_{1}

^{*}

Hereby it studied the fundamental theory created by the London brothers for interpretation of the magnetic field reduction in superconductors. As demonstrated, this theory is not correct. The author of this work has developed the new theory of superconductivity. The equation describing the electron distribution function under the effect of magnetic field explains existence of the Meissner-Ochsenfeld effect. It is shown that the critical field equation matches the width of a potential well in the kinetic energy dependence of mean electron energy. As a result, the supercurrent density formula has been derived. Existence of magnetic fields is explained by two steady-state Maxwell equations. There are magnetic fields that are found to be created in individually shaped metals. Penetration of the external magnetic field in a superconductor has bee n explained.

Kamerlingh Onnes discovered the phenomenon of superconductivity at Leiden Laboratory, Holland, in 1991 [

A superconductor is immersed in liquid helium. Initially, weak current is supplied from a battery. Then, temperature is reduced. When temperature falls below the defined value, the superconductor circuit is shorted. The superconductor circuit current sustains its steady state as long as it can. A magnetic needle provided as a detector finds some persistent current in the superconductor, thus indicating to the magnetic field produced in the solenoid. The pattern of temperature T dependence of specific resistance ρ in a superconductor is shown in _{c} is named for critical temperature. This means that we cannot measure resistance of the superconductor. The matter is that the superconductor has the property that makes impossible to measure any specific resistance.

Shortly thereafter, it was discovered that such superconductivity disappears when a test piece is placed in a relatively weak magnetic field. This phenomenon was discovered by Meissner and Ochsenfeld [_{m} of the magnetic field strength in which superconductivity is disrupted is called a critical field. The temperature dependence of the critical field is described by the following empirical formula:

where

Brothers Fritz and Heinz London developed the first macroscopic theory of superconductivity in 1935 [

Here B is magnetic induction inside a superconductor. Such facts have been accepted a priori. But why does the specific resistance go to zero? There might be other superconducting factors that make it impossible to measure ρ. Why does it occur that the magnetic induction inside a superconductor gets equal to zero? The facts accepted a priori should be proved.

Let’s take B for magnetic induction of the external field. The London brothers have derived the following external magnetic field equation:

where

Value

Let us assume the superconductor occupies half-space_{o} runs along the interfacial area (see

The solution of the Equation (1.5) is formulated as follows:

It should be noted that this function does not agree with the Meissner-Ochsenfeld effect. When magnetic field strength at the surface of the conductor exceeds critical value

There are two magnetic fields in the superconductor. One magnetic field is created by the supercurrent and another external field is induced from other sources. The compass needle shown in _{c}:

wherein the strength of the external magnetic field at the surface of the superconductor is less than that of the critical field:

In other cases the superconductor will represent ordinary metal properties.

The new density-matrix based superconductivity theory has been developed and described in the works [_{kk}' with wave vectors k and k’ has been derived in the work [

where I and J are energy dimension constants, _{k}:

where function

Here G is a number of valence states specified in one crystal lattice point, N is a number of electrons within the lattice.

The equations (2.2) may be easily solved by means of a computational modeling method. At temperatures

The plot of the distribution function w(ε, τ) that complies with temperature T = 0 and parameters

Let us denote the values of multiple-valued function

There can be the anisotropic solution made when functions

Actually, electrons exhibit their lowest energy macro-state. At T = 0, the energy will be minimized to the state expressed by the following formula:

This function obtains anisotropy at

The mean electron energy is:

The kinetic energy ε dependence

Here,

The plots demonstrated have the following specific features. At temperature

Value

according to which the “well” edges graphically specified by dependence

the function. This means that there is a “gap” in the range of values of the electron energy_{c} to zero. The well width

also grows up from zero to I value while temperature is decreased down to T = 0.

Let us assume that a superconductor is in the magnetic field which strength will be denoted by the H value. The electron energy values have been found in the works [

where

On minimizing thermodynamic potential Ω with account for the energy values (3.1), we can obtain the nonlinear equation to be applied for determining wave vector electron distribution function

If

than the Equation (3.4) will have the previous solution:

Let’s assume that

It is necessary to find the lowest electron energy (3.1) for obtaining the real-valued distribution function. Now, let us study the case when a magnetic field destroys superconductivity at T = 0.

Let

The lowest energy (3.1) may be found when function w_{k} is expressed as follows:

As it is shown above, function w_{k} is to be isotropic?i.e. the superconductive state disappears. The distribution function plot is shifted to the right by value I while the magnetic field destroys the superconductive state that consequently disappears (see

Let’s assumed that value Λ is equal to I. In this case, the distribution function is shifted to the right. We shall find the critical magnetic field strength at T = 0 using the Formulas (3.2) and (3.7).

If the

where

The relation

Now, we shall compare the plot of the critical field (see

If no magnetic field is applied, the width of the well is to be equal to

We shall now substitute value

When strength H of the magnetic field gets its critical value

Now, we shall use the formula derived in the works [

where

Basically, equilibrium values of the magnetic field

For finding the solution to this problem, it is necessary to know the current density

Let us study the magnetic field applied to the planar surface of the superconductor when the H-vector induces the effect acting paralleled to this surface. The x axis is placed along the conductor and parallel to the H-vector, the y axis is arranged perpendicularly to the surface and the z axis faces us (see

In this case, the Equation (6.1) is expressed as follows:

Since the supercurrent is created by a negative component

where

The superposition principle-based H-vector is equal to the sum of the external field and magnetic field as being induced by supercurrent flows:

The projection of the complete field will be represented by the following expression:

With the above equation substituted into the Formula (6.4), the following expression is formulated:

Let’s assume that any supercurrent to be produced in a substance is flowing along the z axis and the external magnetic field is not applied:

Such current may flow along the surface of the superconductor for an unlimited duration. The supercurrent creates the

The supercurrent-induced field strength applied to the superconductor surface will be equal to zero:

The strength that satisfies such condition will be expressed by the formula:

The plot of this function is demonstrated in

Now, we shall create the x-directed external magnetic field with its strength denoted by the

Let us assume that the magnetic field satisfies the original condition:

The solution of this equation will be formulated as follows:

Hence, we have obtained some kind of decay of the superconductor inwardly directed external magnetic field strength. But the character of decay does not match the function predicted by the London equation:

The plot of the function (6.14) is shown in

The complete field (6.7) will take up the following form:

Now, we shall find the supercurrent density by the Formula (5.3) as a function of the y coordinate. For this purpose, we shall substitute the Formula (6.16) into the Equation (5.3). As a result, we shall obtain:

where

As it is seen from the above formula, the current density exponentially decays in the direction off the superconductor surface. When the strength of the external field goes to that of the critical field

Let us study the case when the external field strength is applied in the opposite direction:

This will result in failed application the Meissner-Ochsenfeld effect. For correcting such condition it should be noted that the external field actually changes direction of supercurrent flow. Hence, it should be noted that the direction of the superconductor self-magnetic field H-vector matches that of the external magnetic field strength. As

provided by the above calculations, the external magnetic field decays inside a conductor when being exposed to the supercurrent-induced magnetic field.

We have discussed the absolutely correct solutions of the Maxwell’s and current density equations in this Section. Some approximate expressions for supercurrent and magnetic fields will be discussed below.

Let the superconductor be a flat disk. An external magnetic field is perpendicular to the plane of the disk (see

The superconducting current density

where r is the radial coordinate, R is the radius of the superconducting disk. The moduli of tension of self-magnetic field produced by superconducting current and the external magnetic field inside the drive will be approximately equal, if these formulas are constructed like Formulas (6.11) and (6.14):

If you turn off the external magnetic field, a superconducting current will flow through the disk, the magnitude of which is equal to

and the magnitude of the tension of the self-magnetic field remains the same.

Let’s discuss the behavior of the H magnetic field inside and outside a spherically shaped superconductor. The conductor with no superconducting properties induced at temperature

When temperature drops down below the critical value:

of the critical field:

Let’s implement spherical coordinates

where R is a sphere radius. For the purpose of current density, conditions

External magnetic field

The above formula demonstrates the influence of the external field on the supercurrent density. As soon as

These fields are shown in

Now, we switch off the external magnetic field. In this case, its modulus applicable to the conductor surface is to be equal to zero

will be much the same as before. The pattern of self-magnetic field lines only will obtain a few changes (see

Let us discuss about a superconducting wire coil. Current flows through such conductor passing each circular loop. Much the same current flows over a thin-coat disk surface (see

and coil plain:

Now, let’s discuss about a superconducting solenoid. Current may flow passing through a circular loop similar to that described in the previous section. But the solenoid will have exceeded self-magnetic field. If there are N coils in the solenoid, then the strength of the magnetic field within its core will go to

Hence, we can explain the cause of behavior by reference to external magnetic field

If to compare moduli of such fields, we can seen that the external magnetic field modulus is characterized by H_{o} and self-magnetic field?by

Bondarev, B.V. (2016) New Theory of Superconductivity. Does the London Equation Have the Proper Solution? No, It Does Not. Self-Generated and External Magnetic Fields in Super- conductors. Open Access Library Journal, 3: e3148. http://dx.doi.org/10.4236/oalib.1103148