Question

A metal ball of radius 12 cm is charged to potential 200 V. A point charge 3 nC is moved from infinity to a point near the ball surface. In doing so, work 100 nJ is performed. Find the distance from the final position of the charge to the ball surface. The answer should be expressed in centimeters [cm]. The ball is in a vacuum.

Answer 1

The charge of the ball with radius $R$ can be expresses thru its potential:

$$\phi_1=\frac{kq_1}{R},\\\space\\ q_1=\frac{R\phi_1}{k}.$$

As we know, the field outside of the ball behaves as a field of a point charge.

The work done by a charge approaching from infinity in a potential field created by another charge:

$$W=\frac{kq_1q_2}{r},\\\space\\ r=\frac{kq_1q_2}{W}=\frac{\phi_1q_2R}{W}=0.72\text{ m}.$$

Thus, the distance to the surface of the sphere is

$$d=r-R=72-12=60\text{ cm}.$$