Question

An infinitely large thin plane sheet has a uniform surface charge density $+\sigma$. Obtain the expression for the amount of work done in bringing a point charge q from infinity to a point, distant r, in front of the charged plane sheet.

Answer 1

Let PQRS is an infinitely plane charged sheet and $+\sigma$ be the distribution of charge over the sheet.Let the charge $q$ is bringing from infinity to the distance $r$ from the sheet.

Then, Electric field at the distant r from the sheet

$\varphi =E.△Scos\theta$

$\therefore \varphi =E.△S(\theta =0)$----------(1)

and $\varphi =\frac{\sigma }{{ϵ}_o}$-------------(2)

From equation (1) and (2)

$E=\frac{\sigma }{{ϵ}_o\times △S}$

$E=\frac{\sigma }{{ϵ}_o\times 2}$

We know that,

$W=F.r$

$W=q.\frac{\sigma }{{ϵ}_o\times 2}.r$