# Electric Potential Energy

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 (q2) from infinity to a point in the field against electrostatic force is called electrostatic Potential Energy .

Electric field at A  due to fixed charge q1 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 (q2) 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 Q1 , Q2 , Q3 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 q1 and q2 located at two points A and B having position vectors r1 and r2 respectively.
Let V1 and V2 be the potentials due to external sources at the two points respectively.

The work done in bringing the charge q1 from infinity to the point A is W1 = q1V1

In bringing charge q2 , the work to be done not only against the external field but also against the field due to q1.

The work done in bringing the charge q2 from infinity to the point B is W2 = q2 V2 .

While bringing q2 from infinity to position r2 , the work done on q2 against the field due to q1 is

$\large W_3 = \frac{1}{4\pi \epsilon_0}\frac{q_1 q_2}{r_{12}}$  ; where r12 is the distance between q1 and q2.

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

$\large U = W_1 + W_2 + W_3$

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