Flux and conducting shells. A charged particle is held at the center of two concentric conducting spherical shells. Figure 23-39ashows a cross section. Figure 23-39b gives the net flux $\mathbf{\varphi }$through a Gaussian sphere centered on the particle, as a function of the radius rof the sphere. The scale of the vertical axis is set by$\mathbf{\varphi }\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}{\mathbf{m}}^{\mathbf{2}}\mathbf{/}\mathbf{C}$.What are (a) the charge of the central particle and the net charges of (b) shell A and (c) shell B?

1. A charged particle is held at the center of the two concentric conducting shells.
2. The scale of the vertical axis, ${\mathrm{\varphi }}_{\mathrm{s}}=5.0\times {10}^5\mathrm{N}·{\mathrm{m}}^2/\mathrm{C}$

Using the concept of the enclosed charge within a Gaussian surface, we can get the required values of the e charges for their respective flux through the use of the graph.

Formula:

The enclosed charge using Gauss law due to the flux,

${\mathrm{q}}_{\mathrm{enc}}={\mathrm{\epsilon }}_0\mathrm{\varphi }$ (1)

The value for the central region is given as:$\mathrm{\varphi }=-9.0\times {10}^5\mathrm{N}·{\mathrm{m}}^2/\mathrm{C}$

So, for small r, the enclosed charge using equation (1) is given as:

role="math" localid="1657345352842" $$\begin{gathered} {\mathrm{q}}_{\mathrm{centre}}=\left(8.85\times {10}^{-12}{\mathrm{N}}^{-1}·{\mathrm{m}}^{-2}·{\mathrm{C}}^{-1}\right)\left(-9\times {10}^5\mathrm{N}·{\mathrm{m}}^2·{\mathrm{C}}^{-1}\right) \\ =-7.79\times {10}^{-6}\mathrm{C}\left(\frac{1\mathrm{\mu C}}{{10}^{-6}\mathrm{C}}\right) \\ \approx -8.0\mathrm{\mu C} \end{gathered}$$

Hence, the value of the charge is $-8.0\mathrm{\mu C}$.

The next (nonzero) r-value that takes the flux value is given as: $\mathrm{\varphi }=4.0\times {10}^5\mathrm{N}·{\mathrm{m}}^2/\mathrm{C}$,

Thus, using the given value in equation (i), we get the enclosed charge value as:

But we have already accounted for some of that charge in part (a), so the charge of shell A is given as:

$$\begin{gathered} {\mathrm{q}}_{\mathrm{A}}={\mathrm{q}}_{\mathrm{enc}}-{\mathrm{q}}_{\mathrm{centre}} \\ =11.5\times {10}^{-6}\mathrm{C} \\ \approx 12\mathrm{\mu C} \end{gathered}$$

Hence, the value of the charge is $12\mathrm{\mu C}$.

Finally, with the large r-value, the flux is given as:

$\mathrm{\varphi }=-2.0\times {10}^5\mathrm{N}·{\mathrm{m}}^2/\mathrm{C}$,

So, the enclosed charge using the above data in equation (1) is given as:

$$\begin{gathered} \mathrm{qenc}=\left(8.85\times {10}^{-12}{\mathrm{N}}^{-1}·{\mathrm{m}}^{-2}·{\mathrm{C}}^2\right)\left(-2\times {10}^5\mathrm{N}·{\mathrm{m}}^2·{\mathrm{C}}^{-1}\right) \\ =-1.77\times {10}^{-6}\mathrm{C}\left(\frac{1\mathrm{\mu C}}{{10}^{-6}\mathrm{C}}\right) \\ =-1.77\mathrm{\mu C} \end{gathered}$$.

Thus, the charge of the shell B is given as:

$$\begin{gathered} {\mathrm{q}}_{\mathrm{B}}={\mathrm{q}}_{\mathrm{total}\mathrm{enc}}-{\mathrm{q}}_{\mathrm{A}}-{\mathrm{q}}_{\mathrm{centre}} \\ =-5.3\mathrm{\mu C} \end{gathered}$$

Hence, the value of the charge is $-5.3\mathrm{\mu C}$.