2 Electric Potential and Capacitance

Potential, energy, and electric fields

Electric potential energy describes the energy associated with a charge’s position in an electric field. Electric potential expresses this energy per unit charge:

Potential difference is also called Voltage. Its SI unit is the volt, where $1\ \mathrm{V}=1\ \mathrm{J/C}$. Potential is a scalar, so contributions from several charges add algebraically. The zero of potential is a choice; for an isolated point charge, it is customary to set the potential to zero at infinity.

For a point charge $Q$, the potential at distance $r$ is

For several point charges, add the potential contributions:

A charge $q$ at potential $V$ has potential energy $U=qV$, measured relative to the chosen zero. With zero potential energy at infinite separation, like charges have positive interaction energy and unlike charges have negative interaction energy.

The electric field relates potential at two locations. In a uniform field directed from point $A$ to point $B$ over distance $d$,

More generally,

The field points toward decreasing potential. The work done by the field on a charge is $W_{\text{field}}=-\Delta U=-q\Delta V$. Moving the charge slowly against the field requires an equal amount of external work.

Example. If a $+2.0\ \mathrm{\mu C}$ charge moves through a potential difference of $-5.0\ \mathrm{V}$, then

The potential energy decreases by $10\ \mathrm{\mu J}$, so the field does positive work of $10\ \mathrm{\mu J}$.

Takeaway: Potential is energy per unit charge, and the sign of $q\Delta V$ tells whether a charge’s potential energy increases or decreases.

Conductors at equilibrium

A conductor contains mobile charges. If an electric field existed inside its conducting material, those charges would move. In Electrostatic equilibrium, charge has redistributed so that:

• The electric field inside the conducting material is zero.

• The conductor’s surface and interior have the same potential; the conductor is an equipotential.

• Any excess charge resides on the surface.

• Just outside the surface, the electric field is perpendicular to it. A tangential field component would move surface charges.

These properties explain electrostatic shielding: an empty cavity within a conductor has no electric field when the conductor is in electrostatic equilibrium and no charge is placed inside the cavity. Sharp regions can have especially strong fields because surface charge is more concentrated there.

Takeaway: In electrostatic equilibrium, a conductor’s interior has no electric field, and its entire surface is at one potential.

Capacitors and capacitance

A capacitor consists of two conductors separated by an insulating gap. Equal and opposite charges, $+Q$ and $-Q$, establish a potential difference $\Delta V$. Capacitance describes the charge stored per unit potential difference:

Capacitance depends on the conductors’ geometry and the material between them, not on the particular values of $Q$ and $\Delta V$. Its SI unit is the farad, with $1\ \mathrm{F}=1\ \mathrm{C/V}$.

For two large parallel plates of area $A$, separated by distance $d$, and with small edge effects,

Increasing plate area increases capacitance; increasing the separation decreases it. In this approximation, the field between the plates is nearly uniform, with

Takeaway: A capacitor’s geometry determines how much charge it stores for a specified potential difference.

Energy stored in a capacitor

Charging a capacitor requires work, which is stored as electric potential energy in its electric field. For charge magnitude $Q$, capacitance $C$, and potential difference $\Delta V$, the stored energy can be calculated in three equivalent ways:

Choose the form that uses the quantities known in a problem. For example, a capacitor with capacitance $4.0\ \mathrm{\mu F}$, charged to $12\ \mathrm{V}$, stores

For a uniform electric field in vacuum, the field energy density, meaning energy per unit volume, is

This expresses that stored energy is associated with the electric field, not simply with charge sitting on the plates.

Takeaway: The three energy formulas are equivalent; select one based on whether charge, capacitance, or potential difference is known.

Dielectrics and what remains constant

A Dielectric is an insulating material placed between a capacitor’s conductors. Its molecules or atoms become polarized in the electric field: bound charges shift slightly, creating a field that opposes the original field within the material.

If a uniform dielectric completely fills the gap, the capacitance becomes

where $C_0$ is the vacuum capacitance and $\kappa$ is the dielectric constant, or relative permittivity. For common linear dielectrics, $\kappa>1$, so the capacitance increases.

The resulting changes depend on whether the capacitor remains connected to a battery:

• Disconnected capacitor: Its charge $Q$ stays constant. As $C$ increases, $\Delta V=Q/C$ decreases, and the stored energy $U=Q^2/(2C)$ decreases.

• Battery-connected capacitor: Its potential difference $\Delta V$ stays constant. As $C$ increases, the capacitor draws more charge, and its stored energy $U=\frac{1}{2}C(\Delta V)^2$ increases.

Always identify what is held constant before comparing charge, potential difference, or energy. Energy can be exchanged with the battery or through mechanical forces as the dielectric enters the gap.

Takeaway: Inserting a dielectric increases capacitance, but the changes to charge, potential difference, and energy depend on whether charge or voltage is held constant.