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 :

Options

A. ` ( q lamda ) /( pi epsi _ 0 ) log _ e sqrt 2 `
B. ` ( q lamda ) /( 4 pi epsi _ 0 ) log _ e sqrt 2 `
C. ` ( q lamda ) /( 4pi epsi _ 0 ) log _ e 2 `
D. ` ( q lamda ) /( 2pi epsi _ 0 ) log _e "" ( 1 ) /(2 ) `

Answer

A

Explanation

As work done by electric force is independent of path so the work done to bring the charge from C to A along arc CA is same as that along the path CBA as shown in figure.

Due to infinite line charge field is along the line AB and hence work done is moving the charge from C to B is zero as ` vec F _ e` and displacement are perpendicular to each other.
So required work done.
` W = U _ A - U_ B = q ( V_A- V_B ) `
` W = q xx int - vecE. d vecr = - q int _ (2a ) ^ a ( lamda dr ) / ( 2 pi in _ 0 r ) `
` = ( q lamda ) /( 2 pi in _ 0 ) ` ln 2 = ` ( q lamda ) /( pi in _ 0 ` ln ` ( sqrt 2 ) `