7.5: Electric Potential and Potential Energy

Learning Objectives

• Define electric potential and electric potential energy.
• Describe the relationship between electric potential difference and electric field.
• Describe the relationship between electric potential and electrical potential energy.

When a free positive charge \(q\) is accelerated by an electric field, such as shown in Figure \(\PageIndex{1}\), it is given kinetic energy. The process is analogous to an object being accelerated by a gravitational field. It is as if the charge is going down an electrical hill where its electric potential energy is converted to kinetic energy. Let us explore the work done on a charge \(q\) by the electric field in this process, so that we may develop a definition of electric potential energy.

Figure \(\PageIndex{1}\): A charge accelerated by an electric field is analogous to a mass going down a hill. In both cases potential energy is converted to another form. Work is done by a force, but since this force is conservative, we can write \(W=-\Delta \mathrm{PE}\).

The electrostatic or Coulomb force is conservative, which means that the work done on \(q\) is independent of the path taken. This is exactly analogous to the gravitational force in the absence of dissipative forces such as friction. When a force is conservative, it is possible to define a potential energy associated with the force, and it is usually easier to deal with the potential energy (because it depends only on position) than to calculate the work done directly from force \(W=F d\), where d is displacement and F is the force).

We use the letters PE to denote electric potential energy, which has units of joules (J). The change in potential energy, \(\triangle \mathrm{PE}\), is crucial, since the work done by a conservative force is the negative of the change in potential energy; that is, \(W=-\Delta \mathrm{PE}\). For example, work \(W\) done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative \(\triangle \mathrm{PE}\). There must be a minus sign in front of \(\triangle \mathrm{PE}\) to make \(W\) positive. PE can be found at any point by taking one point as a reference and calculating the work needed to move a charge to the other point.

Definition: POTENTIAL ENERGY

\(W=-\Delta \mathrm{PE}\). For example, work \(W\) done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative \(\Delta \mathrm{PE}\). There must be a minus sign in front of \(\Delta \mathrm{PE}\) to make \(W\) positive. PE can be found at any point by taking one point as a reference and calculating the work needed to move a charge to the other point.

Gravitational potential energy and electric potential energy are quite analogous. Potential energy accounts for work done by a conservative force and gives added insight regarding energy and energy transformation without the necessity of dealing with the force directly. It is much more common, for example, to use the concept of voltage (related to electric potential energy) than to deal with the electric field (related to Coulomb force) directly.

Given some conservative force \(F\) and displacement \(d\) under the force, the work done and the change in potential energy can be calculated as, \(W=\langle\boldsymbol{F}, \boldsymbol{d}\rangle\) and \(\Delta \mathrm{PE}=-\mathrm{W}=\langle-\mathbf{F}, \boldsymbol{d}\rangle\). For electric force, the force is given by the product of electric charge and the electric field, \(\boldsymbol{F}=q \boldsymbol{E}\), where \(q\) is the charge experiencing the force and \(E\) is the electric field at the location of the charge. So the potential energy change due to work done by electric force is \(\Delta \mathrm{PE}=q(\langle-\mathbf{E}, \boldsymbol{d}\rangle)\). If we define change in electric potential \(V\) as \(\Delta V=\langle-\mathbf{E}, \boldsymbol{d}\rangle\), then the electric potential energy PE is simply expressed in terms of electric potential, \(\mathrm{PE}=q V\), or,

\[V=\frac{\mathrm{PE}}{q}, \nonumber \]

electric potential energy per charge.

Definition: ELECTRIC POTENTIAL

Electric potential is the electric potential energy per unit charge.

\[V=\frac{\mathrm{PE}}{q} \nonumber\]

With potential energy, the case often is that its value at a single point has no significant meaning but what is important is the difference in potential energy. From the difference in potential energy, we are able to calculate other quantities, such as change in kinetic energy (if no force other than the conservative force acts) or work needing to be done by other forces (if other forces act). So likewise, rather than the electric potential itself, we are often interested in difference in electric potential \(\Delta V\) between two points, where,

\[\Delta V=V_{\mathrm{B}}-V_{\mathrm{A}}=\frac{\Delta \mathrm{PE}}{q}. \nonumber \]

The potential difference between points A and B, \(V_{\mathrm{B}}-V_{\mathrm{A}}\), is thus defined to be the change in potential energy of a charge \(q\) moved from A to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (V) after Alessandro Volta.

\[1 \mathrm{~V}=1 \frac{\mathrm{J}}{\mathrm{C}} \nonumber \]

Definition: POTENTIAL DIFFERENCE

The potential difference between points A and B, \(V_{\mathrm{B}}-V_{\mathrm{A}}\), is defined to be the change in potential energy of a charge q moved from A to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (V) after Alessandro Volta.

\[1 \mathrm{~V}=1 \frac{\mathrm{J}}{\mathrm{C}} \nonumber\]

The familiar term voltage is the common name for potential difference. Keep in mind that whenever a voltage is quoted, it is understood to be the potential difference between two points. For example, every battery has two terminals, and its voltage is the potential difference between them. More fundamentally, the point you choose to be zero volts is arbitrary. This is analogous to the fact that gravitational potential energy has an arbitrary zero, such as sea level or perhaps a lecture hall floor.

In summary, the relationship between potential difference (or voltage) and electrical potential energy is given by

\[\Delta V=\frac{\Delta \mathrm{PE}}{q} \text { and } \Delta \mathrm{PE}=q \Delta V. \nonumber \]

POTENTIAL DIFFERENCE AND ELECTRICAL POTENTIAL ENERGY

The relationship between potential difference (or voltage) and electrical potential energy is given by

\[\Delta V=\frac{\Delta \mathrm{PE}}{q} \text { and } \Delta \mathrm{PE}=q \Delta V. \nonumber\]

The second equation is equivalent to the first.

Voltage is not the same as energy. Voltage is the energy per unit charge. Thus, a motorcycle battery and a car battery can both have the same voltage (more precisely, the same potential difference between battery terminals), yet one stores much more energy than the other since \(\Delta \mathrm{PE}=q \Delta V\). The car battery can move more charge than the motorcycle battery, although both are 12 V batteries.

Example \(\PageIndex{1}\): Calculating Energy

Suppose you have a 12.0 V motorcycle battery that can move 5000 C of charge, and a 12.0 V car battery that can move 60,000 C of charge. How much energy does each deliver? (Assume that the numerical value of each charge is accurate to three significant figures.)

Strategy

To say we have a 12.0 V battery means that its terminals have a 12.0 V potential difference. When such a battery moves charge, it puts the charge through a potential difference of 12.0 V, and the charge is given a change in potential energy equal to \(\Delta \mathrm{PE}=q \Delta V\).

So to find the energy output, we multiply the charge moved by the potential difference.

Solution

For the motorcycle battery, \(q=5000 \ \mathrm{C}\) and \(\Delta V=12.0 \mathrm{~V}\). The total energy delivered by the motorcycle battery is

\[\begin{aligned}
\Delta \mathrm{PE}_{\text {cycle }} &=(5000 \mathrm{C})(12.0 \mathrm{~V}) \\
&=(5000 \mathrm{C})(12.0 \mathrm{~J} / \mathrm{C}) \\
&=6.00 \times 10^{4} \mathrm{~J}.
\end{aligned} \nonumber\]

Similarly, for the car battery, \(q=60,000 \ \mathrm{C}\) and

\[\begin{aligned}
\Delta \mathrm{PE}_{\mathrm{car}} &=(60,000 \mathrm{C})(12.0 \mathrm{~V}) \\
&=7.20 \times 10^{5} \mathrm{~J}.
\end{aligned} \nonumber\]

Discussion

While voltage and energy are related, they are not the same thing. The voltages of the batteries are identical, but the energy supplied by each is quite different. Note also that as a battery is discharged, some of its energy is used internally and its terminal voltage drops, such as when headlights dim because of a low car battery. The energy supplied by the battery is still calculated as in this example, but not all of the energy is available for external use.

Note that the energies calculated in the previous example are absolute values. The change in potential energy for the battery is negative, since it loses energy. These batteries, like many electrical systems, actually move negative charge—electrons in particular. The batteries repel electrons from their negative terminals (A) through whatever circuitry is involved and attract them to their positive terminals (B) as shown in Figure \(\PageIndex{2}\). The change in potential is \(\Delta V=V_{\mathrm{B}}-V_{\mathrm{A}}=+12 \mathrm{~V}\) and the charge \(q\) is negative, so that \(\Delta \mathrm{PE}=q \Delta V\) is negative, meaning the potential energy of the battery has decreased when \(q\) has moved from A to B.

Figure \(\PageIndex{2}\): A battery moves negative charge from its negative terminal through a headlight to its positive terminal. Appropriate combinations of chemicals in the battery separate charges so that the negative terminal has an excess of negative charge, which is repelled by it and attracted to the excess positive charge on the other terminal. In terms of potential, the positive terminal is at a higher voltage than the negative. Inside the battery, both positive and negative charges move.

Section Summary

• Electric potential is potential energy per unit charge.
• The potential difference between points A and B, \(V_{\mathrm{B}}-V_{\mathrm{A}}\), defined to be the change in potential energy of a charge \(q\) moved from A to B, is equal to the change in potential energy divided by the charge, Potential difference is commonly called voltage, represented by the symbol \(\Delta V\).

  \[.\Delta V=\frac{\Delta \mathrm{PE}}{q} \text { and } \Delta \mathrm{PE}=q \Delta V \nonumber\]

Glossary

electric potential

potential energy per unit charge

potential difference (or voltage)

change in potential energy of a charge moved from one point to another, divided by the charge; units of potential difference are joules per coulomb, known as volt