A block having charge `q` mass m is resting on a smooth horizontal surface at a distance `d` from the wall as shown. Discuss the motion of the block when a uniform electic field E is applied horizontally towards the wall assuming that collision of the block with the wall is completely elastic

Explanation

The situation is shown in Figure. Electric fforce produced by sheet will accelerate the block towards the sheet producting an acceleration. Acceleration will be uniform because electric field E due to the sheet is uniform.
`a=F/m =(QE)/m`, where `E=sigma//2epsi_(0)`
As initially the block is at rest and acceleration is constant, from second equation of motion, time taken by the block to reach the wall
`d=1/2" at"^(2)` i.e., `t=sqrt((2L)/a)=sqrt((2md)/(QE))=sqrt((4mdepsi_(0))/(Q sigma))`
As collision with the wall is perfectly elastic, the block will rebond with same speed and as now its motion is opposite to the acceleration, it will come to rest after traveling same distance d in same time t.
after stopping, it will again be accelerated towards the wall and so the block will execute oscillatory motion with 'span' d and time period.
`T=2t=2 sqrt((2md)/(QE))=2 sqrt((4md epsi_(0))/(Q sigma))`
However, as the restoring force `F = QE` is constant and not proportional to displacement x, the motion is not simple harmonic.