k q_1 q_2 \left( \frac{1}{r_f} - \frac{1}{r_i} \right)$$

Work Done by an External Agent

Sometimes a charge is moved slowly so that its kinetic energy does not change much. In that case, the work done by an external agent equals the increase in electric potential energy:

$$W_{\text{ext}} = \Delta U$$

For example, bringing two positive charges closer together requires positive external work. Pulling apart a positive and a negative charge also requires positive external work.

If the electric force itself moves the charges naturally, then the external work may be negative or zero.

Visualizing Energy Changes

A simple way to think about electric potential energy is to imagine a hill or valley. Like charges close together are like an object high on a hill. If released, they move apart, going toward lower energy. Opposite charges close together are like an object deep in a valley. Energy must be added to separate them.

Rendering scientific sketch…

Potential energy behavior for two charges

The upper curve corresponds to $q_1 q_2 > 0$, and the lower curve corresponds to $q_1 q_2 < 0$.

Relation to Conservation of Energy

Electric potential energy is part of the total mechanical energy of a system. If only electric forces act and no energy is lost, then a decrease in electric potential energy appears as an increase in kinetic energy.

For a simple two-charge system,

$$K_i + U_i = K_f + U_f$$

This is very useful when a charge is released and allowed to move under electric forces.

If only electric forces do work, total mechanical energy is conserved:
$$K + U = \text{constant}$$

Example

Take two point charges, $q_1 = 2.0 \times 10^{-6}\,\text{C}$ and $q_2 = 3.0 \times 10^{-6}\,\text{C}$, separated by $r = 0.50\,\text{m}$. Their electric potential energy is

$$U = k \frac{q_1 q_2}{r}$$

Using $k = 8.99 \times 10^9\,\text{N m}^2/\text{C}^2$,

$$U = (8.99 \times 10^9)\frac{(2.0 \times 10^{-6})(3.0 \times 10^{-6})}{0.50}$$

$$U \approx 0.108\,\text{J}$$

The result is positive because the charges have the same sign.

If one of the charges were negative instead, the magnitude would be the same but the energy would be negative:

$$U \approx -0.108\,\text{J}$$

Many-Charge Systems

If more than two charges are present, the total electric potential energy is the sum of the potential energies for every pair of charges. Each pair contributes once.

For three charges, the total energy is

$$U_{\text{total}} = k \frac{q_1 q_2}{r_{12}} + k \frac{q_1 q_3}{r_{13}} + k \frac{q_2 q_3}{r_{23}}$$

This idea follows the superposition principle. The total energy of the arrangement comes from all pairwise interactions.

Important Physical Meaning

Electric potential energy belongs to the system of charges, not to a single charge by itself. A lone charge does not have electric potential energy unless there is another charge or an electric field present for it to interact with.

This is an important distinction. Force describes what happens at a point, while potential energy describes the energy of a configuration.

Electric potential energy is a property of a system of interacting charges.
It depends on the arrangement of the charges, especially their separations.

Simple Picture of Attractive and Repulsive Cases

Rendering scientific sketch…

Like charges and unlike charges

Final Ideas

Electric potential energy helps us understand how electric forces store and transfer energy. It tells us whether a charge arrangement is energetically costly or energetically favorable. Positive energy usually corresponds to repulsive arrangements, and negative energy usually corresponds to attractive, bound arrangements.

In later discussion, electric potential will describe this same physics in terms of energy per unit charge. Here, the main idea is the energy of charge configurations themselves.

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