I am currently studying the textbook A Student's Guide to Maxwell's Equations by Daniel Fleisch. In a section discussing the integral form of Gauss's law, the author presents the following electric field equations:
Conducting sphere (charge $= Q$):
$$\vec{E} = \dfrac{1}{4 \pi \epsilon_0} \dfrac{Q}{r^2}\hat{r} \ \text{(outside, distance $r$ from center)}$$
$$\vec{E} = 0 \ \text{(inside)}$$
Uniformly charged insulating sphere (charge $= Q$, radius $= r_0)$:
$$\vec{E} = \dfrac{1}{4 \pi \epsilon_0} \dfrac{Q}{r^2} \hat{r} \ \text{(outside, distance $r$ from center)}$$
$$\vec{E} = \dfrac{1}{4 \pi \epsilon_0} \dfrac{Q r}{r_0^3} \hat{r} \ \text{(inside, distance $r$ from center)}$$
I have two questions regarding these equations:
- Why is the "outside, distance $r$ from center" equation for the "conducting sphere" and "uniformly charged insulating sphere" the same?
- Why is $\vec{E} = 0$ "inside" for the conducting sphere, whereas we have that $\vec{E} = \dfrac{1}{4 \pi \epsilon_0} \dfrac{Q r}{r_0^3} \hat{r}$ "inside" for the uniformly charged insulating sphere? I am aware that, if you have a real or imaginary closed surface of any size and shape, and there is no charge inside the surface, the electric flux through the surface must be zero, but I'm unsure of exactly what about these two situations leads to this difference.
I would appreciate it if people would please take the time to clarify these points.
Best Answer
The difference between the two spheres is the charge distribution. By Gauss's law any charge outside the sphere does not distinguish how the charge is distributed as long as it is spherical. Inside the sphere, of course it does matter. For the conductor all charge is on the surface of the sphere. Gauss tells us that the field inside the sphere is zero. For the uniform distribution, the charge inside r is proportional to $r^3/r_0^3$. Combining this with Coulomb's law gives the linear r-dependence.