## Description

Problem 1. In a perfect conductor, the conductivity is infinite, so E = 0, and any net charge resides

on the surface (just as it does for an imperfect conductor in electrostatics). (a) Show that

the magnetic field is constant, i.e. ∂B

∂t = 0, inside a perfect conductor. (b) Show that the

magnetic flux through a perfectly conducting loop is constant.

(2 + 2 marks)

Problem 2. A superconductor is a perfect conductor with the additional property that the (constant)

magnetic field B inside is in fact zero. (This ”flux exclusion”, known as the Meissner effect,

was shown in one of the videos we watched in class.)

Show that the electric current in a superconductor is confined to the surface.

(2 marks)

Problem 3. Consider two light bulbs connected in series around a solenoid (see figure

(A)) producing a sinusoidally varying magnetic field. If we alter the circuit

by connecting a thick copper wire across the circuit (see figure (B)), we find

that the top bulb gets much brighter and the bottom one no longer glows.

Why does this happen?

(2 marks)

Problem 4. Consider a circuit shown in the figure below. The current in a long solenoid

piercing the plane of the circuit is increasing linearly with time, so that

the magnetic flux through the cross-section of the solenoid ΦB = αt. Two

voltmeters are connected to diametrically opposite points A and B, together

with resistors R1 and R2.

Show that the readings on voltmeters are V1 =

αR1

R1+R2

and V2 = −

αR2

R1+R2

,

respectively, i.e. V1 6= V2, even though they are connected to the same

points.

Assume that these are ideal voltmeters that draw negligible current (i.e.

have huge resistance), and that a voltmeter registers R b

a E · dl between the

terminals and through the meter.

(3 marks)

Problem 5. Two coils are wrapped around a cylindrical form in such a way that the same

flux passes through every turn of both coils. (In practice this is achieved

by inserting an iron core through the cylinder; this has the effect of concentrating the flux.) The ”primary” coil has N1 turns and the ”secondary”

has N2. If the current I in the primary is changing, show that the emf in

the secondary is given by E2/E1 = N2/N1, where E1 is the (back emf) of the

primary.

This is a primitive transformer — a device for raising or lowering the emf

of an alternating current source. By choosing the appropriate number of

turns, any desired secondary emf can be achieved.

(2 marks)

Problem 6. Two coils are wrapped around each other as shown in the figure below.

The current travels in the same sense around each coil. One coil has selfinductance L1, and the other coil has self-inductance L2. The mutual inductance of the two coils is M. Show that if the two coils are connected in

parallel, the equivalent inductance of the combination is L =

L1L2−M2

L1+L2−2M .

(3 marks)

Problem 7. A transformer takes an input AC voltage of amplitude V1, and delivers an output voltage

of amplitude V2, which is determined by the turns ratio, V2/V1 = N2/N1. If N2 > N1 the

output voltage is greater than the input voltage. Why doesn’t this violate conservation

of energy? Answer : Power is the product of voltage and current; evidently if the voltage

goes up, the current must come down.

The purpose of this problem is to see exactly how this works out, in a simplified model.

(a) In an ideal transformer the same flux passes through all turns of the primary and of

the secondary. Show that in this case M2 = L1L2, where M is the mutual inductance

of the coils and L1, L2 are there individual self-inductances.

(b) Suppose the primary is driven with AC voltage Vin = V1 cos ωt, and the secondary

connected to a resistor with resistance R. Show that the two currents satisfy the

relations

L1

dI1

dt + M

dI2

dt = V1 cos ωt, L2

dI2

dt + M

dI1

dt = −I2R.

(c) Using the result in (a) solve the equations for I1 and I2. (Assume that I1 has no DC

component.)

(d) Show that the output voltage Vout = I2R divided by the input voltage Vin is equal to

the turns ratio, i.e. Vout/Vin = N2/N1.

(e) Calculate the input power Pin = VinI1 and the output power Pout = VoutI2, and show

that their averages over a full cycle are equal.

(3/2 + 1 + 2 + 1 + 3/2 marks)

Problem 8. In the circuit shown in the figure below, switch S is closed at time t = 0 with no charge

initially on the capacitor.

(a) Find the reading of each ammeter and each voltmeter just after S is closed.

(b) Find the reading of each meter after a long time has elapsed.

(c) Find the maximum charge on the capacitor.

(d) Sketch a qualitative graph of the reading of voltmeter V2 as a function of time.

(4 × 1 marks)