Two point charges `q_(1) and q_(2)` are kept r distance apart in a uniform external electric field `vecE`. Find the amount of work done in assembling this system of charges.

Explanation

Let a system of two point charges `q_(1) and q_(2)` is brought from infinity to the points `vecr_(1) and vecr_(2)` respectively in an external electric field `vecE`, where `|vecr_(1)-vecr_(2)|=r`, the distance between the two charge. We first calculate the work done in bringing the charge `q_(1)` from infinity to `vecr_(1)`. work done in this step is
`W_(1)=q_(1)V(vecr_(1))`, where `V(vecr_(1))` is the electrostatic potential at `vecr_(1)`.
Now we consider the work done in bringing `q_(2)` to `vecr_(2)`. In this step work `W_(2)` is done against the electric field and work `W_(12)` is done against the field due to charge `q_(1)`. Obviously
`W_(2)=q_(2)V(vecr_(2))`, where `V(vecr_(2))` is the electrostatic potential at `vecr_(2)`
and `W_(12)=(q_(1)q_(2))/(4pi in_(0)*r)`, where `r=|vecr_(1)-vecr_(2)|= ` distance between `q_(1) and q_(2)`
`therefore` Total work done in bringing `q_(2)` to `vecr_(2)=W_(2)+W_(12)=q_(2)V(vecr_(2))+(q_(1)q_(2))/(4pi in_(0)*r)`