Question

There are two large parallel metallic plates $S_{1}and{S}_2$ carrying surface charge densities ${\sigma }_{1}and{\sigma }_2$ respectively $({\sigma }_1>{\sigma }_2)$ placed at a distance $d$ apart in vacuum. Find the work done by the electric field in moving a point charge $q$ a distance a $(a<d)$ from $S_{1}towards{S}_2$ along a line making an angle $\pi /4$ with the normal to the plates.

Answer 1

Electric field near a large metallic plate is given by
$E=\sigma /{\epsilon }_0.$ In between the plates the two fields will
be in opposite directions. Hence,
$E_{net}=\frac{{\sigma }_1-{\sigma }_2}{{\epsilon }_0}=E_0(say)$
Now, W = (g) (potential difference)
$=q(E_{0}acos45^0)=(q)\left(\frac{{\sigma }_1-{\sigma }_2}{{\epsilon }_0}\right)$
$=\frac{({\sigma }_1-{\sigma }_2)qa}{\sqrt{2}{\epsilon }_0}$