For an infinite line of charge having charge density lying along … - Vidyadip

MCQ

For an infinite line of charge having charge density $\lambda $ lying along $x-$ axis, the work required in moving charge $q$ from $C$ to $A$ along arc $CA$ is :-

✓
$\frac{{q\lambda }}{{\pi {\varepsilon _0}}}{\log _e}\sqrt 2 $
B
$\frac{{q\lambda }}{{4\pi {\varepsilon _0}}}{\log _e}\sqrt 2 $
C
$\frac{{q\lambda }}{{4 \pi {\varepsilon _0}}}{\log _e} 2 $
D
$\frac{{q\lambda }}{{2\pi {\varepsilon _0}}}{\log _e}\frac{1}{2}$

✓

Answer

Correct option: A.

$\frac{{q\lambda }}{{\pi {\varepsilon _0}}}{\log _e}\sqrt 2 $

a
$\mathrm{B} $ and $ \mathrm{C}$ are equipotential and field is conservative, therefore:

$\therefore \mathrm{W}_{\mathrm{CA}}=\mathrm{W}_{\mathrm{BA}^{-}}=-\int_{2 \mathrm{a}}^{\mathrm{a}} \frac{\lambda}{2 \pi \varepsilon_{0} \mathrm{r}} \mathrm{q} \mathrm{dr}=\frac{\mathrm{q} \lambda}{2 \pi \varepsilon_{0}} \operatorname{In} 2$

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