Capacitance

Electric capacitors:

• Capacitors are charge storing devices.
• Capacitance of a capacitor is the ratio of charge(Q) given and the potential (V) to which it is raised. $C=\frac{Q}{V}$
• The SI unit of capacitance is Farad and its CGS unit is Stat farad.
• 1 stat farad = $\frac{1}{9}\ast 10^{-11}$farad
• The dimensional formula of electrical capacitance is [$M^{-1}{L}^{-2}{T}^{-4}{A}^{-2}$]
• The capacitance of spherical conductor is : $C=4\pi E_{o}R=\frac{R}{9\ast 10^9}$ where R is the radius of the conductor.
• The capacitance of concentric spherical capacitor is: $C=4\pi E_o\frac{ab}{b-a}$where 'b' is the radius of outer sphere and 'a' is the radius of inner sphere.
• The capacitance of the earth is $7.12\ast 10^{-4}$ farad = 712$\mu$F

Parallel plate capacitor:

• It is the simplest capacitor where two conductors are separated by insulator/dielectric.
• The capacitance is given by: $C=\frac{E_{o}A}{d}=\frac{K{E}_{o}A}{d}$where A is the area of each plate, d is distance between them and K is the dielectric constant.
• The capacitance does not depend on material of plates, charge given to plates, potential difference between the plates.
• The value of C depends on size, shape and relative position of the two coatings of the capacitor. It also depends on the medium separating the two coatings.
• When the space between the plates is partially filled with medium of dielectric constant K and thickness t then capacitance becomes: $C^{\prime }=\frac{E_{o}A}{d-t(1-\frac{1}{K})}$

Combination of capacitors:

• Series combination: $\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+.........\frac{1}{C_n}$
• Parallel combination: $C=C_1+C_2+........C_n$

Dielectric in series:

• The capacitance of a parallel plate capacitor having a number of slabs of thickness $t_1,t_2,t_3,.....$and dielectric constants $K_1,K_2,K_3,.......$ respectively in between is: $C=\frac{E_{o}A}{\frac{t_1}{K_1}+\frac{t_2}{K_2}+\frac{t_3}{K_3}+......}$

Dielectric in parallel:

• When a number of dielectric slabs of same thickness(d) and different areas of cross section $A_1,A_2,A_3,....$ having dielectric constants $K_1,K_2,K_3,........$respectively are placed between the plates of a parallel plate capacitor, its capacitance is given by: $C=\frac{E_o(K_1{A}_1+K_2{A}_2+K_3{A}_3+.....)}{d}$

Energy stored in a capacitor: $U=\frac{C{V}^2}{2}=\frac{qV}{2}=\frac{q^2}{2C}$

Energy density: $U_E=\frac{E_o{E}^2}{2}$where E is the electric field intensity

Some important relations:

1. Regrouping of capacitors:

• When like plates are joined:

$C_1{V}_1+C_2{V}_2=(C_1+C_2)V$

$V=\frac{C_1{V}_1+C_2{V}_2}{C_1+C_2}$

Loss in energy = $\frac{C_1{C}_2{(V_1-V_2)}^2}{2(C_1+C_2)}$

• When unlike plates are joined:

$V=\frac{C_1{V}_1-C_2{V}_2}{C_1+C_2}$

Loss in energy = $\frac{C_1{C}_2(V_1+V_2{)}^2}{2(C_1+C_2)}$

1. For charging a capacitor:

$q=q_o(1-e^{\frac{-t}{RC}})$

$V=V_o(1-e^{\frac{-t}{RC}})$

$I=I_o{e}^{\frac{-t}{RC}}$

1. For discharging a capacitor:

$q=q_o{e}^{\frac{-t}{RC}}$

$V=V_o{e}^{\frac{-t}{RC}}$

$I=I_o{e}^{\frac{-t}{RC}}$