Electric Potential

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 🌐 Version en français : Potentiel électrique

📘 Electric Potential

Reference Textbook

📖 Reference Textbook:
University Physics Volume 2: Chapter 7

Theory

Background: The Electric Force is a Conservative Force

The electric force is a conservative force. That means that the work done by or against the electric force depends only on the initial and final positions, not on the path taken.

This allows us to define the change in electric potential energy:

$\mathrm{\Delta }U=U_{\mathrm{f}}-U_{\mathrm{i}}=-W_{\mathrm{e}}$

where $W_{\mathrm{e}}$ is the work done by the electric force.

Since ${\overrightarrow{F}}_{\mathrm{e}}=q\overrightarrow{E}$,

$W_{\mathrm{e}}=\underset{A}{\overset{B}{\int }}{\overrightarrow{F}}_{\mathrm{e}}\cdot d\overrightarrow{s}=\underset{A}{\overset{B}{\int }}q\overrightarrow{E}\cdot d\overrightarrow{s}$

and therefore

$\mathrm{\Delta }U=-\underset{A}{\overset{B}{\int }}q\overrightarrow{E}\cdot d\overrightarrow{s}$

▶ Video: Conservative Forces and Electric Potential Energy

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Electric Potential and Electric Potential Energy

Electric Potential Energy

Electric potential energy is the energy stored in a system of charges due to their positions in an electric field. This is similar to gravitational potential energy, where the position of an object in a gravitational field determines its potential energy.

Electric Potential

The electric potential $V$ at a point is the electric potential energy per unit charge:

$V=\frac{U}{q}$

Electric potential is a scalar quantity. Like potential energy, its value depends on the chosen zero reference.

Electric Potential Difference (Voltage)

The electric potential difference between points A and B is

$\mathrm{\Delta }V=V_{\mathrm{B}}-V_{\mathrm{A}}=\frac{\mathrm{\Delta }U}{q}$

Potential difference is commonly called voltage. Its SI unit is the volt:

$1\mathrm{V}=1\mathrm{J}/\mathrm{C}$

A potential difference of one volt means that the potential energy changes by one joule for each coulomb of charge. Potential difference is often more useful than the potential at a single point because it does not depend on the choice of zero reference.

Finding Electric Potential Energy from Electric Potential

If a charge $q$ moves through a potential difference $\mathrm{\Delta }V$, its change in electric potential energy is

$\mathrm{\Delta }U=q\mathrm{\Delta }V$

If the external electric potential $V$ is measured relative to the same zero reference as $U$, the potential energy of a charge placed at that point is

$U=qV$

The sign of the charge matters. For the same $\mathrm{\Delta }V$, a positive and a negative charge undergo opposite changes in potential energy.

▶ Video: Electric Potential Energy

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▶ Video: Electric Potential (Voltage)

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▶ Video: Calculating Electric Potential Energy from Voltage

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Electric Potential and the Electric Field

The potential difference between two points can be calculated from the electric field:

$\mathrm{\Delta }V=V_{\mathrm{B}}-V_{\mathrm{A}}=-\underset{A}{\overset{B}{\int }}\overrightarrow{E}\cdot d\overrightarrow{s}$

Because the electrostatic force is conservative, this potential difference is independent of the path chosen between A and B.

For a uniform electric field,

$\mathrm{\Delta }V=-\overrightarrow{E}\cdot \mathrm{\Delta }\overrightarrow{s}=-E\mathrm{\Delta }s\cos \theta$

where $\theta$ is the angle between the electric field and the displacement. If the displacement is parallel to the field,

$\mathrm{\Delta }V=-E\mathrm{\Delta }s$

The electric field always points in the direction in which the electric potential decreases most rapidly.

This relationship gives another equivalent unit for electric field:

$1\mathrm{N}/\mathrm{C}=1\mathrm{V}/\mathrm{m}$

Electric Potential Due to Point Charges

Electric Potential Due to a Point Charge at Rest

Choosing the electric potential to be zero at infinity, the potential at a distance $r$ from a point charge $Q$ is

$V=k\frac{Q}{r}$

▶ Video: Electric Potential Due to a Point Charge

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Electric Potential Due to Several Point Charges

The principle of superposition also applies to electric potential. Because potential is a scalar, the contributions are added algebraically:

$V(P)=\sum _i{V}_i(P)=k\sum _i\frac{q_i}{r_i}$

where $r_i$ is the distance from charge $q_i$ to point P. Positive charges make positive contributions and negative charges make negative contributions.

Potential Energy of a System of Charged Particles

The total electric potential energy of a system of charges is the sum of the potential energies between all pairs of charges in the system. For example, for three charges, $q_1,q_2,$ and $q_3$, the potential energy is: $U=k_e(\frac{q_1{q}_2}{r_{12}}+\frac{q_1{q}_3}{r_{13}}+\frac{q_2{q}_3}{r_{23}})$

Equipotential Lines and Surfaces

Equipotential lines connect points having the same electric potential. In three dimensions, the corresponding objects are equipotential surfaces.

• Electric field lines are always perpendicular to equipotential lines or surfaces.
• Electric field lines point from higher potential toward lower potential.
• Moving along an equipotential requires no work by the electric force because $\mathrm{\Delta }V=0$ and $\mathrm{\Delta }U=q\mathrm{\Delta }V=0$.
• Closely spaced equipotential lines indicate a stronger electric field because the potential changes more rapidly with distance.

Around a point charge, the equipotential surfaces are concentric spheres. In a diagram drawn in two dimensions, they appear as concentric circles.

▶ Video: Equipotential Lines

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▶ Video: Electric Potential and Electric-Field Lines

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▶ Video: Finding the Electric Field from the Potential

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The Electronvolt

The electronvolt (eV) is a unit of energy commonly used for atoms and subatomic particles. One electronvolt is the magnitude of the energy change of one elementary charge moving through a potential difference of one volt:

$1\mathrm{e}\mathrm{V}=1.602\times 10^{-19}\mathrm{J}$

The electronvolt is a unit of energy, not a unit of electric potential.

Electric Potential, Current, and Power

A potential difference can drive electric charges through a conductor when a conducting path is available.

Electric current $I$ is the rate at which charge passes through a cross-section:

$I=\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}$

The instantaneous current is $I=dQ/dt$. Its SI unit is the ampere, where $1\mathrm{A}=1\mathrm{C}/\mathrm{s}$.

Power $P$ is the rate at which energy is transferred or converted:

$P=\frac{\mathrm{\Delta }U}{\mathrm{\Delta }t}$

For a device through which a current $I$ passes across a potential difference $\mathrm{\Delta }V$, the magnitude of the electrical power is

$P=I\mathrm{\Delta }V$

▶ Video: Electric Potential, Current, and Power

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Charged-Particle Motion in a Uniform Electric Field

In a uniform electric field, a charged particle experiences the constant force $\overrightarrow{F}=q\overrightarrow{E}$. Its motion can be analyzed using constant-acceleration kinematics, in a way similar to projectile motion under gravity. It can also be analyzed using energy:

$\mathrm{\Delta }K=-\mathrm{\Delta }U=-q\mathrm{\Delta }V$

▶ Example: Charged-Particle Motion in a Uniform Electric Field

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Electric-Potential Simulations

Explore electric potential and equipotential lines with these simulations:

• PhET: Charges and Fields
• PhET: Capacitor Lab: Basics

Additional Resources

• OpenStax Chapter 7 Summary
• OpenStax Chapter 7 Key Equations

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