Question

1. How much excess charge must be placed on a copper sphere$\mathit{25}\mathit{.}\mathit{0}\mathit{}\mathit{\text{cm}}$in diameter so that the potential of its centre, relative to infinity, is$\mathbf{3}\mathbf{.}\mathbf{75}\mathbf{}\mathbf{\text{kV}}$?
2. What is the potential of the sphere’s surface relative to infinity?

Short answer

1. The charge should be $\mathbf{52}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{nC}}$.
2. The potential of the sphere’s surface relative to infinity is $\mathbf{3}\mathbf{.}\mathbf{75}\mathbf{}\mathbf{\text{kV}}$.

Step-by-step solution

Formula for the potential

In the sphere that has charge q and radius R, the potential can be calculated as:

$V=\frac{1}{4\pi {\epsilon }_o}\frac{q}{R}$

Where q is the charge and R is the radius of a sphere

Determine the charge

(a)

Inside the sphere, the electric field is zero everywhere, and the potential is the same at every point inside the sphere and on its surface. So, the potential can be used to get the charge.

$V=\frac{1}{4\pi {\epsilon }_o}\frac{q}{R}$

$q=4\pi {\epsilon }_{o}RV$

Plug the values in the above expression, and we get,

$$\begin{gathered} q=\frac{1}{9.0\times {10}^9}\left(0.125\right)\left(3.75\times {10}^3\right) \\ =52.0\text{nC} \end{gathered}$$

Thus, the charge should be $\mathbf{52}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{nC}}$.

Determine the potential

(b)

As explained in part (a), the potential is constant inside the sphere, on the surface of the sphere, and inside it, so the potential at the surface relative to infinity is $\mathbf{3}\mathbf{.}\mathbf{75}\mathbf{}\mathbf{\text{kV}}$.

Thus, the potential of the sphere’s surface relative to infinity is $\mathbf{3}\mathbf{.}\mathbf{75}\mathbf{}\mathbf{\text{kV}}$.