Electrostatic Potential and Capacitance

In this chapter: electrostatic potential and potential difference, potential of a point charge, a dipole and a system of charges, equipotential surfaces and E = −dV/dr, potential energy of charges and of a dipole in a field, conductors in electrostatics, dielectrics and polarisation, capacitance, the parallel plate capacitor, dielectric slabs, series and parallel combinations, energy stored.

Potential and potential difference

The electrostatic force is conservative, so the work done in moving a charge between two points does not depend on the path. This lets us define potential energy, and from it potential. The potential at a point is the work done by an external agent in bringing a unit positive charge from infinity to that point without acceleration:

Point charge, dipole, system of charges

The dipole potential is zero everywhere on the equatorial plane (θ = 90°) and falls as 1/r². Potential is a scalar, so for a system of charges you add values with their signs; there is no direction to worry about, which is why it is usually easier to find V first and then E.

For a uniformly charged spherical shell of radius R, V = kq/r outside and V = kq/R everywhere inside. The field inside is zero, but the potential inside is not zero; it is constant.

Equipotential surfaces

An equipotential surface has the same potential at every point. No work is done in moving a charge along it, and so the field is always normal to the equipotential surface. For a point charge these surfaces are concentric spheres; in a uniform field they are planes perpendicular to the field.

Where equipotentials are closely spaced, the field is strong. In a uniform field, E = V/d between two surfaces a distance d apart.

Potential energy

The potential energy of a system of point charges is the work needed to assemble them from infinity. For two charges:

Keep the signs of the charges: U is positive for like charges (work must be done to push them together) and negative for unlike charges. In an external field, a single charge q at a point of potential V has energy qV. Energies in atomic physics are often in electron volts: 1 eV = 1.6 × 10⁻¹⁹ J.

Dipole in a uniform field

U is minimum (−pE) at θ = 0, which is stable equilibrium, and maximum (+pE) at θ = 180°, which is unstable. Rotating the dipole from 0 to 180° needs work 2pE.

Conductors in electrostatics

• The field inside a conductor is zero, because free electrons move until the internal field is cancelled.
• Just outside the surface, the field is normal to the surface and equals σ/ε₀.
• There is no net charge in the interior; any excess charge sits on the surface.
• The whole conductor, inside and surface, is at one potential.
• Inside a cavity with no charge in it, the field is zero, whatever charges or fields are outside. This is electrostatic shielding, the reason a car or a metal cage is a safe place during lightning.

Dielectrics and polarisation

A dielectric has no free charges. In an external field its molecules either develop an induced dipole moment (non-polar molecules) or their permanent dipoles partly line up (polar molecules). The dielectric gets a net dipole moment per unit volume, the polarisation P = χ_eε₀E, where χ_e is the electric susceptibility. The result is equivalent to bound charge densities ±σ_p on the faces, which set up a field opposite to the applied field and reduce it. The dielectric constant is K = 1 + χ_e = ε/ε₀.

Capacitors

A capacitor is two conductors carrying charges +Q and −Q. The potential difference V between them is proportional to Q, and the ratio is the capacitance:

C depends only on the geometry and the medium between the conductors, not on Q or V. If the voltage across a capacitor exceeds a limit, the medium ionises and the capacitor leaks; for air this dielectric strength is about 3 × 10⁶ V m⁻¹.

Parallel plate capacitor

For plates of area A separated by d (with d small compared with the plate size), the field between them is E = σ/ε₀ = Q/(ε₀A) and the voltage is Ed. So

A metal slab of thickness t acts like K → ∞, giving C = ε₀A/(d − t).

Combination of capacitors

In series each capacitor carries the same charge and the voltages add; the equivalent is smaller than the smallest. In parallel each has the same voltage and the charges add. Notice that these are the reverse of the rules for resistors.

Energy stored in a capacitor

Charging a capacitor means moving small charges dq against the growing voltage q/C. Adding up the work gives

The energy is stored in the electric field between the plates. Use Q²/2C when the charge is fixed and ½CV² when the voltage is fixed; that choice alone settles most "what happens to the energy" questions.

Worked example: Capacitors of 2 μF, 3 μF and 6 μF are connected in series across a 12 V battery. Find the equivalent capacitance, the charge, the voltage across each and the total energy.
Solution: 1/C = 1/2 + 1/3 + 1/6 = 1, so C = 1 μF. Charge on each: Q = CV = 1 × 12 = 12 μC. Voltages: 12/2 = 6 V, 12/3 = 4 V and 12/6 = 2 V, which add to 12 V as they should. Energy: U = ½ × 1 × 10⁻⁶ × 12² = 72 μJ.

Worked example: A parallel plate capacitor has plates of area 6 × 10⁻³ m² separated by 3 mm of air. Find its capacitance and the charge on each plate when connected to 100 V.
Solution: C = ε₀A/d = (8.85 × 10⁻¹² × 6 × 10⁻³)/(3 × 10⁻³) = 1.77 × 10⁻¹¹ F = 17.7 pF. Charge Q = CV = 1.77 × 10⁻⁹ C.

Common mistakes: (1) Saying the potential inside a charged shell or conductor is zero because the field is zero; it is constant and equal to the surface value. (2) Dropping the sign of the charges in V or U. (3) Using the resistor rules for capacitors. (4) Forgetting whether the battery is connected when a dielectric is inserted or plates are pulled apart. (5) Writing the dipole energy as +pE cos θ; the minus sign is what makes θ = 0 stable.

JEE and NEET focus

• Potential and potential energy of a system of point charges, with signs.
• E = −dV/dr, including finding E from a given V(x) and the idea of equipotential surfaces.
• Work done in rotating a dipole in a uniform field.
• Parallel plate capacitor with dielectric or metal slabs, and the battery connected or disconnected cases.
• Series and parallel networks, charge sharing between two capacitors and energy stored.

Practice questions