(a) Depict the equipotential surfaces for a system of two identical positive point charges placed a distance 'd' apart.
(ii) Deduce the expression for the potential energy of a system of two point charges `q_1 and q_2` brought from infinity of the points `vecr_1` and `vecr_2` respectively in the presence of external electric field `vecE`.

Explanation

(a) Equoipotential surfaces of two identical postive point charges placed at a distance 'd' apart.
(b) (a) Let ` vecE` be the external field.
` therefore` Work done one ` q_2` against the external field ` =q_2.V(r_2)`
Work done on `q_2` against the filed due to
` q_1=(1)/(4 pi epsilon_0) .(q_1q_2)/(r_12)=(q_1q_2)/(4pi epsilon _0r_12)`
where `r_12` is the distance between `q_1 and q_2`.
By the super position principle for fields, we add up the work on `q_2` against the two fields (`vecE` and that due to`q_1`). Therefore, work done in bringing `q_2` to `vecr_2`.
`=q_2.V(vecr_2)+(q_1q_2)/(4piepsilon_0r_12)`
Thus, the potential energy of the system = the total work done in assembling the configuration.
`=q_1.V(vecr_1)+q_2+(q_1q_2)/(4piepsilon_0 r_12)`.