W9_parallel_resonance.eps

192 Week 5: Resistance

Problem 6.

C

Q1i

1 C 2

R

S

+

A pair of capacitorsC 1 andC 2 is connected as shown, with a resistanceRin between them.
Initially, the first capacitor carries a total chargeQ 1 iand the second one is uncharged,Q 2 i= 0. At
t= 0 the switch is closed. Find:

a) The equilibrium (t=∞) charges on the two capacitors,Q 1 fandQ 2 f.

b) Using Kirchoff’s laws for this arrangement, find the time constantfor the equilibration process.

Note that you do NOT have to solve the DE, just formulate it withdtand some arrangement

ofR,C 1 , andC 2 on the other side.

c) (Regular) GUESS what the solution to the DE looks like, based on your answers to a) and

b). To do the latter, try visualizing whatQ 1 (t) andQ 2 (t) will formally look like – it is just

a matter of setting the various constants so that the asymptotic(final) and initial conditions

are correctly represented and the approach to those conditionshas the right time dependence.

d) (Advanced) Solve the DE (it is integrable, although a bit messy) forQ 1 (t) andQ 2 (t). It’s

probably best to solve for just one, and then use conservation ofcharge to find the other,

right?