In this study the electromagnetic theory and quantum mechanics are utilized to find the resistivity in terms of electric and magnetic susceptibility in which the electron is considered as a wave. Critical temperature of the wire at which the resistance vanishes is found. In this case the resistance being imaginary which leads the real part of the resistance to real zero at critical temperature and the material becomes super conductor in this case. If one considers the motion of electron in the presence of inner magnetic field and resistance force, a new formula for the conductivity is to be found; this formula states that the material under investigation becomes a superconductor at critical temperature and depends on the strength of the magnetic field and friction resistance, and the substance conductivity is found to be super at all temperatures beyond the critical temperature.
Susceptibility Superconductivity Critical Temperature Permeability Permittivity1. Introduction
In superconductors, the resistance is zero at temperatures less than the critical temperature [1] ; in this work, the above concept is proved by taking the conductivity as a function of the permittivity and permeability, and within the existence of the conditions that make the resistance be equal to zero or approach to infinity.
2. Resistivity Formula Due to the Permeability and Permittivity
When the temperature of a conductor approach to the absolute zero, the friction resistance can be ignored [2] , if an electron e is induced by an electric field E, then the force on it is given by
Including the position variable in Equation (1) it can be written as
Then
According to the definition of the potential, we get
From Equation (3)
Then
While is constant, and when the potential difference is constant, then the velocity is being also constant.
Using Equation (6) and substituting the value of in the equation of current, that given due to the electron velocity, charges density, and the area, , then the current is found to be
Then the resistance is given
On other hand can be written due to the resistivity, the length, and the crossection area as
Considering the electron as a wave, its velocity becomes [3]
Accordingly the resistivity is given by
3. Critical Temperature at a Changing Permeability µ
If a magnetic field with a flux density, an electric force, besides a friction resistance, and a pressure
force act together, then the centripetal forces which balance this force is given by [4] .
where is the radial velocity, while the friction force and the pressure are given by
where is the friction coefficient.
when the outer magnetic field vanishes, then the radial velocity becomes
And
where denotes the inner magnetic field.
And when an outer magnetic field is applied, then
where is the radial force, and, are the magnetic and the electric forces respectively, which are given by
The equation of motion in the presence of the outer magnetic field is given in the form [5] .
where is the radial velocity, and while then
when is so closed to then
and
where is Larmar frequency, substitute Equation (15) and Equation (19) one gets?
Dividing both sides by we get
The current for one atom with electrons, moving around its nucleus with a frequency f is
where is the atomic number, is the electron charge, and is Larmar frequency.
The magnetic torque for one atom is given by
where is the area surrounded by the current which is equal
And from Figure 1, one get:
But
So the magnetic torque for one atom becomes
Magnetic torque in Z direction
If the number of atoms per unit volume is assumed to be then, the magnetic torque for the matter is
According to the definition of susceptibility then [6] .
Comparing Equations (26) and (27) the susceptibility being
Then the resistivity in Equation (11) becomes
where denotes the photon energy.
The resistivity is imaginary, and the real resistivity vanishes when
or (30)
Accordingly the critical temperature becomes
4. Calculating the Critical Temperature Due to the Conductivity
Assuming that the charges in the conductor are acted by a resistance force, and a magnetic force, besides the electric force, and then the equation of motion becomes [7] .
The previous forces are given by the formulas
where and denotes the density, rigidity coefficient, displacement, mass, velocity, electron charge, magnetic flux density, resolving time, and the electric field respectively.
The equation of motion takes the formula
When the electron moves with a uniform constant velocity, the Equation (33) becomes
And the conductivity is given by
where the electrons density, while denotes the density of the medium atoms, accordingly the conductivity being
And the conductivity approaches to infinity when
According to the Maxwell-Boltzmann statistics the density of the atoms in the medium takes the formula [8] .
Then
Equation (38) represents the critical temperature in which the conductivity becomes very huge, and when
The conductivity also becomes very high, and then
And finally the critical temperature is found to be
5. Discussion
The classical rules of the electron motion in Equation (1) are used to find the classical formula of the resistivity given in Equation (11), and the electron is considered to be a wave according to the quantum principles and this clarified that the resistivity is a function of the electric and magnetic susceptibility.
The interpretation of Equation (28)―in which we derived the magnetic susceptibility from the electron equation of motion, that depend on the friction force within the friction coefficient, the inner magnetic field, the grad of the electrons density, and the atom radius―is that when Equation (28) is substituted in Equation (11) the critical temperature at which the resistance vanishes, was found to be in the form that given by Equation (31), which was completely depends on the inner magnetic field, and the friction, within the coefficient, the radius, and the grad of the electron density.
When we considered the electron motion due to the impact of an inner magnetic field, and a friction resistance, the conductivity was found to be as shown in Equation (36).
The mathematical analysis interprets that the conductivity becomes very high at temperatures less than the critical temperature, which depends on the friction resistance and the inner magnetic field as shown in Equations (41)-(42).
6. Conclusion
The model in which the resistance depends on the electric and magnetic susceptibility, clarifies that the resistance vanishes, and the metal becomes a superconductor at the critical temperature and the temperatures less than it; this relation is not clear in the famous models of the superconductivity.
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