..	You need to be familiar with the following terms before attempting these problems:    Superconductor CriticalTemperature Critical Flux Density  Absolute Zero Phase Transition  Self-Inductance. Try to write a definition for each.

Here Tc is the Critical Temperature in the absence of an externally applied magnetic field and Bo = Bc at 0 Kelvin.

(i) Plot curves (6 data points for each) for the following elements of Bc (y-axis)
against T (x-axis). l
 

(ii) Clearly identify the superconducting and normal regions of your graphs , explaining your reasoning.Comment on the suitability of each as superconducting materials.Are they good conductors at room temperature ?

(i) State with reasons the direction of the magnetic field at point P due to the charge q.

The situation may be quantified by using the Biot-Savart law.For an individual charge carrier this is:
.

(ii) Show that this may be rewritten as follows for a Current Element dl:.
 

(iii) Show that :
 

Hence deduce that the total magnetic field is given by an integral of the form:
.

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(v) For each of the elements Hg , V , Pb at a temperature of 4 Kelvin:
 
 

(a) Write down their critical flux densities Bc at this temperature.

(b) Calculate the maximum current that could be carried by a

2mm diameter superconducting wire made from each of these . materials.Explain your reasoning/assumptions.

3) Ampere's Circuital Law links the total magnetic field B enclosed by a loop G to the current density J of the cross sectional area S(G) that is enclosed by the loop. It may be stated mathematically as:

This is a vector integral equation the left hand side indicating that the integral must be taken around the closed loop (This is termed a path integral) ,the right hand side being a double (area) integral.

(i) Explain why for the single wire shown, that

(ii) Show that the same result as Q2(iv) follows this trivial integration. (Either Ampere's Law or the Biot - Savart law are used as starting points for magnetic field distributions.)

4) (i) Look up and state in simple terms what is meant by the 'Meissner Effect' in superconducting materials.

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The Meissner Effect is most readily studied using a planar geometry. Full vector formalism is required if more complex shaped conductors are to be dealt with.

A large slab of superconducting material with a free surface at z = 0 has an externally applied field Bo = (Bo,0,0) parallel to its surface.

Inside,the field will be of the form B_in = ( B(z),0,0 )

From Maxwell's Equations it may be shown that

is the differential equation governing the magnetic field inside the body, where Lis a parameter termed the 'London Penetration Depth'.

(i) Show that
 

is a solution of this equation and that it remains finite for all z>0.

(ii) What can you deduce about the penetration of the magnetic field into the superconducting slab ? How does this help explain the 'Meissner Effect' ?

(iii) Sketch a graph of this solution and point out the main features.
 

6) A fairly lengthy analysis shows that for a fixed material at fixed temperature, the London Penetration Depth is given by:

where m_c is the mass of a coupled pair of electrons -a 'Cooper Pair'. If a typical superconductor has a value L = 0.5x10^-8 m, then
 

(i) Calculate Nc - the number of Cooper Pairs and comment on your answer.
 

(ii) If the Surface Current Density J obeys Maxwell's Equation
.

then show that

resistance R and self- inductance L decays exponentially from an initial value Io.
 
 

(ii) It may be shown that the self-inductance of a wire ring of radius b, the wire being of diameter 2a is given approximately by

Calculate the self-inductance of a ring of lead if b = 10mm and a = 0.1mm

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(iii) The resistivity of Superconducting lead at 4.2 kelvin is less than 3.6x10^-25Wm.Calculate the resistance of the ring in part (ii). Hence determine that the ratio L/R is approximately 2500 years.
 
 

(v) Sketch a graph of decay of current in the specimen pointing out the main features.
 
 

(vi) A 3000 turn toroid of the same radius and wire diameter is to be used in the same superconducting state.

(a) Show that L =moN²A / c where c is the circumference of the toroid.

(b) Calculate its resistance if c = 20cm.

(c) Calculate its Inductive Time Constant (L/R) and comment.
 

(ii)Find out about Type I and Type II superconductors from this and other websites. Click here for some useful links.