Consider two infinitely large plane parallel conducting plates as shown below. The plates are uniformly charged with a surface charge density $+\sigma$ and $-2\sigma$. The force experienced by a point charge $+q$ placed at the mid point between the two plates will be:
Answer: (B) $\dfrac{3\sigma q}{2\epsilon_0}$
Each large plate produces a uniform field of magnitude (its total charge per area)$/2\epsilon_0$ on either side, however the charge is shared between its two faces.
Between the plates:
- the $+\sigma$ plate (left) gives $\dfrac{\sigma}{2\epsilon_0}$ pointing away from it, i.e. to the right;
- the $-2\sigma$ plate (right) gives $\dfrac{2\sigma}{2\epsilon_0}$ pointing towards it, i.e. also to the right.
$$E = \frac{\sigma}{2\epsilon_0} + \frac{2\sigma}{2\epsilon_0} = \frac{3\sigma}{2\epsilon_0},\qquad F = qE = \frac{3\sigma q}{2\epsilon_0}$$
(The face charges come out as $-\sigma/2,\ +3\sigma/2$ on the left plate and $-3\sigma/2,\ -\sigma/2$ on the right plate; the facing charges $\pm 3\sigma/2$ give the same field $3\sigma/2\epsilon_0$.)
Solution by Sreeraj P, M.Sc Physics