Question

(a) Two point charges $+Q_1$ and $–Q_2$ are placed $r$ distance apart. Obtain the expression for the amount of work done to place a third charge $Q_3$ at the midpoint of the line joining the two charges.

(b) At what distance from charge $+Q_1$ on the line joining the two charges (in terms of $Q_1,Q_2$ and $r$) will this work be done be zero.

Answer 1

($\text{a}$)

To calculate work done, first calculate the potential difference between $\infty$ and $C$ i.e. $(V_C–V_{\infty })$, then

$W=Q_3(V_C-V_{\infty })$ [As $V_{\infty }=0]$

$W=Q_3{V}_c$

$W=Q_3[\frac{1}{4\pi {\epsilon }_0}\frac{Q_1}{\frac{r}{2}}-\frac{1}{4\pi {\epsilon }_0}\frac{Q_2}{\frac{r}{2}}]$

$W=\frac{1}{4\pi {\epsilon }_0}\frac{2Q_3}{r}[Q_1-Q_2]$

($\text{b}$) Work done will be zero, when the potential of point $C$ is zero. Assume the distance of point $C$ from $Q_1$ is $X$.

Net potential at $C$

$K[\frac{Q_1}{x}-\frac{Q_2}{(r-x)}]=0$

$\frac{Q_1}{x}=\frac{Q_2}{(r-x)}$

$Q_{1}r-Q_{1}x=Q_{2}x$

$x=\frac{Q_{1}r}{Q_1+Q_2}$