An isolated conductor has net charge$\mathbf{+}\mathbf{10}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{}\mathbf{C}$and a cavity with a particle of charge$\mathbf{=}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{}\mathbf{C}$What is the charge on (a) the cavity wall and (b) the outer surface?

a) Net charge on the isolated conductor,${\mathrm{q}}_{\mathrm{iso}}=+10\times {10}^{-6}\mathrm{C}$

b) Particle charge in the cavity,$+3.0\times {10}^{-6}\mathrm{C}$

Using the concept of Gaussian surface, the net charge enclosed in a body is zero and thus, it can be defined by the electric flux passing through the enclosed volume of the surface.

Consider a Gaussian surface that is completely within the conductor and surrounds the cavity. Since the electric field is zero everywhere on the surface, the net charge it encloses is zero. The net charge is the sum of the charge q in the cavity and the charge q(w) on the cavity wall,

Thus, the value of the charge on the cavity wall is given as follows:

$$\begin{gathered} \mathrm{q}+{\mathrm{q}}_{\mathrm{w}}=0 \\ {\mathrm{q}}_{\mathrm{w}}=-\mathrm{q} \\ {\mathrm{q}}_{\mathrm{w}}=-3.0\times {10}^{-6}\mathrm{C} \end{gathered}$$

Hence, the value of the charge is$-3.0\times {10}^{-6}\mathrm{C}$

The net charge Q of the conductor is the sum of the charge on the cavity wall and the charge $q_s$on the outer surface of the conductor.

Thus, the value of the charge on the outer surface is given as:

$$\begin{gathered} \mathrm{Q}={\mathrm{q}}_{\mathrm{w}}+{\mathrm{q}}_{\mathrm{s}} \\ {\mathrm{q}}_{\mathrm{s}}=\mathrm{Q}-{\mathrm{q}}_{\mathrm{w}} \\ {\mathrm{q}}_{\mathrm{s}}=\left(10\times {10}^{-6}\mathrm{C}\right)-\left(-3.0\times {10}^{-6^{}}\mathrm{C}\right) \\ {\mathrm{q}}_{\mathrm{s}}=+1.3\times {10}^{-5}\mathrm{C} \end{gathered}$$

Hence, the value of the charge is$+1.3\times {10}^{-5}\mathrm{C}$