Question

The work done by an external force to bring a third point charge $q_3=+e$ from infinity to x=2a is

• A. $\frac{e^2}{4\pi {\epsilon }_{0}a}$
• B. $\frac{e^2}{8\pi {\epsilon }_{0}a}$
• C. $\frac{-e^2}{8\pi {\epsilon }_{0}a}$
• D. $\frac{-e^2}{4\pi {\epsilon }_{0}a}$

Answer 1

Answer: (B)

$\frac{e^2}{8\pi {\epsilon }_{0}a}$

The work that must be done on $q_3$ by an external force ${\overrightarrow{F}}_{ext}$ is equal to the difference between two quantities: the potential energy U associated with $q_3$ when it is at x = 2a and the potential energy when it is infinitely far away.

The second of these two is zero, so the work that must be done is equal to U. The distances between the charges are $r_{13}=2a$ and $r_{22}=a$, so

$W=U=\frac{q_3}{4\pi {\epsilon }_0}(\frac{q_1}{r_{13}}+\frac{q_2}{r_{23}})$
$=\frac{+e}{4\pi {\epsilon }_0}(\frac{-e}{2a}+\frac{e}{a})=\frac{+e^2}{8\pi {\epsilon }_{0}a}$