Two point charges are located on the x - axis : `q_1 = -e` at `x = 0` and `q_2 = + e` at `x = a` and `q_3 = + e` at `x = 2a`.
The total potential energy of the system of three charges.

Options

A. `(e^2)/(4 pi epsilon_0 a)`
B. `(e^2)/(8 pi epsilon_0 a)`
C. `(-e^2)/(8 pi epsilon_0 a)`
D. `(-e^2)/(4 pi epsilon_0 a)`

Answer

C

Explanation

The work that must be done on `q_3` by an external force `vec F_(ext)` is equal to the difference between two quantities: the potential energy `U` associated with `q_3` when it is at `x = 2 a` and the potential energy when it is infinitely far away.
.
The second of these is zero, so the work that must be done is equal to `U`. The distances between the charges are `r_(13) = 2 a` and `r_(23) = a`, so
`W = U (q_3)/(4 pi epsilon_0) (q_(1)/r_(13) + q_(2)/r_(23))`
=`(+e)/(4 pi epsilon_0) ((-e)/(2 a) + (+e)/A) = (+e^2)/(8 pi epsilon_0 a)`
If `q_3` is brought in from infinity along the `+x - axis`, it is attracted byb `q_1` but is repelled more strongly by `q_2,` hence, positive work must be done to push `q_3` to the position at `x = 2a`. The total potential energy of the assemblage of three charges is
`U = (1)/(4 pi epsilon_0) (((-e)(e))/a + ((-e)(e))/(2 a) + ((e)(e))/a) = (-e^2)/(8 pi epsilon_0 a)`.