Question

An arc AB with the centre C and the infinitely long wire having linear charge density $\lambda$ are lying in the same plane. Find the minimum amount of work to be done to move a point charge $q_0$ from point A to B through a circular path AB of radius a.

• A. $\frac{q_0\lambda }{2\pi {\epsilon }_0}ln\frac{3}{2}$
• B. $\frac{q_0\lambda }{2\pi {\epsilon }_0}ln2$
• C. $\frac{q_0\lambda }{2\pi {\epsilon }_0}ln\frac{2}{3}$
• D. $\frac{q_0\lambda }{2\pi {\epsilon }_0}ln3$

Answer 1

The correct option is A $\frac{q_0\lambda }{2\pi {\epsilon }_0}ln\frac{3}{2}$

${\int }_{V_A}^{V_B}dV=-\frac{\lambda }{2\pi {\epsilon }_0}{\int }_{3a}^{2a}\frac{dx}{x}\Rightarrow V_B-V_A=\frac{\lambda }{2\pi {\epsilon }_0}ln\left(\frac{3}{2}\right)thus,W=\frac{q_0\lambda }{2\pi {\epsilon }_0}ln\left(\frac{3}{2}\right)$