The ideas of electric potential, potential difference, and electric potential energy are central to understanding electricity in both technology and biology—but they are not immediately intuitive. We will build these ideas step by step using physical analogies and then connect them to real biological systems.

From Force to Energy: The Big Picture

Previously, we focused on the electric force:

This tells us how strongly an electric field pushes or pulls on a charge. But in many real systems—especially biological ones—we are less interested in the force itself and more interested in energy transfer. For example:

• How much energy is delivered in a defibrillator pulse?
• How much energy does an ion gain when it crosses a membrane?
• How much energy is required to move a charge inside a cell?

These questions are energy questions, not force questions.

When a charge moves in an electric field, the field does work on it. For a conservative force (like gravity or electrostatics),

If the electric field does positive work on a charge, the charge’s electric potential energy decreases. The “lost” potential energy appears as kinetic energy.

Analogy: Gravitational Potential Energy

Consider a ball at the top of a hill. It has gravitational potential energy because of its position. When it rolls downhill:

• Gravitational potential energy decreases.
• Kinetic energy increases.

The height determines how much energy per unit mass is available. Similarly, in electricity:

• Electric potential energy depends on position in an electric field.
• Charge moving “downhill” in electric potential loses potential energy and gains kinetic energy.

The electric field is like the slope of the hill.

Electric Potential Energy (PE)

Electric potential energy (PE) is the energy a charge has because of its position in an electric field. It is measured in joules (J).

If a charge moves from point A to point B, the change in electric potential energy is:

If the electric field accelerates a positive charge, then:

• The field does positive work.
• The charge speeds up.
• Its potential energy decreases.

This is exactly analogous to gravity.

Electric Potential (Voltage)

Because electric potential energy depends on the amount of charge, we define a new quantity that removes this dependence:

Electric potential [latex]V[/latex] is energy per unit charge.

This is extremely important:

• Potential energy depends on how much charge you have.
• Electric potential (voltage) does not.

Just as gravitational potential (height) does not depend on the mass of the object, electric potential does not depend on the test charge.

Potential Difference (Voltage Between Two Points)

In real systems we rarely care about potential at just one point. What matters is the difference in potential between two points:

By definition:

Rearranging:

This equation is one of the most important relationships in this chapter.

Why Voltage Is Not Energy

This is a common point of confusion.

A 12-V motorcycle battery and a 12-V car battery have the same voltage. But the car battery can deliver much more energy.

Why?

Because:

If [latex]\Delta V[/latex] is the same but the car battery can move more charge [latex]q[/latex], then it delivers more total energy.

Voltage tells you how much energy each coulomb of charge would gain—not how many coulombs are available.

Biological Example: Cell Membrane Potential

Cells maintain voltage differences across their membranes (typically around −70 mV for neurons).

If a sodium ion ([latex]\text{Na}^+[/latex]) crosses a membrane with a potential difference of 70 mV:

Since the charge of a single ion is:

The energy change is:

This energy scale is tiny in joules—but extremely significant at the molecular level.

Electron Volt (eV)

Because atomic and molecular energies are so small in joules, we use a more convenient unit: the electron volt (eV).

One electron volt is the energy gained by an electron moving through 1 volt:

Using electron volts makes microscopic processes easier to interpret:

• Chemical bond energies: a few eV
• UV photon energies: several eV
• X-rays: thousands of eV (keV)

This is directly relevant in health sciences:

• UV radiation can damage DNA because its photon energy is comparable to molecular bond energies.
• Ionizing radiation has energies high enough (many eV or keV) to remove electrons from atoms.

In the previous example, we focused primarily on the magnitude of the energy involved. However, in electricity, the sign of the energy change carries important physical meaning. Understanding the sign carefully helps avoid one of the most common conceptual confusions in this topic: how negative charges behave in electric fields and how batteries actually deliver energy.

Recall the fundamental relationship between electric potential energy and potential difference:

This equation tells us that the change in electric potential energy depends on two things:

• The amount and sign of the charge [latex]q[/latex]
• The potential difference [latex]\Delta V[/latex]

Now consider a 12-V car battery connected to a headlight, as shown in Figure 11.2. The battery maintains a potential difference of +12 V between its terminals. By definition,

where B is the positive terminal and A is the negative terminal.

In metal wires, the moving charges are electrons. An electron has charge

Electrons are repelled by the negative terminal and attracted toward the positive terminal. Even though the positive terminal is at higher potential, electrons move toward it because they carry negative charge.

Substituting into [latex]\Delta \text{PE} = q \Delta V[/latex], we see:

The change in potential energy is negative. This means the battery’s stored electric potential energy decreases as electrons move through the circuit. That “lost” potential energy is converted into other forms—light and thermal energy in the headlight.