Question

A spherical conducting shell has a charge of -14 pCon its outer surface and a charged particle in its hollow. If the net charge on the shell is$\mathbf{-}\mathbf{10}\mathit{\mu }\mathit{C}$, what is the charge

(a) on the inner surface of the shell and

(b) of the particle?

Short answer

1. The charge on the inner surface of the shell is $+4\mathrm{\mu C}$.
2. The charge of the particle is $-4\mathrm{\mu C}$.

Step-by-step solution

The given data

1. A spherical conducting shell has a charge of$q_{out}=-14\mathrm{\mu C}$on its outer surface.
2. The net charge on the shell is $Q=-10\mathrm{\mu C}$.

Understanding the concept of the electric field

Using the concept of the Gaussian surface, we can calculate the required charge on a body. Consider the given shell as a Gaussian surface, and then some amount of charge remains at the inner surface while some at the outer surface.

Hence, using this concept, the net charge on the shell is the total value of all charges on the shell.

Formula:

Let,$q_{in}$be the charge on the inner surface and$q_{out}$the charge on the outer surface. The net charge on the shell is given as:

$Q=q_{in}+q_{out}$ (i)

a) Calculation of the charge on the inner surface of the shell

Using the given data in equation (i), the charge on the inner surface of the shell is given as:

$$\begin{gathered} q_{in}=-10\mathrm{\mu C}-(-14\mathrm{\mu C}) \\ =+4\mathrm{\mu C} \end{gathered}$$

Hence, the value of the charge is $+4\mathrm{\mu C}$.

b) Calculation of the charge of the particle

Let q be the charge of the particle. In order to cancel the electric field inside the conducting material, the contribution from the in ${\mathrm{q}}_{\mathrm{in}}=+4\mathrm{\mu C}$on the inner surface must be canceled by that of the charged particle in the hollow. That is, the enclosed charge in the shell surface should be:

$q_{enc}=q-q_i=0$

Thus, the particle’s charge is given as:

$$\begin{gathered} q=-q_{in} \\ =-4\mathrm{\mu C} \end{gathered}$$

Hence, the value of the charge is $-4\mathrm{\mu C}$.