Silsbee rule: An important result of the existence of critical magnetic field is that there is also critical strength of current Ic flowing in the superconductor. Exceeding this limit also causes the disturbance of superconductivity. To derive the relation between critical current field consider a superconductor wire of radius r carrying a current I. This current will produce a magnetic field given by:
H=I/2 π r (Using Ampere’s Circuital law) More >
As we have already derived the London equations in last article. Now let us
explain the flux penetration (Meissner effect) from London equations:
To explain Meissner effect from London equations consider the differential form of Ampere’s circuital law:
del x B = µoJs
where B is magnetic flux density and Js is current density
Take curl on both sides of above equation
del x (del x B) = µo (del x Js) (5)
As del x (del x B)= del(del.B) – del2B
Put above equation and London second equation (equation 4 is derived in last article) in equation (5), we get
del(del.B) – del2B = -[( µo nse2(B)/m]
But del.B = 0 (Maxwell’s second equation or Gauss law for magnetism)
Therefore above equation becomes
del2B = [( µo nse2(B)/m] (6)
del2B = B/λl2 (7)
where λl2 = m/ µo nse2
or λl = (m/ µo nse2)1/2
where λl is known as London’s penetration depth and it has units of length.
The solution of differential equation (7) is
B = B(0)e-x/ λl (8)
Where B(0) is the field at the surface and x is the depth inside the superconductor. More >
As discussed in the Meissner effect that one of the conditions of the superconducting state is that Magnetic flux density (B) = 0 inside the superconductors that is the magnetic flux cannot penetrate inside the superconductor. But experimentally it is not so. The magnetic flux does not suddenly drop to zero inside the surface. The phenomenon of flux penetration inside the superconductors was explained by H. London and F. London.
Derivation of London first equation:
Let ns and vs be the number density (number/volume) and velocity of superconducting electrons respectively. The equation of motion or acceleration of electrons in the superconducting state is given by
m(dvs/dt) = -eE
or dvs/dt = -eE/m (1)
where m is the mass of electrons and e is the charge on the electrons.
Also the current density is given by
Js = -nsevs
Differentiate it with respect to time,
dJs/dt = -nse(dvs/dt)
Put equation (1) in above equation, we get
dJs/dt = (nse2 E)/m (2)
Equation (2) is known as London’s first equation
Derivation of London second equation: More >
Depending upon their behavior in an external magnetic field, superconductors are divided into two types:
a) Type I superconductors and b) Type II superconductors
Let us discuss them one by one:
1) Type I superconductors:
a). Type I superconductors are those superconductors which loose their superconductivity very easily or abruptly when placed in the external magnetic field. As you can see from the graph of intensity of magnetization (M) versus applied magnetic field (H), when the Type I superconductor is placed in the magnetic field, it suddenly or easily looses its superconductivity at critical magnetic field (Hc) (point A).
b). Type I superconductors are also known as soft superconductors because of this reason that is they loose their superconductivity easily.
c) Type I superconductors perfectly obey Meissner effect.
d) Example of Type I superconductors: Aluminum (Hc = 0.0105 Tesla), Zinc (Hc = 0.0054)
2) Type II superconductors: More >
Superconductors are the materials whose conductivity tends to infinite as resistivity tends to zero at critical temperature (transition temperature).
Critical temperature (Tc): The temperature at which a conductor becomes a superconductor is known as critical temperature.
Critical Magnetic Field (Hc): The magnetic field required to convert the superconductor into a conductor is known as critical magnetic field.
Critical magnetic field is related with critical temperature as:
Hc(T) = Hc(0)[1 – T2/Tc2]
Suppose there is a conductor placed in a magnetic field at temperature T (refer figure). Then the temperature is decreased till the critical temperature. See what happened (figure). Lines of force are expelled from the superconductor. This is called Meissner effect.
B is not 0 at T > Tc B=0 at T < Tc
Definition Meissner Effect: The expulsion of magnetic lines of force from a superconducting specimen when it is cooled below the critical temperature is called Meissner effect.
To prove that superconductors are diamagnetic by nature:
B is not 0 at T > Tc B=0 at T < Tc
As B = µ0 (H +M)
Where B is magnetic induction or magnetic flux density,
H is applied magnetic field or magnetic field intensity
And M is intensity of magnetization.
For superconductors B = 0
Thus either µ0 = 0 or H + M = 0
But µ0 can not be zero,
Thus H + M =0
Or M = -H (1)
By definition of magnetic susceptibility
X = M/H
Put equation (1)
Thus X = -1
But magnetic susceptibility is negative for diamagnetic materials, thus it proves that superconductors are diamagnetic by nature.
Note: This article is referred from my authored book “Electrical Engineering Materials” having ISBN 8127234044.
A qualitative discussion of a successful theory of superconductivity was given by Bardeen, Copper and Schrieffer, known as BCS theory after the initials of their names. This theory accounts for all properties of superconductors.
(a) Electron –phonon Interaction. More >