Question

A bullet of mass m and charge q is fired towards a solid uniformly charged sphere of radius R and total charge +q. If it strikes the surface of sphere with speed u, find the minimum speed u so that it can penetrate through the sphere. (Neglect all resistance forces or friction acting on bullet except electrostatic forces)

• A. $\frac{q}{\sqrt{2\pi {\epsilon }_{0}mR}}$
• B. $\frac{q}{\sqrt{4\pi {\epsilon }_{0}mR}}$
• C. $\frac{q}{\sqrt{8\pi {\epsilon }_{0}mR}}$
• D. $\frac{\sqrt{3}q}{\sqrt{4\pi {\epsilon }_{0}mR}}$

Answer 1

Answer: (B)

Step 1, Given data

Mass of the bullet = m

Charge on the bullet = q

The radius of the sphere = R

Step 2, Finding the minimum speed

We know that

Maximum potential energy is at the surface of the sphere

So,

$\frac{1}{2}m{u}^2=v_c-v_s$

Or, $\frac{1}{2}m{u}^2=\frac{3}{2}\frac{k{q}^2}{R}-\frac{k{q}^2}{R}$

Or, $\frac{1}{2}m{u}^2=\frac{k{q}^2}{mR}$

After solving

$u=\frac{kq}{\sqrt{mR}}=\frac{q}{\sqrt{4\pi {\epsilon }_{0}mR}}$

Hence the minimum speed is $\frac{q}{\sqrt{4\pi {\epsilon }_{0}mR}}$