H=I/2 π r                                  (Using Ampere’s Circuital law)

If the current through wire is such that H>H_c then I_c the superconductivity will be destroyed and material will go to normal state. Therefore if I_c is the current for which H=H_c then I_c is called critical current and is given by

I_c=2 π rH_c

This is known as Silsbee’s rule. Critical current density is given by

J_c =I_c/ Area =2 π r H_c/ π r²

Or J_c=2H_c/r

Properties which change in the superconducting Transition:-

(i)      The magnetic properties in superconductors undergo change. In the pure superconducting state practically no magnetic flux is able to enter the metal which thus behaves as if it had zero permeability or strong diamagnetic susceptibility. This effect is called Meissner effect.

(ii)    The specific heat changes discontinuously at the transition temperature. There is small change of volume at transition in the presence of magnetic field.

(iii)   All the thermoelectric effects disappear in the superconducting state.

(iv)  The thermal conductivity changes discontinuously when the superconductivity is destroyed in magnetic field. It is lower in the superconducting state for pure metal but higher for certain alloys.

(v)    The entropy in the superconducting state is lesser comparative to the normal state,  that is the superconductive state is more ordered state.

The post Silsbee rule and other properties in superconductors first appeared on Winner Science.

1. Generation and transmission of electric power.

2. Medical diagnosis

3. Electromagnets (superconducting magnets):

– The type 2 superconducting wires are wound in the form of solenoids to generate a strong magnetic field.

4. In the making of Supercomputers

5. Magnetically levitating the world’s fastest trains.

6. Magnetic Energy storage Devices.

7. Electromagnetic  shielding

8. Superconducting transformers.

9. In the medical industry as superconducting quantum Interferometers (SQUIDS).

10. Bearings:

. The Meissner effect is made use of in the bearings.

11. High power transmission lines:

. The superconducting cables permit high power transmission without power loss.

12. Particle accelerators:

As Superconductors has very poor mechanical strength, therefore superconducting solenoid is already in use to provide the high magnetic fields needed for the large particle accelerators using Nb₃S_n

The above are some of the applications of Superconductors. If you know more, please share in the comment section.

The post Applications of superconductors first appeared on Winner Science.

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 = µ_oJ_s

where B is magnetic flux density and J_s is current density

Take curl on both sides of above equation

del x (del x B) = µ_o (del x J_s)                                                     (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 = -[( µ_o n_se²(B)/m]

But del.B = 0 (Maxwell’s second equation or Gauss law for magnetism)

Therefore above equation becomes

del²B = [( µ_o n_se²(B)/m]                                                            (6)

del²B = B/λ_l² (7)

where λ_l² = m/ µ_o n_se²

or λ_l = (m/ µ_o n_se²)^1/2

where λ_l 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/ λ}_l (8)

Where B(0) is the field at the surface and x is the depth inside the superconductor.

The equation (8) shows that a uniform magnetic field equal to zero can not exist in a superconductor, which is Meissner effect. In the pure superconducting state the only field allowed in the exponentially decreasing field as the flux penetrated from external surface and it is given by equation (8) (Refer figure).

Suppose x = λ_l

Then equation (8) becomes

B = B(0)/e

Definition of London penetration depth: The London penetration depth is the distance inside the surface of a superconductor at which the magnetic field reduces to 1/e times its value at the surface.

The London penetration depth depends strongly on the temperature and becomes much larger as T approaches critical temperature Tc. The relation is

λ_l(T)/ λ_l(0)= [1 – T/T_c)⁴]^-1/2

where λ_l(T) and λ_l(0) are the London penetration depths at temperature T kelvin and 0 k respectively.

The post London equations: explanation of flux penetration first appeared on Winner Science.

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 n_s and v_s 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(dv_s/dt) = -eE

or dv_s/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

J_s = -n_sev_s

Differentiate it with respect to time,

dJ_s/dt = -n_se(dv_s/dt)

Put equation (1) in above equation, we get

dJ_s/dt = (n_se² E)/m                                            (2)

Equation (2) is known as London’s first equation

Derivation of London second equation:

Take curl (that is cross or vector product of del operator with a vector) of London’s first equation, we get

del operator x dJ_s/dt =  [(n_se² )del operator x E]/m                                (3)

By differential form of Faraday’s law of electromagnetic induction (or Maxwell’s third equation)

del x E = -dB/dt

Put this in equation (3), we get

del x dJ_s/dt =  -[(n_se²(dB/dt)/m)

Integrate both sides with respect to time, we get

del x J_s = -[(n_se²(B)/m]                                                             (4)

This is known as London’s second equation.

Note: Read the importance of London’s equations in the article:

London equations: explanation of flux penetration

The post London equations in superconductors: derivation and discussion first appeared on Winner Science.

Let us discuss them one by one:

1) Type I or Soft superconductors:

The following is the definition and Properties of Type I or Soft Superconductors:

a). Type I superconductors are those superconductors that lose their superconductivity very easily or abruptly when placed in the external magnetic field. As you can see from the graph of the intensity of magnetization (M) versus applied magnetic field (H), when the Type I superconductor is placed in the magnetic field, it suddenly or easily loses its superconductivity at the critical magnetic field (H_c) (point A).

After H_c, the Type I superconductor will become a conductor.

b) Type I superconductors are also known as soft superconductors because of this reason that is they lose their superconductivity easily.

c) Type I superconductors perfectly obey the Meissner effect.

d) Example of Type I superconductors: Aluminum (Hc = 0.0105 Tesla), Zinc (Hc = 0.0054)

2) Type II or Hard superconductors:

The following is the definition and Properties of Type II or Hard Superconductors:

a). Type II superconductors are those superconductors that lose their superconductivity gradually but not easily or abruptly when placed in the external magnetic field. As you can see from the graph of the intensity of magnetization (M) versus applied magnetic field (H), when the Type II superconductor is placed in the magnetic field, it gradually loses its superconductivity. Type II superconductors start to lose their superconductivity at the lower critical magnetic field (H_c1) and completely lose their superconductivity at the upper critical magnetic field (H_c2).

b) The state between the lower critical magnetic field (H_c1) and upper critical magnetic field (H_c2) is known as vortex state or intermediate state.

After Hc2, the Type II superconductor will become a conductor.

c) Type II superconductors are also known as hard superconductors because of this reason that is they lose their superconductivity gradually but not easily.

d) Type II superconductors obey the Meissner effect but not completely.

e) Example of Type II superconductors: NbN (Hc = 8 x 10⁶ Tesla), Babi₃ (Hc = 59 x 10³ Tesla)

f) Application of Type II superconductors: Type II superconductors are used for strong field superconducting magnets.

Following is the link of our YouTube Video regarding difference between Type I and Type II Superconductors or Soft and Hard Superconductors:

The post Type I and Type II superconductors first appeared on Winner Science.

a) what are superconductors,

b) what is critical temperature and its importance,

c) What is the critical magnetic field and its importance and the most important

d) What is the Meissner Effect in Superconductivity and the significance of Meissner Effect.

e) We will also prove that all superconductors are diamagnetic by nature. Let us discuss all these things one by one:

Superconductors:

Superconductors are materials whose conductivity tends to be infinite as resistivity tends to zero at critical temperature (transition temperature).

Critical temperature (T_c):

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 a critical magnetic field.

Critical magnetic field is related with critical temperature as:

H_c(T) = H_c(0)[1 – T²/T_c²]

Meissner Effect in Superconductors and its Significance:

Suppose there is a conductor placed in a magnetic field at temperature T (refer left figure and In the figure, the arrow represents the magnetic lines of force passing through the superconducting specimen). When the temperature is decreased till the critical temperature. See what happened ( Right figure). Lines of force are expelled from the superconductor. This is called the 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 the Meissner effect.

To prove that superconductors are diamagnetic by nature:

As B is not 0 at T > Tc      and             B=0 at T < Tc

Also by mathematical expression,  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 the 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

χ = M/H

Put equation (1)

Thus χ = -1

But magnetic susceptibility is negative for diamagnetic materials, thus it proves that superconductors are diamagnetic by nature. 

Do you know why superconductors are superconductors? The same is explained in the BCS theory of superconductivity. Do read the article on BCS theory of Superconductivity.

Following is the link to our YouTube video related to the Meissner effect in superconductors:

The post Superconductors, critical temperature, critical magnetic field and Meissner effect first appeared on Winner Science.

(a)    Electron –phonon Interaction. BCS theory showed that the basic interaction responsible for superconductivity appears to be that of a pair of electrons by means of an interchange of virtual phonons. This is explained as follows:-

Suppose an electron approaches a positive ion core. It suffers attractive coulomb interaction. Due to this attraction ion core is set in motion and thus distorts that lattice. Let a second electron come in the way of distorted lattice and interaction between the two occurs which lowers the energy of the second electron. The two electrons therefore interact indirectly, via lattice distortion or the phonon field, thus lowering the energy of electrons. This type of interaction is called electron-lattice is quantized in terms of phonons the above interaction can also be interpreted as electron –electron interaction through phonons.

Let an electron of wave vector K emits phonon q, which is absorbed by an electron of wave number K .K is thus scattered as K-q and K as K –q the process being a virtual one. The nature of the resulting electron-electron interactions depends on the relative magnitudes of the electronic energy change and the phonon energy. If this phonon energy exceeds electronic energy, the interaction is attractive.

(b) Copper Pairs. The fundamental postulate of BCS theory is that the superconductivity occurs when an attractive interaction mentioned above, between two electrons by means of a phonon exchange, dominate the usual repulsive coulomb interaction. Two such electrons which interact attractively in the phonon field are called copper pair.

(c) Coherence length. The paired electrons are not scattered and can maintain their coupled motion up to certain distance called the coherence length. It is a measure of the distance within which the gap parameter does not change very much in varying magnetic field.

The post BCS Theory of Superconductivity first appeared on Winner Science.