Gauss' Law

Using Gauss' Law to find the electric field: $$\oint E_\perp da = \frac{q_\text{enc}}{\epsilon_0}$$

We're going to use this to find $E_\perp$ (the unknown).

1. Gauss' Law is terribly useful for finding the electric field in situations of high symmetry: If the charge distribution is either
   1. spherical symmetry, or
   2. cylindrical symmetry, or
   3. translational symmetry: e.g. charge distribution is a function of only one coordinate in Cartesian coordinates.
      
      
2. Draw a surface (or surfaces) which is (are) equidistant from the center of the charge distribution, a distance '$r$' away. Connect the equidistant surfaces with other surfaces to make a closed surface. (Not necessary for spherical symmetry). This closed surface is your "Gaussian surface".
   
   
3. Write the flux integral $\oint E_\perp da$ as a sum of integrals for each of the surfaces that make up your closed surface.... $$\oint E_\perp da = \int_{\cal S1} E_\perp da+ \int_{\cal S2}E_\perp da+...$$
   
   
4. In general, you should find that for each surface, you can argue that either
   1. $E_\perp$ is a function of $r$, but $r$ is constant on your equi-distant surface, in which case $\int_{\cal S} E_\perp(r) da = E_\perp(r) \int da = E_\perp(r) A$,
   2. $E_\perp$ is zero (The electric field is completely parallel to the surface) in which case $\int_{\cal S} E_\perp da = 0$.
      
      
5. Now let's work on the right-hand side of the Gauss' Law equation. Find the charge enclosed by your gaussian surface. You might have to do a (hopefully not too difficult...) volume integral. $$q_\text{enc}=\int \rho(r) \,d\tau,$$ where $d\tau$ is the differential of volume. For example, in Cartesian coordinates $d\tau = dx\,dy\,dz$.

For Wednesday: Your practice assignment is to use these steps to solve the test question to find the electric field at locations inside the sphere of constant charge density.

A solid insulating sphere of radius R has a charge Q (positive) distributed uniformly throughout its volume (not merely on the surface). Use Gauss’s law to find the electric field at a distance r from the center of the sphere, when $r \lt R$. (show your reasoning).

You'll need to use some facts like...

• The surface area of a sphere of radius $r$ is $4\pi r^2$.
• The volume of a sphere of radius $r$ is $4\pi r^3/3$.
• The charge density $\rho$ is constant. It is a fraction consisting of the charge (Q) divided by the volume (that of a sphere of radius $R$).
• The amount of charge enclosed within a sphere of radius $r$ is not a constant. It depends on $r$.
• With spherical symmetry, there is a single surface that's equidistant from the center of of the charge distribution: it's a spherical surface. No need to split up the flux integral into different pieces (as in step 3 above).