Positive electric charge is distributed uniformly throughout the volume of an insulating sphere with radius Find the magnitude of the electric field at a point a distance from the center of the sphere.
As in Example 22.5, the system is spherically symmetric. Hence we can use the conclusions of that example about the direction and magnitude of To make use of the spherical symmetry, we choose as our Gaussian surface a sphere with radius concentric with the charge distribution.
From symmetry, the direction of E is radial at every point on the Gaussian surface, so $E_{\perp}=E$ and the field magnitude E is the same at every point on the surface. Hence the total electric flux through the Gaussian surface is the product of E and the total area of the surface $A=4\pi r^2$ that is, $\Phi_E=4\pi r^2 E$
The amount of charge enclosed within the Gaussian surface depends on r. To find E inside the sphere, we choose r < R. The volume charge density $\rho$ is the charge Q divided by the volume of the entire charged sphere of radius R:
$\rho =\frac{Q}{4\pi R^3/3}$
The volume $V_{encl}$ enclosed by the Gaussian surface is $\frac{4}{3}\pi r^3$ so the total charge $Q_{encl}$ enclosed by that surface is
$Q_{encl}=\rho V_{encl} =\left(\frac{Q}{4\pi R^3/3}\right)\left(\frac{4}{3}\pi r^3\right)$
$Q_{encl}=Q\frac{r^3}{R^3}$
Then Gauss’s law, Eq. (22.8), becomes
$4\pi r^2E=\frac{Q}{\epsilon_0}\frac{r^3}{R^3}$ or
$E=\frac{1}{4\pi \epsilon_0}\frac{Qr}{R^3}$ (field inside a uniformly charged sphere)
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| Fig.1: The magnitude of the electric field of a uniformly charged insulating sphere. Compare this with the field for a conducting sphere |
The field magnitude is proportional to the distance r of the field point from the center of the sphere (see the graph of E versus r in Fig. 1).
To find E outside the sphere, we take r > R. This surface encloses the entire charged sphere, so $Q_{encl}=Q$ and Gauss’s law gives
$4\pi r^2E=\frac{Q}{\epsilon_0}$ or
$E=\frac{1}{4\pi \epsilon_0}\frac{Q}{r^2}$ (field outside a uniformly charged sphere)
The field outside any spherically symmetric charged body varies as $1/r^2$, as though the entire charge were concentrated at the center. This is graphed in Fig. 1.
If the charge is negative, E is radially inward and in the expressions for E we interpret Q as the absolute value of the charge.
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