This is a problem from my last E&M exam. Yes, it’s not E&M but it does use all the Gauss’s law stuff. Plus it makes the students go through the derivations. This was given as a closed book, open discussion, take home exam.

1. What is the university law of Gravitation?  How does this compare to Coulomb’s law?
2. Knowing the answer to question 1, derive both an integral and differential form of a Gravitational Gauss’s law if possible.
3. What is the gravitational potential equation?  How does it compare to the Coulomb’s potential?
4. We know that the internal electrical energy is given by the following equation: U ~ Integral of E^2 dV Derive a similar equation for the gravitational field.  (Note: it is not enough to match constants)
5. In a star, the outward pressure of fusion balances the attractive gravitational pull.  Recent measurements have shown that Betelgeuse’s energy output has suddenly dropped to 35% of its original value.  Using the equation you derived in part 4 and the mass density below, determine what the new size would be: assuming it kept its same shape and all the detected energy loss came from reduced fusion output? rho = rho_0 exp (-r^2/a^2)
6. Interestingly up close images of Betelgeuse have shown that it doesn’t keep the same shape??  It started in a regular shape as discussed in part 5 but then went wonky.  Come up with an idea using what you learned in E&M (or other class) about what happened?

After grading the exams, it was interesting to see that the mechanics of the derivations were fine. However, students wanted to break up the charge density. They felt that there was an inside and outside region. They solved the problem as if it was two separate problems. I found that interesting.

Also you have to do the integrals numerically. MuHaHaHa! Take home exams are fun!