W9_parallel_resonance.eps

154 Week 4: Capacitance

Homework for Week 4

Problem 1.

Physics Concepts

Make this week’s physics concepts summary as you work all of the problems in this week’s
assignment. Be sure to cross-reference each concept in the summary to the problem(s) they were
key to. Do the work carefully enough that you can (after it has been handed in and graded) punch
it and add it to a three ring binder for review and study come finals!

Problem 2.

Derive the capacitance for:

a) A parallel plate capacitor with cross-sectional areaAand plate separationd;

b) A cylindrical capacitor with inner conductor radiusa, outer conductor radiusb, and lengthL

(whereL≫b−a);

c) A spherical capacitor with inner conductor radiusaand outer conductor radiusb.

Show in the latter two cases that the capacitance is approximately:

C≈ǫ^0 A

d

whereAis the area of the cylinder/sphere andd=b−a≪a(“small” separation). You will need
to use the power series expansion ln(1 +x)≈x+O(x^2 )...to first order to do the cylinder.

Problem 3.

Prove that the energy stored on the capacitor can be written aseitherside of:

U=

1

2

QV=

∫

V

1

2

ǫ 0 E^2 dV

for all three geometries (where the integral is over the volumeVbetween the plates).