Physics 230 Feb 8 2002
 
  Problem: Find the electric field produced by a uniformly charged sphere with total charge Q and radius a. Find the electric field  inside (r<a)  the shell.
 
 

1. Draw the field-line diagram. Do you have enough symmetry to use Gauss' law? Discuss!!

Since the system is spherically symmetrical, and can be rotated through any angle without changing anything, then there is enough symmetry to use Gauss' law. Notice that some  field lines begin inside the sphere of radius a.


 

2. Construct a Gaussian surface that you can use to apply Gauss' law in order to find the electric field at a point P with r>a. Describe how you chose your volume. Draw it on the diagram. Label its dimensions.

The surface is chosen to be perpendicular or parallel to the electric field lines at each point. Thus it is also a sphere, inside the original sphere,  with radius r < a.
 

3. Find the flux through your surface. Be sure to explain each step clearly and carefully.
 
 

From here the solution follows Example 24.1 exactly, and the example we did in class.  Because of the symmetry, E is radial and the r-component has a constant value over the surface that we have chosen, so the flux is 4(pi)r2Er .

4. Now find the charge inside the surface.
The charge density inside the whole sphere is Q/(4pi a3/3) and the charge inside our Gaussian surface is

Qinside=Q(r/a)3

5. Apply Gauss' law to find the electric field vector. Comment on your result, and how it corresponds to your field line diagram.
 Setting flux through surface = charge inside/epsilon we get

Er = kQr/a3
Equivalently, this is Er = (rho)r/(3epsilonzero).
Thus the field strength increases from 0 to r = a, and at r = a we get kQ/a2, as expected.  The increasing field strength is consistent with the idea that field lines start on the charges inside the sphere.