Two square metal plates have sides of length L and thickness `t ( lt lt lt L)`. They are arranged parallel to each other with their inner faces at a separation of `(5)/(2)`t. One of the plates is given a charge `-Q` and the other one is given a charge `+ Q` . A third rectangular metal plate of sides L and x, having thickness `(t)/(2)` is inserted between the plates as shown. The third plate is equidistant from the two plates and parallel to them. Neglect edge effects.

(a) Find the charge density on lower plate at points 1 and 2 shown in Figure.
(b) Find potential difference between the upper plate and the middle plate.
(c) Find electric field between the two outer plates in space where the third plate is not present (i.e., at a point above point 1.)
(d) Find the capacitance of the system across two outer plates.

Answer

(a) `sigma_(1)=(4Q)/(L(4L+x));sigma_(2)=(5Q)/(L(4L+x))`
(b) `(5Qt)/(in_(0)L(4L+x))`
(c) `(4Q)/(inL(4L+x))`
(d) `(in_(0)L(4L+x))/(10t)`