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**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=VQ $
- The SI unit of capacitance is Farad and its CGS unit is Stat farad.
- 1 stat farad = $91 ∗10_{−11}$farad
- The dimensional formula of electrical capacitance is [$M_{−1}L_{−2}T_{−4}A_{−2}$]
- The capacitance of spherical conductor is : $C=4πE_{o}R=9∗10_{9}R $ where R is the radius of the conductor.
- The capacitance of concentric spherical capacitor is: $C=4πE_{o}b−aab $where 'b' is the radius of outer sphere and 'a' is the radius of inner sphere.
- The capacitance of the earth is $7.12∗10_{−4}$ farad = 712$μ$F

**Parallel plate capacitor:**

- It is the simplest capacitor where two conductors are separated by insulator/dielectric.
- The capacitance is given by: $C=dE_{o}A =dKE_{o}A $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_{′}=d−t(1−K1 )E_{o}A $

**Combination of capacitors:**

- Series combination: $C1 =C_{1}1 +C_{2}1 +.........C_{n}1 $
- 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=Kt +Kt +Kt +......E_{o}A $

**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=dE_{o}(K_{1}A_{1}+K_{2}A_{2}+K_{3}A_{3}+.....) $

**Energy stored in a capacitor: **$U=2CV_{2} =2qV =2Cq_{2} $

**Energy density: **$U_{E}=2E_{o}E_{2} $where E is the electric field intensity

**Some important relations:**

- Regrouping of capacitors:

- When like plates are joined:

$C_{1}V_{1}+C_{2}V_{2}=(C_{1}+C_{2})V$

$V=C_{1}+C_{2}C_{1}V_{1}+C_{2}V_{2} $

Loss in energy = $2(C_{1}+C_{2})C_{1}C_{2}(V_{1}−V_{2})_{2} $

- When unlike plates are joined:

$V=C_{1}+C_{2}C_{1}V_{1}−C_{2}V_{2} $

Loss in energy = $2(C_{1}+C_{2})C_{1}C_{2}(V_{1}+V_{2})_{2} $

- For charging a capacitor:

$q=q_{o}(1−e_{RC−t})$

$V=V_{o}(1−e_{RC−t})$

$I=I_{o}e_{RC−t}$

- For discharging a capacitor:

$q=q_{o}e_{RC−t}$

$V=V_{o}e_{RC−t}$

$I=I_{o}e_{RC−t}$