## MCAT Physics and Math Review

## Chapter 5: Electrostatics and Magnetism

### 5.4 Electrical Potential

Electrical potential, discussed here, and electrical potential energy, discussed above, sound like the same (or nearly the same) thing. They are not, although they are very closely related. In fact, **electrical potential** is defined as the ratio of the magnitude of a charge’s electrical potential energy to the magnitude of the charge itself.

**Equation 5.4**

where *V* is the electrical potential measured in **volts (V)** and Even if there is no test charge *q*, we can still calculate the electrical potential of a point in space in an electric field as long as we know the magnitude of the source charge and the distance from the source charge to the point in space in the field. By dividing by *q*, we get:

**Equation 5.5**

Electrical potential is a scalar quantity, and its sign is determined by the sign of the source charge *Q*. For a positive source charge, *V* is positive, but for a negative source charge, *V* is negative. For a collection of charges, the total electrical potential at a point in space is the scalar sum of the electrical potential due to each charge.

**MCAT EXPERTISE**

These are essential equations for Test Day. Know how they relate to each other and when to use them. By memorizing Coulomb’s law, you should be able to recreate the table through mathematical manipulation. From left to right, multiply by *r*; from top to bottom, divide by *q*.

Because electrical potential is inversely proportional to the distance from the source charge, a potential difference will exist between two points that are at different distances from the source charge. If *V*_{a} and *V*_{b} are the electrical potentials at points a and b, respectively, then the **potential difference** between them, known as **voltage**, is *V*_{b} – *V*_{a}. From the equation for electrical potential above, we can further define potential difference as:

**Equation 5.6**

where *W*_{ab} is the work needed to move a test charge *q* through an electric field from point a to point b. The work depends only on the potentials at the two points a and b and is independent of the actual pathway taken between a and b. Like gravitational force, the electrostatic force is a conservative force.

**MNEMONIC**

The “plus” end of a battery is the high-potential end, and the “minus” end of a battery is the low-potential end. Positive charge moves from + to – (the definition of current) while negative charge moves from – to +.

We’ve already seen that charges, if allowed, will move spontaneously in whatever direction results in a decrease in electrical potential energy. For a positive test charge, this means moving from a position of higher electrical potential to a position of lower electrical potential. The voltage,Δ*V* = *V*_{b} – *V*_{a}, is negative in this case; because *q* is positive (for a positive test charge), *W*_{ab} must also be negative, which represents a decrease in electrical potential energy.

**KEY CONCEPT**

Electrical potential is the ratio of the work done to move a test charge from infinity to a point in an electric field surrounding a source charge divided by the magnitude of the test charge.

Now let’s consider a negative test charge. A negative test charge will spontaneously move from a position of lower electrical potential to a position of higher electrical potential. The voltage, Δ*V* = *V*_{b} – *V*_{a}, is positive in this case; because *q* is negative (for a negative test charge), *W*_{ab} must also be negative, which again represents a decrease in electrical potential energy. The takeaway: positive charges will spontaneously move in the direction that *decreases* their electrical potential (negative voltage), whereas negative charges will spontaneously move in the direction that*increases* their electrical potential (positive voltage)—yet, in both cases, the electrical potential *energy* is decreasing.

**BRIDGE**

Create analogies between mechanics and electrostatics to familiarize yourself with these concepts. Electric field is like a gravitational field, and it exerts forces on charges much like a gravitational field exerts forces on masses. A test charge has a particular electrical potential energy at a given electrical potential, depending on the magnitude of its charge, much like a mass has a particular gravitational potential energy, depending on the magnitude of its mass. These mechanics concepts are discussed in Chapters 1 and 2 of *MCAT Physics and Math Review*.

**MCAT Concept Check 5.4:**

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

1. What is the difference between electrical potential and voltage?

2. How will a charge which is placed at a point of zero electrical potential move relative to a source charge?

3. True or False: The units of electrical potential energy and electrical potential are different.