Question

A particle of mass $m$ and negative charge $q$ is thrown in a gravity free space with speed $u$ from the point $A$ in the large nonconducting charged sheet with surface charge density $\sigma$, as shown in figure. The maximum distance from $A$ on sheet where the particle can strike is $\frac{x{ϵ}_0{u}^{2}m}{q\sigma }$. Find $x$.

Answer 1

Electric potential due to large non conducting sheet = $\frac{\sigma }{2{ϵ}_o}$

Force on charge = qE = $\frac{\sigma q}{2{ϵ}_o}$ towards sheet

acceleration, a = F/m = $\frac{\sigma q}{2m{ϵ}_o}$

initial velocity perpendicular to sheet = $u\sin \theta$

velocity parallel to sheet = $u\cos \theta$

Time of flight = $\frac{2u\sin \theta }{a}$

Distance traveled parallel to sheet = $\frac{2u\sin \theta u\cos \theta }{a}$ = $\frac{u^2\sin \left(2\theta \right)}{a}$

So distance travelled will be maximum for $\theta ={45}^{\circ }$ and equal to

$\frac{u^{2}2m{ϵ}_o}{\sigma q}$ =$\frac{2u^{2}m{ϵ}_o}{\sigma q}$

therefore x = 2