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)
