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 = ***4πε _{0}R**

This follows from *C = Q/V* where *V* = (1/4πε_{0} )(*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 = ***ε _{0}A/d**

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

*C = ***ε _{0 }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 = ***ε _{0 }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 *K*_{1}, *K*_{2},* K*_{3},* K*_{4} etc. with thicknesses *t*_{1},* t*_{2},* t*_{3},* t*_{4} etc., the effective capacitance (*C*) is given by

*C = *ε_{0 }*A/*[*d *– (*t*_{1} +* t*_{2} *+ t*_{3}* + t*_{4} + …) + (*t*_{1}* /K*_{1}) +(*t*_{2}* /K*_{2}) + (*t*_{3}* /K*_{3}) + (*t*_{4}* /K*_{4}) +…. ]

Since (*t*_{1} +* t*_{2} *+ t*_{3}* + t*_{4} + …) = *d*, we obtain

*C = ***ε _{0 }A/[(t_{1} /K_{1}) +(t_{2} /K_{2}) + (t_{3} /K_{3}) + (t_{4} /K_{4}) +…]**

(4) Effective capacitance (*C*) of a *series *combination of *n *capacitors of capacitance *C*_{1}, *C*_{2}, *C*_{3}, *C*_{4},…..* C*_{n}* *is given by the reciprocal relation,

**1/ C = 1/C_{1} + 1/C_{2} + 1/C_{3} + 1/C_{4} + ....... + 1/C_{n}**

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

*C* = *C*_{1} + *C*_{2} +* C*_{3} +* C*_{4} + ....... +* C*_{n}

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

*U =*(½)*CV*^{2}

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

*U = Q*^{2}/2*C* = (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)**ε _{0}**

*E*

^{2}

*Ad*[You will get it from *U = Q*^{2}/2*C* on substituting *Q = A*σ and *E *= σ/ε_{0} where σ is the surface density of charge on the plates].

The ** energy density** in the space between the plates is

**(½)**

**ε**

_{0}

*E***since**

^{2}*Ad*is the volume of the space between the plates.

(8) Capacitance of a cylindrical capacitor is given by

*C = ***2πε _{0} 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 = ***4πε _{0} r_{a} r_{b} /(r_{b }– **

*r*

_{a}**)**where

**and**

*r*_{a}**are respectively the radii of the inner and outer spherical shells.**

*r*_{b}
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