An infinite line charge of uniform charge density $λ$ is held fixed along the Z-axis of a cartesian coordinate system. A point electric dipole initially kept at the point A(+a, 0, 0) with the dipole moment $\vec{P}$ along the negative X-direction is moved to the point B(+2a, 0, 0) and rotated so that its dipole moment is along the positive X-direction. The work done by the external agent on the dipole in the entire process is

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$- \frac{λp}{2 π ε_{0} a}$

$\frac{λp}{2 π ε_{0} a}$

$- \frac{3 λp}{4 π ε_{0} a}$

$\frac{λp}{4 π ε_{0} a}$

answer is C.

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Detailed Solution

Field due to infinite line of charge $E = \frac{λ}{2 {πε}_{0} r}$ electric dipole initially kept at the point A(a,0,0).

field $E = \frac{λ}{2 {πε}_{0} a}, U_{i} = - PEcos θ$

at (a, 0, 0), $U_{initial} = \frac{{Pλ}^{2}}{2 π ε_{0} a} and$

at(2a, 0, 0) $U_{final} = \frac{- Pλ}{2 π ε_{0} a}$

$\begin{array}{l} W = ΔU = U_{initial} - U_{final} \\ \Rightarrow W = - \frac{3}{4} \frac{λP}{{πϵ}_{0} a} \Rightarrow W = - \frac{3}{4} \frac{λP}{{πϵ}_{0} a} \end{array}$

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