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.

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

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

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

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

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).

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.