Physics 15b Assignment #5

Read Chapter 4 of Purcell by Monday March 7.

Q&A questions to be answered on the Physics 15b website before 11pm on Monday,March 7:

5QA-1. Which pair of numbers below is a good answer to the two questions in Problem 4.6 inPurcell?

A : 256/81 and 4

B : 64/27 and 4

C : 16/9 and 2

D: 4/3 and 2

E : None of the above.

5QA-2. Which answer below is the best answer to Problem 4.16 in Purcell?

A : R1 = (√

5− 1)R0/2

B : R1 = R0/√

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C : R1 = (√

3− 1)R0

D: R1 = R0

E : None of the above.

In addition, there are some survey questions and feedback questions.

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Problems due at the beginning of class on Thursday, March 10—

5-1. Problem 4.21 in Purcell.

In the circuit, all five resistors have the same value,100 ohms, and each cell has anelectromotive force of 1.5 volts. Find the open-circuit voltage and the short-circuitcurrent for the terminalsA andB. Then findE0 andR0 for the Th́evenin equivalentcircuit.

5-2. Problem 4.32 in Purcell.

Some important kinds of networks are infinite in extent. The figure shows a chain ofseries and parallel resistors stretching off endlessly to the right. The line at the bottomis the resistanceless return wire for all of them. This is sometimes called an attenuatorchain, or a ladder network. The problem is to find the “input resistance,” that is, theequivalent resistance between terminalsA andB. Our interest in this problem mainly

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concerns the method of solution, which takes an odd twist and which can be used inother places in physics where we have an iteration of identical devices (even in infinitechain of lenses in optics). The point is that the input resistance which we do not yetknow — call it R — will not be changed by adding a new set of resistors to the frontof the chain to make it one unit longer. But now, adding this section, we see thatthis new input resistance is justR1 in series with the parallel combination ofR2 andR. We get immediately an equation that can be solved forR. Show that, if voltageV0 is applied at the input to such a chain, the voltage at successive nodes decreasesin a geometric series. What ratio is required for the resistors to make the ladder anattenuator that halves the voltage at every step? Obviously a truely infinite ladderwould not be practical. Can you suggest a way to terminate it after a few sectionswithout introducing any errors in its attenuation?

5-3.

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Q0

C1

R1

C2

R2

In the circuit shown above, capacitorC1 has chargeQ0 on the upper plate and capacitorC2 isuncharged and no current is flowing. At timet = 0, the switch is closed.

a. Find the chargeQ on capacitorC1 as a function of time.

b. Find the energy stored in each capacitor and the power dissipated in each resistor as functionsof t.

5-4. A standard physics joke starts “Consider a spherical cow ...” In this problem, we considera spherical resistor.

5-4a. Suppose a conducting sphere of radiusa is centered at the origin and surrounded bymaterial with conductivityσ out to radiusb. At radiusb, the whole thing is covered with anotherconductor. Now we attach leads to the inner and outer conductors and measure the resistance.What do we get? Assume that somehow we can attach the lead to the inner conductor withoutdisturbing the nice spherical symmetry of the system.

5-4b. Use the result of part5-4a to find an approximate value for the resistance of a systemof two spherical conductors with radiia andb in an infinite sea of material with conductivityσ,where the distanced between the conductors is very large compared toa andb, and explain anyapproximations you make.

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