There is 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 that there is no effect of hollow pipe on the electrostatic interaction between the charges Q and q. Then,

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

The kinetic energy of the particle when its acceleration is maximum is $\frac{Q q}{4 π \in_{0} R} (1 - \sqrt{\frac{2}{3}})$

The maximum acceleration of the ball in the pipe is $\frac{Q q}{6 \sqrt{3} π \in_{0} m R^{2}}$

The maximum acceleration of the ball in the pipe is $\frac{Q q}{3 \sqrt{3} π \in_{0} m R^{2}}$

The maximum electric field exists along the axis at a distance of $x = \frac{R}{\sqrt{2}} .$

answer is A, C, D.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

$V = \frac{Q}{4 π \in_{0} \sqrt{R^{2} + x^{2}}}; E = - \frac{d V}{d x} = \frac{Q x}{4 π \in_{0} {(R^{2} + x^{2})}^{\frac{3}{2}}}$
$\frac{d E}{d x} = 0 \Rightarrow x = \frac{R}{\sqrt{2}}$
$E_{\max} a t x = \frac{R}{\sqrt{2}} i s \frac{Q}{6 \sqrt{3} π \in_{0} R^{2}}$
$∴ a_{\max} = \frac{q E_{\max}}{m} = \frac{Q q}{6 \sqrt{3} π \in_{0} m R^{2}}$
$k_{\max} = q Δ V_{\max} = q [\frac{Q}{4 π \in_{0} R} - \frac{Q \sqrt{2}}{4 π \in_{0} R \sqrt{3}}]$

Watch 3-min video & get full concept clarity

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

How to Join IIT after 10th

🔥 Start Your JEE/NEET Prep at Just ₹1999 / month - Limited Offer! Check Now!

Get Expert Academic Guidance – Connect with a Counselor Today!

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

How to Join IIT after 10th