Question

A large solid sphere with uniformly distributed positive charge has a smooth narrow tunnel through its centre. A small particle with negative charge, initially at rest far from the sphere, approaches it along the line of the tunnel, reaches its surface with a speed $v$, and passes through the tunnel. Its speed at the centre of the sphere will be

• A. $0$
• B. $v$
• C. $\sqrt 2 v$
• D. $\sqrt 1.5 v$

Answer 1

Answer: (D)

$\sqrt 1.5 v$

Answer 2

Potential at $\infty, V_{\infty}=0$ Potential at the surface of the sphere, $V_{s}=k \frac{Q}{R}$ Potential at the centre of the sphere, $V_{c}=\frac{3}{2}k \frac{Q}{R}$ Let $m$ and $-q$ be the mass and the charge of the particle respectively Let $\upsilon_{0}$ = speed of the particle at the centre of the sphere $\frac{1}{2}m\upsilon^{2}=-q\left[V_{\infty}-V_{s}\right]=qk \frac{Q}{R}$ $\frac{1}{2}m\upsilon_{0}^{2}=-q \left[V_{\infty}-V_{c}\right]=q\cdot\frac{3}{2}k \frac{Q}{R}$ Dividing, $\frac{\upsilon_{0}^{2}}{\upsilon^{2}}=\frac{3}{2}=1.5$ or $\upsilon_{0}=\sqrt{1.5\upsilon}$