A block having mass m and charge Q is resting on a frictionless plane at a distance d from fixed large non-conducting infinite sheet of uniform charge density `-sigma` as shown in Figure. Assuming that collision of the block with the sheet is perfectly elastic, find the time period of oscillatory motion of the block. Is it SHM ?

Explanation

The situation is shown in figure. Electric force produced by sheet will accelerate the block towards the sheet producing an acceleration. Acceleration will be uniform because electric field E due to the sheet is uniform.
`a = (F)/(m) = (qE)/(m)`, where `E = sigma//2 epsilon_(0)`

As initially the block is at rest and acceleration is constant, from second equation of motion, time taken by the block to each the wall
`L = (1)/(2) a t^(2)`
i.e., `t = sqrt((2L)/(a)) = sqrt((2mL)/(aE)) = sqrt((4mL epsilon_(0))/(a sigma))`
As collision with the is perfectly elastic, the block will rebound with same speed and as now its motion is opposite to the acceleration, it will come to rest after travelling same distance L in same time t. After stopping it will be again accelerated towards the wall and so the block will execute oscillatory motion with 'span' L and time period.
`T = 2t = 2 sqrt((2mL)/(aE)) = 2sqrt((4mL epsilon_(0))/(a sigma))`
However, as the restoring force F = qE is constant and not proportional to displacement x, the motion is not simple harmonic.