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

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$\frac{qλ}{2 {πε}_{0}} \log_{e} 2$

$\frac{qλ}{4 {πε}_{0}} \log_{e} 2$

$\frac{qλ}{4 {πε}_{0}} \log_{e} \sqrt{2}$

$\frac{qλ}{2 {πε}_{0}} \log_{e} \frac{1}{2}$

answer is A.

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Detailed Solution

$As E = \frac{2 λ}{4 {πε}_{0} r} so F = \frac{2 qλ}{4 {πε}_{0} r} W = \int F dr = \int_{a}^{2 a} \frac{2 qλ}{4 {πε}_{0} r} dr = \frac{qλ}{2 {πε}_{0}} \int_{a}^{2 a} \frac{dr}{r} = \frac{qλ}{2 {πε}_{0}} {(In r)}_{a}^{2 a}$

$= \frac{qλ}{2 {πε}_{0}} (In 2 a - In a) = \frac{qλ}{2 {πε}_{0}} In (\frac{2 a}{a}) = \frac{qλ}{2 {πε}_{0}} In 2$

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