Superconductor in a magnetic field.

In summary, the conversation discusses the behavior of a superconductor under the influence of an external magnetic field and the equation for the coexistence line between the superconducting and normal state. It is shown that the parameter 'a' in the equation must be 0 and the parameter 'b' can be expressed in terms of the critical field and temperature. The latent heat per molecule in the transition from superconductor to normal state is calculated and it is also shown how to find the jump in heat capacity per molecule when crossing the coexistence line in a constant field. The validity of the solution is questioned and it is suggested that B does not necessarily equal zero in the superconducting state.
  • #1
MathematicalPhysicist
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Problem Statement:
when a superconductor of the first order is under the influence of an external magnetic field, the field is repelled from the the supercondutin material until the field reaches a critical value called the critical field [tex]H_c(T)[/tex]. for H>H_c the material turns into regualr metal (normal state) and for H<H_C the material turns into a superconductor.
The coexsitence line separating between the two phases is a parabola (as a function of the temprature) given by:
[tex]H_c(T)=H_0+aT+bT^2[/tex]

1.Get an analogous equation to clausius -clpareon equation for the coexistence line and show that 'a' must be 0, in the two phases the ground state is non degenrate, and gibbs energy is G=U-TS-HM where H is the magnetic field, U is the thermal energy, T temp, S entropy, M magentization, and dU=TdS+Hdm+[tex]\mu[/tex]dN.

2.from the codnition H_c(T_c)=0 express b with H_0 and T_c.

3. use the fact that B=0 in the superconductin state, and the connection [tex]B=H+4\pi m[/tex] (this is a vectorial equation), to compute the latent heat per molecule in the transition from superconductor to normal state as function of T, you can assume that the magnetization at normal state is zero.

4. use the connection: [tex]c_H=T\frac{dS}{dT}_H[/tex] to find the jump in heat capacity per molecule when crossing the coexistence line in a constant field.


My attempt at solution

for (1) I think the equation is something like this:
[tex]\frac{dH}{dT}=\frac{(\frac{d\mu}{dT}_H_{super}-\frac{d\mu}{dT}_H_{Normal})}{(\frac{d\mu}{dH}_T_{Normal}-\frac{d\mu}{dH}_T_{Super})}[/tex]
which form what is given I think that [tex]dG=\mu dN-SdT-mdH[/tex]
I guess the equation in question shoule look something like this:
[tex]\frac{dH}{dT}=\frac{S_{super}-S_{normal}}{m_{normal}-m_{super}}[/tex].

And now a=0 cause dH/dT=a+2bT and at T=0 the entropies are equal while the magnetizations aren't so we get that a=0.

for (2), ofcourse it should be: b=-H0/T_c^2.

for (3), here I'm using the fact L=T_c*(S_sup-S_nor)/N (N the number of particles) is our latent heat and H=4pi*m in the superconductor, so we get that:
[tex]dH/dT=-L/(T_c*m_{super})=2bT_c[/tex]
and we know what is m_super, it equals H_c/4pi=H0+bT_c^2
after plugging the equations I get: [tex]L=-2bT^2_c(H_0+bT^2_c)/4\pi[/tex], is this even correct?

for (4), for c_H I'm just taking the derivative of L wrt T ( I'm repalcing T_c with some T in the coexistnece line), and after some rearrangements I get to what is [tex]\delta c_H[/tex].


So what do you think of this essay of mine, any good?
My solution is valid?

thanks in advance.
 
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  • #2
P.S for the last question I think that there B doesn't necessarily eqauls zero, not sure, any hints, thanks in advance.
 

1. What is a superconductor?

A superconductor is a material that can conduct electricity with zero resistance when cooled below a certain temperature, known as the critical temperature.

2. What is a magnetic field?

A magnetic field is a region in space where magnetic forces are present due to the movement of electrically charged particles.

3. How does a superconductor behave in a magnetic field?

A superconductor exhibits the Meissner effect when placed in a magnetic field, causing it to expel the magnetic field lines and create a perfect diamagnetism. This means that the superconductor has zero resistance to the flow of electricity and no energy is lost as heat.

4. What is the critical magnetic field for a superconductor?

The critical magnetic field is the maximum strength of an external magnetic field that a superconductor can tolerate before it loses its superconducting properties and becomes a normal conductor.

5. What are the applications of superconductors in a magnetic field?

Superconductors in a magnetic field have various applications, including in the medical field for MRI machines, in transportation for magnetic levitation trains, and in research for creating strong magnetic fields for experiments. They also have the potential for use in energy-efficient power transmission and energy storage systems.

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