Whenever a charge is moved in an electrostatic field, work is done by electrostatic forces. An electrostatic field is a conservative force field. Therefore, any work done against the field is stored as potential energy.

**Electrostatic Potential Energy :** The work done in moving a charge (q_{2}) from infinity to a point in the field against electrostatic force is called electrostatic Potential Energy .

Electric field at A due to fixed charge q_{1} is

$\large E = \frac{1}{4\pi \epsilon_0} \frac{q}{x^2} $ , (here OA = x )

Small amount of work done in bringing a charge (q_{2}) from A to B is

$\large dW = \vec{F}.\vec{dx} = F dx cos180^o = – F dx$

$\large dW = -\frac{1}{4 \pi \epsilon_0}\frac{q_1 q_2}{x^2} dx$

Total amount of work done in in moving a charge from infity to Point P is

$\large W = -\frac{q_1 q_2}{4 \pi \epsilon_0} \int_{\infty}^{r} \frac{1}{x^2} dx$

$\large W = -\frac{q_1 q_2}{4 \pi \epsilon_0} [-\frac{1}{x}]_{\infty}^{r}$

$\large W = \frac{q_1 q_2}{4 \pi \epsilon_0} [\frac{1}{r} – \frac{1}{\infty}]$

$\large W = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r} $

By definition ;

Potential Energy of the System is

$\large U = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r} $

The potential energy of a two-charge system is taken to be zero, when the distance between the charges is infinity.

i.e. U = 0 if r = ∞

For like charges U is +ve & for unlike charges U is −ve. In gravitation, potential energy U is always −ve.

**Potential energy of System of Charges: **

(i) Two charges Q1 and Q2 are separated by a distance ‘d’. The P.E. of the system of charges is

$\large U = \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{d} $

(ii) Three charges Q_{1} , Q_{2} , Q_{3} are placed at the three vertices of an equilateral triangle of side ‘a’.

The P.E. of the system of charges is

$\large U = \frac{1}{4 \pi \epsilon_0} [\frac{Q_1 Q_2}{a} + \frac{Q_2 Q_3}{a} + \frac{Q_3 Q_1}{a}]$

**NOTE :** (i) A charged particle of charge Q2 is held at rest at a distance ‘d’ from a stationary charge Q1. When the charge is released, the K.E. of the charge Q2 at infinity is $\large \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{d} $.

(ii)If two like charges are brought closer, P.E of the system increases.

(iii)If two unlike charges are brought closer, P.E of the system decreases.

(iv)For an attractive system U is always NEGATIVE.

(v)For a repulsive system U is always POSITIVE.

(vi)For a stable system U is MINIMUM.

i.e. F = -dU/dx = 0 (for stable system)

### Potential Energy of a System of two Charges in an External Field:

Consider two charges q_{1} and q_{2} located at two points A and B having position vectors r_{1} and r_{2} respectively.

Let V_{1} and V_{2} be the potentials due to external sources at the two points respectively.

The work done in bringing the charge q_{1} from infinity to the point A is W_{1} = q_{1}V_{1}

In bringing charge q_{2} , the work to be done not only against the external field but also against the field due to q_{1}.

The work done in bringing the charge q_{2} from infinity to the point B is W_{2} = q_{2} V_{2} .

While bringing q_{2} from infinity to position r_{2} , the work done on q_{2} against the field due to q_{1} is

$\large W_3 = \frac{1}{4\pi \epsilon_0}\frac{q_1 q_2}{r_{12}}$ ; where r_{12} is the distance between q_{1} and q_{2}.

The total work done in assembling the configuration or the potential energy of the system is

$\large U = W_1 + W_2 + W_3 $

$\large U = q_1 V_1 + q_2 V_2 + \frac{1}{4\pi \epsilon_0}\frac{q_1 q_2}{r_{12}}$