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/ \varepsilon_0. $ In between the plates the two fields will
be in opposite directions. Hence,
$\, \, \, \, \, \, \, E_{net}= \frac {\sigma_1-\sigma_2}{\varepsilon_0}=E_0 (say) $
Now, W = (g) (potential difference)
$\, \, \, \, \, \, \, =q(E_0acos45^0)=(q)\bigg (\frac {\sigma_1-\sigma_2}{\varepsilon_0}\bigg ) $
$\, \, \, \, \, \, \, =\frac {(\sigma_1- \sigma_2)qa}{\sqrt 2 \varepsilon_0} $