Question

There are two large parallel metallic plates $P_1$ and $P_2$ , carrying surface densities ${\sigma }_1$ and ${\sigma }_2({\sigma }_1>{\sigma }_2)$ respectively, placed at a distance $d$ apart in vaccum. Determine the work done by the electric field in moving a point-charge $q$ from $P_1$ to $P_2$ along a line of length $a(a<d)$ making an angle of $\pi /4$ with the normal to the plate.

Answer 1

Let the electric fields due to plates $P_1$ and $P_2$ are $E_1$ and $E_2$.

The net electric field between the plates is given as,

$E=E_1-E_2$

$=\frac{{\sigma }_1}{{\epsilon }_0}-\frac{{\sigma }_2}{{\epsilon }_0}$

The force on the charge due to the electric field is given as,

$F=qE$

$=\frac{q\left({\sigma }_1-{\sigma }_2\right)}{{\epsilon }_0}$

The work done by the electric field is given as,

$W=F\times d\times \cos \theta$

$W=\frac{q\left({\sigma }_1-{\sigma }_2\right)}{{\epsilon }_0}\times d\times \cos \left(\frac{\pi }{4}\right)$

$W=\frac{qd\left({\sigma }_1-{\sigma }_2\right)}{{\epsilon }_0\sqrt{2}}$

Thus, the work done by the electric field is $W=\frac{qd\left({\sigma }_1-{\sigma }_2\right)}{{\epsilon }_0\sqrt{2}}$.