AP Physics B & C – Electrostatics – Capacitors – Equations to be remembered

The following points are to be noted by AP Physics B as well as AP Physics C aspirants:

(1) Capacitance (C) of a capacitor is given by

C = Q/V where Q is the magnitude of charge on one of the plates of the capacitor and V is the potential difference between the plates. Q is in coulomb, V is in volts and C is in farad.

In a charged capacitor the charge on one plate is positive and the charge on the other plate is negative; but the charges are of equal magnitude so that the total charge on the two plates taken together is zero. But when you say “charge on a capacitor”, you mean the magnitude of charge on one of the plates.

[Often you may be required to calculate the capacitance of a single conductor such as a sphere. What you mean here is the ratio of the charge given to the conductor to the potential to which it is raised: C = Q/V]

(2) Capacitance (C) of a spherical conductor of radius R is given by

C = 4πε₀R

This follows from C = Q/V where V = (1/4πε₀ )(Q/R), which is the potential on the surface of a spherical conductor carrying charge Q.

(3) Capacitance (C) of a parallel plate capacitor having air (or vacuum) as dielectric, with each plate of area A and with separation d between the plates is given by

C = ε₀A/d

If the dielectric is a material of dielectric constant (relative permittivity) K, the capacitance is K times and is given by

C = ε₀ KA/d

If a dielectric slab of dielectric constant K and thickness t is introduced in between the plates of a parallel plate air capacitor, the capacitance (C) is given by

C = ε₀ A/[d – t + (t/K)]

If the space between the plates of a parallel plate capacitor is completely filled with different dielectric slabs of dielectric constants K₁, K₂, K₃, K₄ etc. with thicknesses t₁, t₂, t₃, t₄ etc., the effective capacitance (C) is given by

C = ε₀ A/[d – (t₁ + t₂ + t₃ + t₄ + …) + (t₁ /K₁) +(t₂ /K₂) + (t₃ /K₃) + (t₄ /K₄) +…. ]

Since (t₁ + t₂ + t₃ + t₄ + …) = d, we obtain

C = ε₀ A/[(t₁ /K₁) +(t₂ /K₂) + (t₃ /K₃) + (t₄ /K₄) +…]

(4) Effective capacitance (C) of a series combination of n capacitors of capacitance C₁, C₂, C₃, C₄,….. C_n is given by the reciprocal relation,

1/C = 1/C₁ + 1/C₂ + 1/C₃ + 1/C₄ + ....... + 1/C_n

(5) Effective capacitance (C) of a parallel combination of n capacitors is given by

C = C₁ + C₂ + C₃ + C₄ + ....... + C_n

(6) Energy (U) stored in a charged capacitor is given by

U =(½)CV²

Since Q = CV, the energy can be written also as

U = Q²/2C = (½)QV

Note that the energy in a charged capacitor is stored in the electric field between the plates.

The following points are meant for AP Physics C aspirants only:

(7) If E is the electric field between the plates of a parallel plate capacitor the expression for the energy can be written in terms of the electric field E between the plates as

U = (½)ε₀E²Ad

[You will get it from U = Q²/2C on substituting Q = Aσ and E = σ/ε₀ where σ is the surface density of charge on the plates].

The energy density in the space between the plates is (½)ε₀E² since Ad is the volume of the space between the plates.

(8) Capacitance of a cylindrical capacitor is given by

C = 2πε₀ L / ln(r_b – r_a) where L is the length of the cylinders and r_a and r_b are respectively the radii of the inner and outer cylinders.

(9) A spherical capacitor is made of two concentric spherical conducting shells. Capacitance of a spherical capacitor is given by

C = 4πε₀ r_a r_b /(r_b – r_a) where r_a and r_b are respectively the radii of the inner and outer spherical shells.

In the next post we will discuss questions in this section. Meanwhile, find a useful post containing multiple choice questions (with solution) in this section at physicsplus