## MCAT Physics and Math Review

## Chapter 5: Electrostatics and Magnetism

### 5.3 Electrical Potential Energy

We have already defined potential energy as stored energy that can be used to do something or make something happen. There are different types of potential energy; gravitational, elastic, and chemical are three forms that you will need to know for Test Day. A fourth form is **electrical potential energy**. Similar to gravitational potential energy, this is a form of potential energy that is dependent on the relative position of one charge with respect to another charge or to a collection of charges. Electrical potential energy is given by the equation

**Equation 5.3**

If the charges are like charges (both positive or both negative), then the potential energy will be positive. If the charges are unlike (one positive and the other negative), then the potential energy will be negative. Remember that work and energy have the same unit (the joule), so we can define electrical potential energy for a charge at a point in space in an electric field as the amount of work necessary to bring the charge from infinitely far away to that point. Because and *W* = *Fd* cos *θ*, if we define *d* as the distance *r* that separates two charges and assume the force and displacement vectors to be parallel, then:

Consider two charges: a stationary negative source charge and a positive test charge that can be moved. Because these two charges are unlike, they will exert attractive forces between them. Therefore, the closer they are to each other, the more stable they will be. Opposite charges will have negative potential energy, and this energy will become increasingly negative as the charges are brought closer and closer together. Increasingly negative numbers are actually decreasing values because they are moving farther to the left of 0 on the number line. This decrease in energy represents an increase in stability.

**KEY CONCEPT**

Electrical potential energy is the work necessary to move a test charge from infinity to a point in space in an electric field surrounding a source charge.

Now let’s consider two positive charges. As like charges, these will exert repulsive forces, and the potential energy of the system will be positive. Because like charges repel each other, the closer they are to each other, the less stable they will be. Remember that unlike gravitational systems, the forces of electrostatics can be either attractive or repulsive. In this case, the like charges will become more stable the farther apart they move because the magnitude of the electrical potential energy becomes a smaller and smaller positive number.

**KEY CONCEPT**

The electrical potential energy of a system will increase when two like charges move toward each other or when two opposite charges move apart. Conversely, the electrical potential energy of a system will decrease when two like charges move apart or when two opposite charges move toward each other.

**Example:**

If a charge of +2*e* and a charge of −3*e* are separated by a distance of 3 nm, what is the potential energy of the system? (Note: *e* is the fundamental unit of charge equal to 1.6 × 10^{−19} C, and *k* is the electrostatic constant equal to )

**Solution:**

The equation for potential energy is From the question stem, we know that the charges are +2*e* and −3*e*, and *r* = 3 nm = 3 × 10^{−9} m. Plugging into the equation, we get:

**MCAT Concept Check 5.3:**

Before you move on, assess your understanding of the material with these questions.

1. How does a change in electrical potential energy from –4 J to –7 J reflect on the stability of a system?

2. Compare the relationship between electrical potential energy and Coulomb’s law to the relationship between gravitational potential energy and the universal law of gravitation.

3. How does electrical potential energy change between two particles as the distance between them increases?