Physics 272: Advanced Honors Physics II
Spring 2018

HW3 due in class Monday, February 5, 2018
Reading: Purcell 2.1-12 and
1.15

Note: the solution for each problem and your name should be written on a separate page (sorry, trees!) to facilitate the grading. Paperclip the pages together.
I have put the answers at the bottom of this page. You can use them to check your own answers -- if they don't agree, you know you need to go back and look for at least one mistake.

1. *1.60 Field from a hollow cylinder (Gauss' law)
Follow the steps we used in class to use Gauss's law to show that the E field inside a uniform hollow sphere is zero. For the square cross section, don't do a calculation; just give a plausible justification for your answer.

2. **1.65 Building a sheet from rods (field of an infinitely uniformly charged rod, setting up integral)

Slice the sheet into bars of width dx and treat the bar as an infinitely uniformly charged rod. You might find the integral table in Appendix K.2 useful.
 
3. **1.73 Sphere in a cylinder (field inside a charged cylinder, field inside a charged sphere, superposition)
The difficult part of this problem is adding up the field from the charged cylinder (pointing radially away from the axis at x = 0) and the field from the charged sphere

(pointing radially away from the center of the sphere at x = a). I suggest converting both to Cartesian coordinates (remember vec r = x i-hat + y j-hat in the xy plane)
.

4. *2.34 Extremum of phi
Use other arguments to show that Ex and Ez are zero at all points on this line. Sketch the potential along the z axis, which will give you Ey at all points along the axis. For the last question, don't do a calculation, but just make a rough estimate by numerical "guess and check" for a few values of y.

5. *2.38 Interstellar dust (potential of a spherical shell)
The number of electrons has to be an integer. Assume that all numbers are given to 2 sig figs.

6. **2.43 Potential from a rod
You might find the integral table in Appendix K.2 useful.

7. **2.55 Hole in a disk (potential of a uniform disk; superposition OR integral)
Use the usual form for the kinetic energy (1/2)mv2 even though you will see that v is getting close to the speed of light and so you might worry that you need to use special relativity.

8. *2.58 Energy of a shell

9. **2.61 Dipole field on the axes

10. **2.70 Triangular E
To clarify the statement of the problem, assume Ey = Ez = 0 and the E in the plot is Ex.

OPTIONAL 11. ***2.74 Oscillating exponential phi


ANSWERS
1. proof; no.
2. proof.
3. proof.
4. explanation, estimate.
5. 31 electrons, -5 x 105 V/m
6. (lambda ln 3)/(4 pi epsilon0); x= Sqrt(3) d
7. -11,300 V; 6.3 x 107 m/s
8. proof
9. proof
10. rho(a/2,0,0)=epsilon0 E0/a  ; phi(a/2,0,0)=-3 E0 a/8