Electric Potential

You’ve seen voltage written on batteries: 1.5 V on a AA, 9 V on a transistor battery, 12 V on a car battery. But what does that number actually mean? And how is “voltage” different from the “potential energy” we covered in the previous section?

This is one of the trickiest distinctions in physics, and the MCAT loves to test it. The good news: there’s a single mountain-and-hiker analogy that makes it click forever. Let’s break it down.

The Big Idea: Voltage Is Like Elevation

Imagine you are hiking and you look at a topographic map. The map shows the elevation at every point on the landscape - 500 meters here, 800 meters there - regardless of whether anyone is actually standing at those spots. The elevation is a property of the location, not of any particular hiker.

Voltage (electric potential) is the electrical version of elevation.

Just as a map can tell you the height at every point on a mountain, voltage tells you the “electrical height” at every point in space around a charged object. This electrical height exists whether or not any other charge is sitting there to experience it.

Building Up from What You Know

In the previous section, you learned about electric potential energy (U) - the energy stored when two charges are positioned relative to each other. That formula was:

$\displaystyle U = \dfrac{kq_1 q_2}{r}$

(energy depends on BOTH charges)

Now we want to separate out the contribution of just ONE charge - the source charge that creates the “electrical landscape.” We define:

$\displaystyle V = \dfrac{kq}{r}$

(voltage depends on only the SOURCE charge)

The voltage $V$ tells you what the source charge has done to space. If you later bring in a second charge, you can easily find the potential energy by multiplying:

$\displaystyle U = qV$

(energy = charge × voltage)

Key Properties of Voltage

• Voltage is a scalar - it has magnitude but no direction. Unlike electric field (a vector), you don’t need to worry about components or angles.
• Voltage is created by source charges - it is a property of a location in space
• Voltage is positive near positive charges and negative near negative charges
• Voltage decreases with distance from a positive charge (since V = kq/r, bigger r means smaller V)
• Voltage is zero at infinity - we define our reference point there

The Critical Distinction: V vs. U

This is the most important concept in this section. The MCAT tests it repeatedly because students often confuse these two quantities.

The Analogy That Makes It Click

Think about a mountain:

1. The mountain’s height (elevation) exists whether or not anyone is climbing it. A peak might be 4,000 meters high - that’s a fact about the location itself.

2. A climber’s potential energy depends on both the mountain’s height AND the climber’s mass. A 100 kg climber at 4,000 m has twice the gravitational PE of a 50 kg climber at the same height.

Now translate this to electricity:

1. Voltage (V) is like the mountain’s height. It’s a property of the location, created by source charges. It exists whether or not any test charge is there.

2. Potential energy (U) depends on both the voltage AND the charge you place there. U = qV - the energy depends on the “height” (V) and the “mass” (q).

Side-by-Side Comparison

Potential Difference: What Batteries Actually Measure

When you see “9V” on a battery, what does that mean? It’s the potential difference - the difference in voltage between the two terminals.

In most physics problems, what matters isn’t the absolute voltage at a point, but the difference in voltage between two points:

$\displaystyle \Delta V = V_{\text{final}} - V_{\text{initial}}$

This difference tells you how much energy per coulomb is available to move charges from one point to another.

How Charges Move in Voltage Differences

Think about what you learned about potential energy and conservation of energy. When an object moves from high potential energy to low potential energy, that lost PE becomes kinetic energy.

The same happens with charges:

• A positive charge naturally moves from high voltage to low voltage (like a ball rolling downhill)
• A negative charge naturally moves from low voltage to high voltage (attracted toward the positive region)

Energy from Voltage Differences

The change in potential energy when a charge moves through a voltage difference is:

$\displaystyle \Delta U = q \Delta V$

If a positive charge ($q > 0$) moves from high $V$ to low $V$ ($\Delta V < 0$), then $\Delta U$ is negative - the charge loses potential energy. That energy becomes kinetic energy. The charge speeds up.

How Voltage Connects to Electric Field

You learned in Section 5.3 that electric fields point from positive charges toward negative charges. Here’s another way to think about it:

Electric fields point from high voltage to low voltage.

This makes sense: the field points in the direction a positive charge would naturally move, which is “downhill” on the voltage landscape.

The mathematical relationship in a uniform field (like between parallel plates) is:

$\displaystyle E = \dfrac{V}{d}$

where $V$ is the voltage across the plates and $d$ is the distance between them. The field strength tells you how quickly the voltage changes with distance.

The Electron-Volt: A Convenient Energy Unit

When dealing with single electrons or atoms, the joule is an inconveniently large unit. It’s like measuring your commute in millimeters.

The electron-volt (eV) is a more practical unit at atomic scales:

$1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$

An electron-volt is the energy an electron gains when it accelerates through a voltage difference of 1 volt. Since the electron has charge $q = 1.6 \times 10^{-19}$ C:

$\displaystyle \text{Energy} = qV = (1.6 \times 10^{-19} \text{ C})(1 \text{ V}) = 1.6 \times 10^{-19} \text{ J} = 1 \text{ eV}$

You will see this unit again in atomic physics and nuclear physics.