Electrostatic Potential and Capacitance

The One Thing

Electric potential gives an energy-based description of an electric field, while capacitance describes the charge stored per unit potential difference. Conductors, equipotential surfaces, capacitors and dielectrics are understood through the relationships among potential, electric field, charge and stored energy.

Reactions, Processes and Experiments

Key Terms

• Electric potential: The work done per unit positive test charge in bringing a test charge from infinity to a point without acceleration; $V=W/q$.
• Potential difference: The work done per unit charge in moving a positive test charge between two points; $V_B-V_A=W_{AB}/q$.
• Electric potential due to a point charge: The potential at distance $r$ from charge $q$, taking potential at infinity as zero: $V=(1/4\pi {\epsilon }_0)(q/r)$.
• Superposition of potential: The total potential due to several charges is the algebraic sum of the potentials produced by individual charges.
• Equipotential surface: A surface on which electric potential is constant at every point.
• Electric field and potential: The electric field points in the direction of decreasing potential; for a uniform field, $E=-dV/dr$.
• Electrostatic equilibrium: A state in which the electric field inside a conductor is zero and the entire conductor has the same potential.
• Capacitance: The charge stored per unit potential difference, $C=Q/V$.
• Parallel-plate capacitor: A capacitor consisting of two large conducting plates separated by distance $d$; in vacuum or air, $C={\epsilon }_{0}A/d$.
• Dielectric: An insulating material that becomes polarised in an electric field and can increase the capacitance of a capacitor.
• Dielectric constant: The relative permittivity $K$, defined by $K=\epsilon /{\epsilon }_0$; when it completely fills a capacitor, the capacitance becomes $K{C}_0$.
• Polarisation: The slight separation or alignment of positive and negative charges within a dielectric placed in an electric field.
• Capacitors in parallel: A combination for which $C_{eq}=C_1+C_2+C_3+...$, with the same potential difference across each capacitor.
• Capacitors in series: A combination for which $1/C_{eq}=1/C_1+1/C_2+1/C_3+...$, with the same charge on each capacitor.
• Energy stored in a capacitor: The electrostatic energy stored in a capacitor, $U=1/2C{V}^2=1/2QV=Q^2/(2C)$.
• Energy density: The energy stored per unit volume of an electric field, $u=1/2\epsilon E^2$.
• Volt: The SI unit of electric potential and potential difference; $1\mathrm{V}=1\mathrm{J}{\mathrm{C}}^{-1}$.
• Farad: The SI unit of capacitance; $1\mathrm{F}=1\mathrm{C}{\mathrm{V}}^{-1}$.
• Permittivity of free space: ${\epsilon }_0\approx 8.85\times 10^{-12}\mathrm{F}{\mathrm{m}}^{-1}$, with $1/(4\pi {\epsilon }_0)\approx 9\times 10^9\mathrm{N}{\mathrm{m}}^2{\mathrm{C}}^{-2}$.

Easily Confused

• Electric potential and electric field: Potential is a scalar quantity measured in volts, whereas electric field is a vector quantity related to the spatial decrease of potential by $E=-dV/dr$.
• Potential difference and potential energy: Potential difference is work done per unit charge, whereas potential energy is the energy of a particular charge at a potential, $U=qV$.
• Equipotential surface and electric field line: An equipotential surface has constant potential and no work is done along it; an electric field line follows the field direction and is perpendicular to an equipotential surface.
• Capacitance and charge: Capacitance is $C=Q/V$ and is determined by geometry and medium, whereas charge is the amount stored on the capacitor.
• Capacitors in series and parallel: Series capacitors carry the same charge and have an equivalent capacitance smaller than the smallest individual capacitance; parallel capacitors have the same potential difference and an equivalent capacitance greater than the largest individual capacitance.
• Dielectric insertion with a connected battery and with an isolated capacitor: With the battery connected, potential difference remains constant and charge increases; with the capacitor isolated, charge remains constant and potential difference decreases.
• Electric field inside and outside a charged spherical conductor: The field is zero inside the conductor; outside, it behaves as if the entire charge were concentrated at the centre.
• Steady direct current and capacitor charging current: A capacitor blocks steady direct current through its dielectric but permits charging and discharging through an external circuit.

What Gets Asked

• Define electric potential, potential difference and capacitance, including $V=W/q$, $V_B-V_A=W_{AB}/q$, and $C=Q/V$. Marks are lost by omitting the “per unit charge” meaning or confusing potential with potential energy.
• Calculate the potential due to a point charge, an electric dipole, or several charges. Marks are lost by treating potential as a vector instead of adding it algebraically.
• Explain equipotential surfaces and their relationship with electric fields. Marks are lost by failing to state that no work is done along an equipotential surface or that the electric field is perpendicular to it.
• Apply conductor results to an isolated spherical conductor. Marks are lost by assigning a non-zero electric field inside the conductor or failing to distinguish the constant internal potential from the external field.
• Derive or use parallel-plate capacitance and dielectric relations, including $C={\epsilon }_{0}A/d$ and $C=K{\epsilon }_{0}A/d=K{C}_0$. Marks are lost by treating capacitance as depending directly on charge or potential rather than on geometry and medium.
• Analyse capacitor combinations and dielectric insertion. Marks are lost by interchanging the series and parallel rules, or by failing to distinguish the battery-connected case from the isolated-capacitor case.