reising1 said:
If there is a sphere with uniform charge (not specified if the sphere is a conductor or a nonconductor) of radius A concentric with a shell (So there is a sphere sitting inside a shell with an inner radius B and an outer radius C where C>B>A), why is the electric field on the inside of the sphere (at a distance from the center, for example, of A/2) not zero?
The easiest way to explain this using Gauss' law (in particular because the charge distribution is spherically symmetrical in your description). If we make a spherical Gaussian surface to be inside the sphere (for example, A/2), there is charge enclosed within the closed surface! And since the electric field is perpendicular to the Gaussian surface (due to the spherical symmetry of the charge distribution), there is an electric field inside the Gaussian surface! And since the surface and the electric field are always perpendicular and symmetric, the magnitude of the electric field can be pulled out from under the integral.
In this case, mostly in part due to the spherical symmetry discussed in the above paragraph,
<br />
\oint _S \vec E \cdot \vec{dA} = \frac{Q_{enc}}{\epsilon _0} <br />
reduces to
<br />
E \int _{\theta = 0} ^{\pi} \int _{\phi = 0} ^{2\pi} r^2 sin\theta d\theta d\phi = \frac{Q_{enc}}{\epsilon _0} <br />
E4\pi r^2 = \frac{Q_{enc}}{\epsilon _0}
or
<br />
\vec E = = \frac{Q_{enc}}{4\pi \epsilon _0} \frac{1}{r^2}\hat r
But also note that the enclosed charge is a function of the volume within the Gaussian surface, and the charge density. Noting the that volume of a sphere is (4/3)\pi r^3, and assuming a uniform volume charge density \rho,
Q_{enc} = \rho \frac{4}{3}\pi r^3
Combining these these equations gives us a relationship for the electric field within a solid, uniformly charged sphere.
(And of course, that's assuming that there are no other charges around, *unless* those other charges are spherically symmetric with respect to the Gaussian surface. In other words, concentric, uniformly charged shells on the outside have no effect on the results. On the other hand, a point charge somewhere off to the side will effect the results, so it is assumed that there are none of those.)
Is this because the uniform charge is distributed not only on the surface but on the inside as well, since the problem statement only specified "a sphere with uniform charge." Thus, there is charge on the inside.
Yes, that's right. But it would be more clear if the problem states something like "uniform charge across the volume" for volume charge or "uniform charge distribution about the surface" for surface charge.
If the problem states that the non-conducting sphere is solid and has a "uniform charge," you can probably assume that the question means uniform
volume charge distribution. But it wouldn't hurt the authors of the problem statement to be more clear.
Also, can someone explain why the Electric field at a distance r = 0 (at the center of the sphere) is 0? According to the equation of electric field, would that not make the electric field undefined, rather than 0. Since we are dividing by r^2, or 0?
Go ahead and combine the above equations, and you will find that the electric field does in fact go to zero as r goes to zero.