Three point charges, + Q + 2Q and – 3Q are placed at the vertices of an equilateral triangle ABC of side l. If these charges are displaced to the mid-point A_{1}, B_{1} and C_{1}, respectively, find the amount of the work done in shifting the charges to the new locations.

#### Solution

q_{1}= +Q

q_{2}= +2Q

q_{3}= -3Q

r = l (for each side)

Intial potential energy of system

`U_1=1/(4piin_0l)[(q_1xxq_2)+(q_2xxq_3)+(q_3xxq_1)]`

`U_1=1/(4piin_0l)[(Qxx2Q)+(2Qxx(-3Q))+((-3Q) xxQ)]`

`U_1=(-7Q^2)/(4piin_0l`

These charges displaced to mid points then final potential energy of system,

`U_2=1/(4piin_0l/2)[(q_1xxq_2)+(q_2xxq_3)+(q_3xxq_1)]`

`U_2=2/(4piin_0l)[(Qxx2Q)+(2Qxx(-3Q))+((-3Q)xxQ)]`

`U_2=(-7Q^2)/(2piin_0l)`

Work done, W = U_{2} - U_{1}

`W=(-7Q^2)/(2piin_0l)-(-7Q^2)/(4piin_0l)`

`W=(7Q^2)/(piin_0l)[(-1)/2-((-1)/4)]=(7Q^2)/(piin_0l)[(-1)/2+1/4]`

`W=(-7)/4(Q^2/(piin_0l))`