Physics 321 -- Spring 2002

Homework #8, due Wednesday March 27

1. (4) Marion & Thornton: problem 5-1. To answer the last question in this problem, think about electrostatics.

2. (4) Marion & Thornton, problem 5-2. Hint: Use the differential form of Gauss’s Law, which is Eq. 5.37 in the textbook. To calculate the divergence of a vector field in spherical coordinates, see Eq. F.18 in Appendix F.

3. (3) Marion & Thornton, problem 5-3.

4. a) (3) Marion & Thornton, problem 5-7. To do the integral, look at Appendix E.

b) (3) From your result in part (a), use the Taylor series expansion to evaluate the potential very close to the rod, i.e. in the limit R << l. You should expand the quantity (1+(2R/l)²)^1/2 to first order in (2R/l)². Then take the gradient of the potential to find the gravitational field. The result should look familiar from electrostatics --the electric field of an infinitely long charged rod.

5. (4) Marion & Thornton, problem 5-15. Hint: First use the Shell Theorem to calculate the gravitational force on the mass as a function of its distance from the center of the Earth. Then use Newton’s 2^nd Law to obtain the equation of motion of the mass.

6. (4) Marion & Thornton, problem 5-16. Use the method of Example 5.3 in the book. This integral you can do yourself.

If you want to check your answer, you can use Gauss’s Law to calculate the gravitational field g due to the infinite sheet of charge. From that you can find the force on the sphere. By Newton’s 3^rd Law, the force on the sphere is equal and opposite to the force on the sheet.