Question

Find the work done by external agent in moving a $1C$ charge from point P to Q in presence of an infinitely long uniformly charged line of linear charge density $\lambda$ as shown.

• A. $\frac{\lambda }{2\pi {\epsilon }_0}\ln 1.5$
• B. $\frac{\lambda }{2\pi {\epsilon }_0}$
• C. $\frac{\lambda }{2\pi {\epsilon }_0}\ln 5$
• D. $\frac{7\lambda }{2\pi {\epsilon }_0}$

Answer 1

The correct option is A $\frac{\lambda }{2\pi {\epsilon }_0}\ln 1.5$

The electric field due to an infinitely long line of charge at a distance r from it is given by
$|\overrightarrow{E}|=\frac{\lambda }{2\pi {\epsilon }_{0}r}$

We know that $\Delta V=-{\int }_{r_1}^{r_2}Edr$

$\Delta V=V_Q-V_P=-{\int }_1^{1.5}\frac{\lambda }{2\pi {\epsilon }_{0}r}dr=-\frac{\lambda }{2\pi {\epsilon }_0}(\ln 1.5-\ln 1)=-\frac{\lambda }{2\pi {\epsilon }_0}\ln 1.5$

Change in potential energy:
$\Delta U=\Delta V/q$
$\Delta U=-\frac{\lambda }{2\pi {\epsilon }_0}\ln 1.5$

Since, work done is negative of change in potential energy.
$W=-\Delta U=\frac{\lambda }{2\pi {\epsilon }_0}\ln 1.5$