**Answer the following question.**

Two-point charges q_{1} and q_{2} are kept at a distance of r_{12} in air. Deduce the expression for the electrostatic potential energy of this system.

#### Solution

Electrostatic potential energy of a system is defined as the total amount of the work done in bringing the various charges to their respective position from infinitely large mutual separation.

Let us consider a charge q_{1} at a postion vector r_{1} and q_{2} at infinity which is to be brought at point P_{2} having position vector r_{2}. and dW is the small amount of work done in moving a charge to a distance dx.

`dW = vec"F". vecdx`

`dW = -(kq_1q_2)/(r^2). dx`

`W = -int_∞^(r12) (kq_1q_2)/(r^2).dx`

`W = -Kq_1q_2 int_∞^(r12) 1/r^2. dx`

`W = -Kq_1q_2 [-1/r]_∞^(r12)`

`W = Kq_1q_2[1/(r_12)]`

`W = U = (Kq_1q_2)/(r_12)`.