91影视

Figure 23-23 shows, the central metal ball with charge $\mathrm{Q}$ and two spherical metal shells ($3Q$ and $5Q$), and three Gaussian surfaces of radii $\mathrm{R}$, $2R$, and $3\mathrm{R}$ are given.

Here, the total charge within a Gaussian charge with its given area is used to calculate the electric field at any point of the Gaussian surface.

Formula:

The electric field at any point of a Gaussian surface,

$\mathrm{E}=\frac{{\mathrm{q}}_{\mathrm{enc}}}{{\mathrm{础蔚}}_0}$ 鈥�.. (i)

Here, $\mathrm{E}$ is the electric field, ${\mathrm{q}}_{\mathrm{enc}}$ is the enclosed charge, $A$ is the area, and ${蔚}_0$ is the permittivity of the free space.

Now, using the data from the figure, the charge enclosed by Gaussian surface 1 can be given as:

${\mathrm{q}}_{\mathrm{enc}}=\mathrm{Q}$

Thus, using the above charge value in equation (i), the electric field at any point of Gaussian surface 1 of radius$\mathrm{R}$can be given as:

${\mathrm{E}}_1=\frac{\mathrm{Q}}{4{\mathrm{蟺搁}}^2{\mathrm{蔚}}_0}(鈭�\text{Surfaceareaofasphere,}\mathrm{A}=4{\mathrm{蟺搁}}^2)$

Now, using the data from the figure, the charge enclosed by Gaussian surface 2 can be given as:

$\begin{array}{c}{\mathrm{q}}_{\mathrm{enc}}=\mathrm{Q}+3\mathrm{Q}\\ =4\mathrm{Q}\end{array}$

Thus, using the above charge value in equation (i), the electric field at any point of Gaussian surface 2 of radius$2R$can be given as:

$\begin{array}{c}{\mathrm{E}}_2=\frac{4\mathrm{Q}}{4\mathrm{蟺}{\left(2\mathrm{R}\right)}^2{鈭�}_0}(鈭�\text{Surfaceareaofasphere,}\mathrm{A}=4{\mathrm{蟺搁}}^2)\\ =\frac{\mathrm{Q}}{4{\mathrm{蟺搁}}^2{鈭�}_0}\end{array}$

Now, using the data from the figure, the charge enclosed by Gaussian surface 3 can be given as:

$\begin{array}{c}{\mathrm{q}}_{\mathrm{enc}}=\mathrm{Q}+3\mathrm{Q}+5\mathrm{Q}\\ =9\mathrm{Q}\end{array}$

Thus, using the above charge value in equation (i), the electric field at any point of Gaussian surface 3 of radius$3R$can be given as:

$\begin{array}{c}{\mathrm{E}}_3=\frac{9\mathrm{Q}}{4\mathrm{蟺}{\left(3\mathrm{R}\right)}^2{鈭�}_0}(鈭�\text{Surfaceareaofasphere,}\mathrm{A}=4{\mathrm{蟺搁}}^2)\\ =\frac{\mathrm{Q}}{4{\mathrm{蟺搁}}^2{鈭�}_0}\end{array}$

Hence, the rank of the rank of the Gaussian surfaces is ${\mathrm{E}}_1={\mathrm{E}}_2={\mathrm{E}}_3$.