Obtain the equation of electric potential energy of a system of two electric charges in external electric field.

Explanation

Two electric charges `q_(1)` and `q_(2)` are brought from infinity distance through distance `r_(1)` and `r_(2)` relative to origin in external electric field.

The work done in bringing the charge `q_(1)`, from infinity through `r_(1)`, distance,
`W_(1)=q_(1)V(vecr_(1))`
and work done in bringing `q_(2)` through distance `r_(2)` . Here work is done not only against the external field `vecE` but also against the field due to `q_(1)`
`W_(2) = q_(2)V(vecr_(2))`
Work done on `q_(2)` against the field due to `q_(1)`.
`W_(3)=(kq_(1)q_(2))/(r_(12))`
where `r_(12)` is the distance between `q_(1)` and `q_(2)` The total potential energy of the system = the total work done in assembling the configuration or potential energy
`U=q_(1)V(vecr_(1))+q_(2)V(vecr_(2))+(kq_(1)q_(2))/(r_(12))`