# Capacitor & Capacitance

###### Capacitor:

Capacitor is a device for storing electric charge. It consists of a pair of conductors carrying equal and opposite charges (generally). Magnitude of this charge is known as the charge on the capacitor. Potential difference (V) between the two conductors (known as the potential across the capacitor) is proportional to the charge on the capacitor (Q)

Q ∝  V

Q = CV

Here the proportionality constant C is known as the capacitance of the capacitor. The value of capacitance depends on the geometry of the two conductors, their relative position and the medium between them.

### Parallel plate capacitor:

(i) Electric field between the plates :

Electric field due to the positive plate,

$\displaystyle E_1 = \frac{\sigma}{2 \epsilon_0} = \frac{Q}{2 \epsilon_0 A}$ ; from positive to negative plate.

Net Field , $\displaystyle \vec{E} = \vec{E_1} + \vec{E_2}$

$\displaystyle E = \frac{Q}{2 \epsilon_0 A} + \frac{Q}{2 \epsilon_0 A}$

$\displaystyle E = \frac{Q}{ \epsilon_0 A} = \frac{\sigma}{\epsilon_0}$

Where σ = Q/A = Surface Charge density

P.d across the plates :

$\displaystyle V = E d = \frac{Q}{ \epsilon_0 A} d$

Capacitance $\displaystyle C = \frac{Q}{V} = \frac{Q}{ \frac{Q}{ \epsilon_0 A} d }$

Capacitance $\displaystyle C = \frac{ \epsilon_0 A}{ d }$

If Space between the plates is filled with dielectric medium of relative permittivity εr then ,

Capacitance $\displaystyle C = \frac{ \epsilon_0 \epsilon_r A}{ d }$

Capacitance of a parallel plate capacitor is :

(i) directly proportional to the area of the plates and

(ii) inversely proportional to the distance of separation between them

Force acting between the plates:

Force acting between the plates = (Charge on one plate) x (Electric field due to the other plate)

$\displaystyle F = Q (\frac{Q}{ 2 \epsilon_0 A } )$

$\displaystyle F = \frac{Q^2}{ 2 \epsilon_0 A }$

##### Energy stored in a capacitor:

Energy stored in a capacitor (U) = Amount of work done in charging the capacitor from initial uncharged state to the given charge state.
For a parallel plate capacitor it is the work done in increasing the separation of the charged plates from zero to d.

$\displaystyle W = \int_{0}^{d} F dx$

$\displaystyle = \frac{Q^2 d}{2 \epsilon_0 A}$

$\displaystyle U = \frac{Q^2 }{2 C}$

$\displaystyle U = \frac{1}{2 } C V^2$

$\displaystyle U= \frac{1}{2 } Q V$

Energy Density (u):

It is Energy stored per unit volume of the field .

$\displaystyle u = \frac{U}{volume}$

$\displaystyle U = \frac{1}{2 } C V^2$

Capacitance $\displaystyle C = \frac{ \epsilon_0 A}{ d }$ and V = E d

$\displaystyle U = \frac{1}{2 }(\frac{ \epsilon_0 A}{ d }) (E d)^2$

$\displaystyle u = \frac{U}{A d} = \frac{1}{2 }\epsilon_0 E^2$

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