Electrodynamics REN Xincheng, Postdoctoral , ProfessorTel2331505; 13244118078Email:ydxxxyrenxch@163.com

Chapter 3. magetostatic field

The magetostatic field excited by the constant current distribution is discussed in this chapter. In the constant current problems, the electric field also exists, because on the one hand, it demands a certain field that maintain the electric current in conductors, on the other hand, there are charge distribution on the electrode generated current and the conductor surface, they also generates an electric field outside the conductor. But it can be seen by the Maxwells equation that the electric and magnetic fields have not direct contact in steady conditions, thus they can be discussed separately. As mentioned above, steady electric field and electrostatic field has the same properties, it can be treated as electrostatic field. Together with the electrostatic field is called Coulomb field.

The magetostatic field can also be described by potential. Corresponding the scalar potential of the electrostatic field, the vector potential of magetostatic field is an important concept. The vector potential of the magnetic field is introduced and the solving method of magnetic field boundary value problems is illustrated in first section of this chapter. Although the describing the magnetic field with vector potential is universal, but solving some practical problems is often more complex. That the magetostatic field can also be describe using the scalar potential under certain conditions is illustrated in second section of this chapter. The expanded formula in the distance of field excited by the current distribution within local range is calculated and the concept of magnetic multipole moment is introduced in third section of this chapter. The A-B effect and the electromagnetic properties of superconductors is described in the last two sections.

3.1 Vector potential and its differential equation1. The introduction of vector potentialMagnetic field of steady currentThis A is the vector potential of magnetic field that we introduce, which can describe the property of the magnetic field, it is same as the A that introduced in deriving basic differential equations of the magetostatic field in first chapter. Discussion1The physical meaning of vector potential AVector potential A is different from the scalar potential , vector potential A has no direct physical meaning, the circulation of A only have the physical meaningthe circulation of A along any closed loop representatives the magnetic flux through any surface with the boundary of the loop.

2Vector potential A has a non-uniqueness, gauge transformation, normality conditionB can uniquely determined in the condition that A is given, but A can not be uniquely determined in the condition that B is given.Example It is obvious that there are an infinite number of group A corresponding to B.In fact

The randomicity of vector potential roots in only defining the curl of vector potential, but the divergence of it without any constraints. (To determine a vector field must also determine its divergence and curl.)So, to uniquely determine the vector potential A, we must add some restrictive condition to its divergence. And different restrictive condition form a different norm. For example:For convenience that dealing with the problem, different norm form can be selected, the gauge transformation and normality condition will be discussed in later chapters. Coulomb norm is selected below, this selection can always be done, if a solution A does not satisfy the Coulomb norm, set up

2. The derivation of differential equation of Vector potential A To uniform, linear and non-ferromagnetic materialSo, there is

This is a satisfying differential equation of A. It is a vector differential equation, the concrete computation can chose different system of coordinates depending on the problem, under rectangular coordinate system , the components Ai of rectangular coordinate meet with scalar potential identical Poisson equation,that is Therefor, By contrast with scalar potential can get some results about vector potential A.

The vector potentialof external magnetic field

Electrostatic fieldSteady current field

When the charge distribution in a limited area can get

orWhen the charge distribution in a limited area can get

or

We can go through like the electrostatic fieldFind the vector potential A(Example 1,Example 2P104~107)But the same can not solve the general problem,This is because the general problem of the current and the magnetic field is mutual restraint.They can only be obtained by solving the Poisson equation with boundary conditions that the solution to complete the boundary value problem.3. Boundary value relationship of the vector potential ABoundary value relationship for the magnetic induction BHere consider the general conductor=0Only perfect conductors or superconductors only need to consider the surface current distribution.

By the additional conditionsSo That is

Such steady current field solver comes down for the sake of the following boundary value problem

Example 1(P104) Infinitely long straight wire carrying current I seek magnetic vector potential and magnetic induction. This problem should be selected as the reference zero point of a finite far point.Example 2(P105) current-carrying wire ring Is radius is a, and seek vector potential and magnetic induction.Solution:The vector potential generated by the coil current isCoordinate with the ballSeen by the symmetry A only has Componentand A is only4 Examples

And rely on with nothing to . A calculation can be selected at a point P in xz plane, at the point

getIt can be obtained by Oval Tables. in the book P106 has gotsimilar results under conditions. Example 3 in Figure find infinitely long cylindrical conductor inside and outside of the vector potential distribution.Solution: 1) According to the boundary conditions select cylindrical coordinate system2) According to the symmetry, we know3) Poisson equation inside and outside of the column is

Select a finite far point as the reference point zero in conductor (current in the infinite region). This determines a constant. Body bounded A can be determined by a constant, while the value of the relationship determine two unknown constants .Consider r 0, A1 limited b = 0.When R = a, satisfy the boundary relations.b, c, d, e are undetermined coefficients, can be determined by boundary conditions.

Task(P131)13

3.2 Scalar magnetic potential For the magnetic field, although the description of the vector potential is universal, the solution of vector potential equations is much more complex than the solution of scalar potential equations. Especially for some coordinate system. Here we can see that, under certain conditions, describing the magnetic field can also be introduced scalar potential - magnetic scalar potential. This would greatly simplify the problem.1. Conditions for the introduction of the magnetic scalar potentialThe basic equation of the static magnetic field

In general, because , that H is non-conservative force field, the magnetic scalar potential can not be introduced. However, in many practical problems, the discussed local region there is no current distribution.(For example the entire magnetic field of the permanent magnet, in addition to the two electrodes solenoid magnetic field, the magnetic field in a more general non-current area, etc.) In any loop of these areas it will not be chain link by current , then will always be guaranteed Thus this area can introduce a magnetic scalar potential. Boils down to the introduction of the magnetic scalar potential condition is In any loop of these areas it will not be chain link by current . Equivalence argument isNo single connected region of freedom current distribution.

Note: s corresponding differential form is But in a region J = 0, can not guarantee that any of the loop region of the I = 0. So not necessarily be able to introduce magnetic scalar potential, J = 0 of the area to be able to introduce magnetic scalar potential, this area must also be a simply connected region. For example: The digging up area after the ring current region is not simply connected regions.2. The introduction of the magnetic scalar potential and the differential equationIn the case satisfied the above conditions are

From (2), (3), we obtainCompared with the electrostatic field is just no free magnetic charge (described magnetic monopoles)

Due to the symmetry of the above, we can extended a series of electrostatic field conclusions to the magnetic scalar potential by analogy. The corresponding solving methods of Electrostatic field can also be used to solve the magnetic field.3. Analogies of electric potential and magnetic scalar potential

Explanation1Their following corresponding amounts 2When introducing the magnetic scalar potential, molecular currents replaced by magnetic charge. Therefore, this method does not consider the molecular current. But after getting the results, you can calculate magnetizing current (molecular current);3The above boundary value relationship was only established to uniform linear non-ferromagnetic media , which is due toGeneral boundary value relationship is

4. ExamplesExample1(P.83) Prove surface of magnetic material is equal magnetic potential surfaceUsing the relationship of the magnetic field boundary value to prove outside the magnetic material, the magnetic field strength perpendicular to the surface.Application: Select the appropriate shape with the magnetic pole surface can obtain the magnetic fields of different forms.Example 2(P.83-85) Find magnetic field generated by the uniformly magnetized iron ball that magnetization vector is M0 .Inside and outside of the ball

get the general solution (using the boundary conditions at infinity and conditions of the center of the sphere is limited boundary )Using the boundary value relationship when R = R0Undetermined coefficients can be calculated

Potential solution isAs can be seen from the results, the magnetic field outside the sphere is generated by the dipole, dipole moment isB line is closed as shown , H-line is not. Both are very different. H is the secondary volume.

Example3(P.85) Find a magnetic scalar potential generated by current coil.Let current coil containing current I. It can be seen as a surface coil around the many small current-carrying coils I. Let the point coil located on a small area is .Its magnetic moment is .

The small magnetic field generated by the coil magnetic scalar potential is a dipole magnetic scalar potential.Magnetic scalar potential generated by the entire coil current issolid angle that coil subtended to the field point x .Discussion: If x points surrounded above the surface of the coil > 0 If x points surrounded under the surface of the coil

3.3 Magnetic multipole momentsIf the current distribution is in a small areaAnd to study its far-field, we can do multipole expansion the same as the electrostatic field vector potential . And the introduction of the concept of a magnetic multipole moments.1. Vector potential multipole expansionGiven the current distribution, vector potential of spatial magnetic isIf the current distribution is in a small area The field point x and very far away from the current area, you can do multipole expansion to A.

Take a point O in the current region as coordinate origin.

1Since no magnetic monopoles, with no corresponding vector potential point charge items, namelyIn addition, as the current line is closed, the result is also described by virtual closed tube current . A the integral is applied current totube

For small current coil m = IS (corresponding to a pair of magnetic charge, forming a magnetic dipole). The following to prove special circumstances for the results . Steady current can be divided into many closed current tube , considered one of the current tube

To general conditionsDose not discussed Higher order.

2. Field of the magnetic dipole moment and magnetic scalar potentialNo current distribution in a simply connected domain magnetic scalar potential can be introducedUsing the first chapter Problem 6

This is the magnetic dipole scalar potential. A small current coil in the distance can be seen as a magnetic dipole (magnetic dipole moment m=ISn).The magnetic field of magnetic dipole moment

3. The outside energy field of the current distribution in small area Let vector potential Ae in an external magnetic field Be.The current distribution J (x) in the interaction energy of external magnetic field isTo coil current IIf we take the appropriate point where the coil on the coordinate origin in the region ,regional lines of the magnetic field is much smaller than the line of the occurrence of a significant change. You can expand Be (x) in the neighborhood of the origin

The results in the magnetic dipole case is precisely established.Electric dipole in an external field energy is In contrast, the negative difference. If the direction of m is the same as the direction of , Wi> 0, representing the potential increase. At this point whether the force acting? Further, does that mean that when the magnetic field effect ,magnetic dipoles will tend to reverse with the external magnetic field ? Not true. Because these results are obtained on the exported assumption that conditions of the coil current I and the external magnetic field generated under the current do not change . Now specifically as follows:Let the external magnetic field is generated by another coil Ie with current Le, the interaction can be written as

When the coil movement, if keep the current I and Ie constant, magnetic energy is changed toHowever, due to the change of the magnetic flux, the induced electromotive force in the coil current acting it, and will change the value of I and Ie. To maintain I and Ie unchanged, It must be provided energy by the power supply to resist the work done by Induced electromotive force. The induced electromotive force on the coil L and Le was

In the time t ,the work done by the induced electromotive force is Must provide power to resist this induced electromotive force energy to maintain I and Ie constant. Under the conditions, the magnetic energy constant when I and Ie exists separately . Therefore, changing the total magnetic energy is equal to the change in the interaction W.Now the system includes three aspects of interaction: external power supply, electromagnetic fields and currents on the two coils. Application of the law of conservation of energy on the system , have:energy power supply should be equal to the total magnetic energy.

Change the sum with magnetic field the work done to the coil . ThatThat magnetic field made power to coils, it equal to the increments rather than its reduction(The key is to assume that I and Ie unchanged; thus appeare the energy provided by power).If in the definition of mechanics, the potential energy function U make doing work equal to the potential function reduction , should have In the external magnetic field ,potential function of Magnetic dipole is

The force of magnetic dipole in the external magnetic field is The moment of magnetic dipole in the external magnetic field isIts vector form isTask(P.109)15

3.4 Aharonov-Bohm effectIn classical electrodynamics, the basic physical electromagnetic is field strength E and the magnetic induction B. Potential and vector potential A is only an auxiliary capacity to facilitate the processing of mathematical introduced. They have no direct physical significance. However, in quantum theory, the potential A and have physical effects can be observed.In 1959, Aharonov and Bohm proposed that two new Effect (it is referred to as the A-B Effect).This effect was subsequently verified by tests.

Experimental apparatus: the right figure is a verification test apparatus of A-B effect (Electron double-slit interference experiment).Experimental phenomena: When the solenoid through current, it moved relative to no through current interference fringesExperimental Analysis: The experiment was carefully excluded solenoid external magnetic field, make electrons through the space B = 0, flux is only focused on the internal solenoid. This means that mobile of electronic interference fringes was not due to B caused by the space through which. The magnetic interactions are local interaction, that is, a point charge and the current only by playing the role of the neighborhood at that point.Therefore, the tube B can not be directly applied to the electron tube.

A-B illustrate the physical effects of the magnetic field Effect can not be fully described by B.When the solenoid magnetic flux , although the external space of the electrons pass through B = 0, but A 0.This is because to any one of the inner closed loop surrounding the solenoid haveTherefore, A-B effect indicates that A is a observable physical Effect .It may play a role in the electron ( impact e-beam phase).So that the movement occurrence of interference fringe .

A-B effect is a quantum effect. In quantum mechanics, free movement of electronic states is described by a plane wave function. Normalization factor is omitted.This wave function is:When the solenoid is not energized, the two beams of electrons reach the screen from the center point y, there is a phase difference ofWhen the solenoid is energized, the tube A 0, the wave function of the electron

Momentum Applications insteading by Canonical momentum.Electron wave function isDue A 0 , phase difference between two beams of electrons becomes

Phase change causes movement of the interference fringes. Results experimentally observed confirm the above formula.In quantum mechanics, the position in which the vector potential A is much more important than in classical mechanics. A-B effect showed that only the magnetic field B is not enough to describe. However, due to the non-uniqueness of A, description of the magnetic field with A is clearly excessive. Through the above analysis, the magnetic field that can be entirely appropriate to describe the physical quantity is relative factorIn the formula, C is either a closed path.To a problemC can

shrink to a little infinitesimal path, so that the integral value not changed,thenTherefore, the description of phase factor is equivalent to the description of local field B (x). Otherwise, the information relative to the physical factors which can not be contained in the local field B (x) to description (such as A-B effect).

3.5 Electromagnetic properties of superconductorsSince 1911, have found a lot of elements, compounds, alloys and other materials, when the temperature falls below a certain critical temperature Tc or less, the resistivity becomes zero. This phenomenon is called After 1986, they found a series of successively higher critical temperature of the superconducting material (may be higher than the liquid helium temperature ~ 80K), making the application of superconducting materials have broad prospects. Superconductivity is an important area of research in modern physics. In this section we discusses the basic superconductor power dynamics.

1. The basic electromagnetic properties of superconductors1) superconductivitySuperconductivit, critical temperature, the superconducting state and normal state.Different materials have different critical temperatures. Figure is variation curve on resistance of Mercury sample in temperature , its critical temperature is Tc = 4.2K.When an object is in the superconducting state, if coupled with the magnetic field, when the strength of magnetic field is big enough to a critical value Hc, superconductivity is destroyed, superconductor transition from the superconducting state to the normal state.

Critical field Hc related to temperature. Hc (T) for the empirical formula isCritical field Hc and temperature curve shown right2Meissner effect(Perfect diamagnetism)1933 Meissner discovered inside the superconductor magnetic induction B = 0, and independent of the history of superconductor it through. If the original object is in the superconducting state, when applying a magnetic field, the magnetic field strength as long as no more than Hc, then B can not enter the superconductor;

If the object is placed in the normal state in a magnetic field, when the temperature drops to an object into a super when the guide states, B is discharged outside the superconductor.That is, in any case, the object in the superconducting state, the total internal B = 0.Meissner effect of superconductors and zero resistance are independent of the two important characteristics. Meissner effect indicates that the superconductor can not simply be regarded as a conductor when the resistivity is limited . This can be described as follows:Typically conductor J = E, when and J is limited This obviously can not explain conclusion that B is discharged outside the superconductor.

2. Equations of the electromagnetic properties of superconductorsMaxwell equations of electromagnetic phenomena is generally established.Different substances exhibit different magnetic properties of electromagnetic phenomena from the equation of matter. Here's to illustrate the unique electromagnetic properties of superconductors equation.1) The first equation in LondonSuperconductivity is a quantum phenomenon. When the object is in the superconducting state, a portion of the electrons cohesion in the quantum state. In order to become fully exercise, it scatter from the lattice without resistance effect. The rest is still normal electron electron, which is a two-fluid model of superconductors. Pursuant to which, the phenomenological descriptions superconductivity.

Let conduction electron density n of superconductor is the sum of superconducting electron density ns and normal electron density nn , namelyAccordingly, the current density J in the superconductorNormal current Jn= EOhm's Law.The superconducting current obey the new regularity. No damping its movement, acceleration in the electric field E,superconducting electron velocity represented by , there

Next use the first London equation to export zero resistance properties In the case of superconductors constant .In the constant case Therefore, in the constant case, current of superconductors entirely come from within the superconducting electron motion.No resistance effect.In the case of alternatingThen two types of electron are involved in sports, so there are resistive losses.But in general the low frequency alternating current, the loss is very small. In this regard it may be calculated as follows:

2The second equation in LondonThe first London equation is not sufficient to describe all electromagnetic properties of superconductors. Considering Meissner effect, in superconductors inside B = 0 . Known by the Maxwell equations in the superconductor J = 0.The following illustrate superconductor in a magnetic field in the superconducting state. Tangential magnetic field must exist within the superconductor surface layer.If there is a surface current ( = 0) and the magnetic field, the application of boundary value relationship H1t = H2t, the inner surface of the magnetic field is zero, and the outer surface of the magnetic field is not zero, which is obviously impossible. So, in the superconducting state of a superconductor in a magnetic field. Tangential magnetic field must exist within the superconductor surface layer.

Then, in internal superconductors must exist mechanism of mutual restraint on the current and the magnetic field. So that they can only exist in a thin surface layer, but can not deep inside the superconductor. In addition to London assuming Maxwell equationsHere we can see that the first and second equations and Maxwell equations in London are compatible.On both sides of the first London equation, take the curl and integral of time., there is mutual restraint relationship of another field in the superconductor and the current . This is London's second equation. That

The second equation in London is results assuming the above equation f (x) = 0. London two equations summarize the electromagnetic properties of superconductors. By the two equations can say Mingmai Knysna effect, because for both sides to take the curl of a constant currentFor general superconductors, L on the order of 10-7m.

Seen by the following, L represents a line of occurrence of significant changes in the value of B superconductor.Considering the superconductor filled z> 0 of the half-space,and set B along the x direction, Bx = B (z), in this simple case, the above equation becomes

Its solution isWhen z into several L ,B (z) is substantially zero.Therefore, L marked lines degree of the penetration of the magnetic field within the superconductor. It is called the penetration depth. For current distribution within the superconductor can also be obtained (Both sides of the second London equation to take the curl and using the first London equation and

) It is the same form with the equation B. So superconducting current can only exist within the thin layer thickness ~ L surface of the superconductor. Superconducting current excited by magnetic field inside Superconductor offset the external magnetic field .To make the body magnetic induction B = 0. Using the value of the relationship to the outer edge of the superconductor boundary B and interface tangent.3. Superconducting as completely antimagnetsUnder steady conditions, current within superconductors include superconducting current Js and molecular magnetization current Jm. Magnetizing current and the magnetizing current within magnetic media is usually the same .

Thus, superconductor magnetic permeabilityand permeability in the normal state have . Previous section is on a description of superconductors with this view.

In this view, Meissner effect in superconductors is not from the nature of superconductors as a special magnetic media, but from the shielding effect of the superconducting current.Here we adopt another point of equivalent view description of superconductors. The basic physical field is B. It is always linked to the current density J. As for, How to divide the total current to freedom current and magnetization current , and How appropriate B decomposition of H and M, then with a certain arbitrariness. Press the front section of the opinion, superconducting current is free current, associated with the H, and the magnetizing current is associated with the magnetization M

But in many problems, the view will be more convenient to use, i.e., the superconducting current can also be seen as the magnetizing current, which was associated with M. According to this view, when the superconductor is placed outside the magnetic field.It is magnetized and induced by the magnetization current, making superconductors with magnetic moment M. For simplicity, we omit the molecular magnetizing current of superconductor (usually small), it isIf it is described by the surface superconducting current density , there is

According to this view, B = 0, is a special magnetic properties of superconductors. By theThus the magnetic susceptibility of superconductorsCorresponding permeabilityThis means that the superconductor is perfectly diamagnetic.Example(P127)Superconductor is placed in a uniform external magnetic field H0, find current distribution of magnetic and superconducting surface .Using the second view magnetic to solve scalar potential method .

4. magnetic flux quantization inside the superconductor

At T> Tc, the normal state to a superconducting ring is placed in an external magnetic field. When the temperature is lowered to T

The law of electromagnetic induction was applied on C2magnetic flux is quantized According to the quantum nature of superconductivity, superconducting electrons are in a quantum state, when around completed circuit a week. The phase change will be an integer multiple of 2

(which is described in the quantum state of the wave function of the required single value). By the 3.4 shows that the phase change around C for a week isThe formula -2e is charge of two electron . It is condensed in the source of the basic unit of quantum states. Its an electron pair - Cooper pairs.

Visibility, flux is quantized. This phenomenon is also confirmed by experiments. The quantum of magnetic flux plays an important role in the physics of superconductivity. This phenomenon once again illustrates that the vector potential A has a real physical meaning. Because in the superconductor circuit C, B = 0, but A 0, the vector potential A affects the phase of superconducting electron wave function. Resulting magnetic flux quantum phenomenon.Task (P134)13

*5. Nonlocal theory First class and second class of superconductorLondon equation is the local equation. The described relationship between the superconducting current and the electromagnetic field is localized. The superconducting current at a point and Electromagnetic field at the neighborhood of the point effects directly . This local picture is too simple for a superconductor. Since superconducting electronics in units of electrons pairs cohesion in a quantum state. Electronics on the different spatial points has a strong correlation with each other,leading to Superconducting current with the effective interaction of electromagnetic field is generally not localized. that a certain line of the electromagnetic field and the range of current occurs effective role.Electromagnetic fields and current lines within a range of degree occurs effective role.

Just in case 0 London equation is correct. Pippard first proposed the concept of coherence length . And made a nonlocal correction to London theory . According to nonlocal theory, superconductors exist in two characteristic scale:London penetration depth L and the coherence length . Both the length of relative size determines the nature of the superconductor.1The first class of superconductorsL

When the magnetic field H> Hc (T), the superconducting state becomes a normal state.2) The second class of superconductors L>>In a weak magnetic field, the London equation was established. Superconductors exhibit Meissner effect. In the interior B = 0. When the magnetic field increase to exceed the first threshold magnetic field HC1 (T). Quantum of magnetic flux lines start (radius ~ L) form into the superconductor. In the normal state of flux lines, between the lines of magnetic flux remains superconducting state. Since the superconducting region is connected, the object remains superconductivity.

When the magnetic field is further enhanced over the second critical field HC2 (T) .Field covered inside the object, the entire object converted to normal state. Such a superconductor has a high critical magnetic field HC2 (T) (as shown), through a strong superconducting current, and therefore have important practical applications.