Question

Charge $+q$ is uniformly distributed throughout the volume of a fixed non-conducting solid sphere of radius $R$. Two identical small particles each of charge $+q$ and mass $m$ are projected towards each other along a narrow tunnel inside the sphere as shown in the figure. (Neglect gravity). Choose the correct option(s).

• A. Linear momentum of system of particles is conserved.
• B. Linear momentum of system of particles is not conserved.
• C. Minimum value of u for which particles will just reach at the surface of the sphere is $\sqrt{\frac{5k{q}^2}{3mR}}$
• D. Minimum value of u for which particles will just reach at the surface of the sphere is $\sqrt{\frac{2k{q}^2}{mR}}$

Answer 1

Answer: (A), (C)

Linear momentum of system of particles is conserved.
Minimum value of u for which particles will just reach at the surface of the sphere is $\sqrt{\frac{5k{q}^2}{3mR}}$

Since no external force is acting on the system, the linear momentum of the system is conserved.
Using energy conservation,
$K{E}_{initial}+P{E}_{initial}=K{E}_{final}+P{E}_{final}$
$1/2m{u}^2\times 2+k{q}^2/6r+kq/3r\times q\times 2=k{q}^2/2r+kq/r\times q\times 2$
$m{u}^2+5k{q}^2/6r=5k{q}^2/2r\Rightarrow m{u}^2=5k{q}^2/3r$