Question

There is a fixed semicircular ring of radius R, lying in y - z plane, with centre of arc at origin and it is uniformly charged with charge Q. There is a long hollow pipe of very small radius, inner surface of pipe is smooth and it is made of insulated material. Pipe is fixed along x-axis from origin. A small ball with charge q and mass m is projected into the pipe with negligible velocity. Ball can smoothly move in pipe. Whole arrangement lies in gravity free space. Assume thatthere is no effect of hollow pipe on the electrostatic interaction between the charges Q and q. Then,

• A. The maximum acceleration of the ball in the pipe is $\frac{Qq}{6\sqrt{3}{ϵ}_{0}m{R}^2}$
• B. The maximum acceleration of the ball in the pipe is $\frac{Qq}{3\sqrt{3}{ϵ}_{0}m{R}^2}$
• C. The maximum electric field exists along the axis at a distance of $x=\frac{R}{\sqrt{2}}$.
• D. The kinetic energy of the particle when its acceleration is maximum is $\frac{Qq}{4{ϵ}_{0}R}(1-\sqrt{\frac{2}{3}})$

Answer 1

Answer: (A), (C)

The correct options are
A The maximum acceleration of the ball in the pipe is $\frac{Qq}{6\sqrt{3}{ϵ}_{0}m{R}^2}$
C The maximum electric field exists along the axis at a distance of $x=\frac{R}{\sqrt{2}}$.

$V=\frac{Q}{4\pi {ϵ}_0\sqrt{R^2+x^2}};E=-\frac{dV}{dx}=\frac{Qx}{4\pi {ϵ}_0{\left(\sqrt{R^2+x^2}\right)}^{\frac{3}{2}}}$
$\frac{dE}{dx}=0\Rightarrow x=\frac{R}{\sqrt{2}}$
$E_{max}atx=\frac{R}{\sqrt{2}}Is\frac{Q}{6\sqrt{3}\pi {ϵ}_0{R}^2}$
$\therefore a_{max}=\frac{q{E}_{max}}{m}=\frac{Qq}{6\sqrt{3}{ϵ}_{0}m{R}^2}$
$k_{max}=q\Delta V_{max}=q[\frac{Q}{4\pi {ϵ}_{0}R}-\frac{Q\sqrt{2}}{4\pi {ϵ}_{0}R\sqrt{3}}]$