In the previous section, we examined the relationship between voltage and energy. We now connect voltage to something more fundamental: the electric field. This connection is extremely important conceptually. Voltage is most closely related to energy, while the electric field is most closely related to force. Understanding how they are connected allows us to move freely between energy-based reasoning and force-based reasoning.

A useful physical example is a uniform electric field produced between two large parallel metal plates, as shown in Figure 12.1. One plate is positively charged and the other negatively charged. The potential difference between the plates creates a uniform electric field between them (except near the edges).

Let us carefully derive the relationship between voltage and electric field in this simple but important case.

Suppose a positive test charge [latex]q[/latex] moves from plate A (higher potential) to plate B (lower potential). The electric field does work on the charge. From the definition of potential energy,

Since [latex]\Delta V = V_B - V_A[/latex], we can write the potential difference between the plates in magnitude form as

The work done by the field becomes

Now let us compute the same work using force. In a uniform electric field,

If the charge moves a distance [latex]d[/latex] in the direction of the field, then

Since both expressions describe the same work,

The charge cancels, leaving the fundamental relationship:

or equivalently,

This result is extremely important. It tells us that the electric field is the rate at which voltage changes with distance in space. A large voltage applied across a small distance produces a strong electric field. A small voltage across a large distance produces a weak electric field.

The units also confirm this relationship. Since

the unit of electric field can be written as volts per meter (V/m). We already know electric field can be measured in newtons per coulomb (N/C). Therefore,

These two units describe the same physical quantity.

Physical Interpretation

Conceptually, the electric field tells us how rapidly electric potential changes in space. If voltage changes very quickly over a small distance, the field is strong. If voltage changes slowly over a large distance, the field is weak.

This idea is deeply relevant in biology and medicine. For example:

• Cell membranes maintain voltage differences (~70 mV) across extremely small thicknesses (~5 nm), creating very strong electric fields inside membranes.
• Defibrillators apply large voltages across the chest to create electric fields strong enough to reset cardiac electrical activity.
• Neural stimulation devices rely on controlled electric fields to influence ion movement.

Even relatively small voltages can create large electric fields if applied across very small distances.

Example: Calculating Electric Field from Voltage

Suppose two parallel plates are separated by 2.0 mm and a potential difference of 500 V is applied. What is the electric field strength between the plates?

First convert distance to meters:

Using

This is a very strong electric field. It illustrates how high voltages over small distances can produce intense electric forces on charges.

In summary, voltage and electric field are two complementary ways to describe the same physical situation. Voltage relates to energy per unit charge. Electric field relates to force per unit charge. In a uniform field, they are directly connected through

Understanding this connection allows us to translate between energy-based reasoning and force-based reasoning in electrical systems.

So far we have considered the special case of a uniform electric field. In more general situations, the electric field varies from point to point. Even then, an important relationship holds:

This equation tells us that the electric field equals the rate at which voltage changes with distance. The minus sign indicates that the electric field points in the direction of decreasing potential. A positive charge naturally moves “downhill” in potential.

In continuously varying fields, calculus replaces finite differences, and the electric field is defined as the negative gradient of the electric potential. Conceptually, however, the meaning remains the same: electric fields measure how rapidly voltage changes in space.