You need to be familiar with the following terms before attempting these problems:
Critical Flux Density
Try to write a definition for each.
These questions will help to give you a more thorough
physical understanding of how these quantities are linked.
1) The formula for Critical Magnetic Flux Density Bc as a function of temperature, T, is an empirical one, based on experimental evidence rather than a solid theoretical foundation:
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 ?
2) A long metal wire carries a current I. The diagram shows a point charge q moving with drift velocity v. Clearly the total magnetic field experienced at point P will be the sum of all contributions from all the charges in the wire.
(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 :
(a) x dq = dl cosq
(b) dl cos2q = r dq
Hence deduce that the total magnetic field is given by an integral of the form:
(iv) By integrating between suitable limits, show that the magnetic field at P is given by:
(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.
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 Bin = ( 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 L is 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 mc 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
7) (i) Show from first principles that the current It in a conductor with a finite
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
L = mo b loge (b/a)
Calculate the self-inductance of a ring of lead if b = 10mm and a = 0.1mm
(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/Ris 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.
8) (i) Visit the 'Nobel Prize'website at http://www.nobel.se Visit the electronic Nobel laureates museum.. Print out the 'Press Releases' for the 1987 and 1973Nobel Physics prizes. These contain useful information on superconductivity.
(ii)Find out about Type I and Type II superconductors from this and other websites. Click here for some useful links.