Question

The figure shows an infinite line charge of density $C/m$. The work done by the electrostatic force on a unit charge, when it is moved along the path $ABC$, is (plane of the curve $ABC$ contains the line charge)

• A. zero
• B. $\frac{-\sigma }{2\pi {ϵ}_o}ln2$
• C. $\frac{-\sigma }{2\pi {ϵ}_o}ln3$
• D. $\frac{\sigma }{2\pi {ϵ}_o}ln2$

Answer 1

Answer: (C)

$\frac{-\sigma }{2\pi {ϵ}_o}ln3$

Field along $AB=\frac{2k\sigma }{3\sigma }$

$W_{AB}=E.dx=0$ $\because$ displacement perpendicular to displacement

$W_{BC}$

$E=\frac{2k\sigma }{y}$

$dy=dr\cos \theta ,dr=rd\theta$

$y=2r+r\sin \theta$

$\Rightarrow dw=E.dr=\frac{\left(2k\sigma \right)}{2r+r\sin \theta }$ $dr\cos \theta$

$=2k\sigma r$ $\frac{\cos \theta }{(2r+r\sin \theta )}$ $d\theta$

$2r+r\sin \theta =t$

$r\cos \theta d\theta =dt$

$\Rightarrow \int dw=\int 2k$ $\frac{dt}{t}$

$w=2k\sigma ln2r+r\sin \theta$

$=2k\sigma$ $ln\frac{r}{3r}$

$=\frac{-2\sigma }{4\pi {\in }_0}ln3$

$=-\frac{\sigma }{2\pi {\in }_0}$ $ln3$