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## Monday, July 14, 2008

### 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,….. Cn is given by the reciprocal relation,

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

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

C = C1 + C2 + C3 + C4 + ....... + Cn

(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

[You will get it from U = Q2/2C on substituting Q = Aσ and E = σ0 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(rb ra) where L is the length of the cylinders and ra and rb 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 ra rb /(rb ra) where ra and rb 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