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8
Magnetoquasistatic
Fields: Superposition
Integral and Boundary
Value Points of View

8.0 Introduction

\titleshortheadMQS Fields: Superposition Integral and Boundary Value Views We now follow the study of electroquasistatics with that of magnetoquasistatics. In terms of the flow of ideas summarized in Fig. 1.0.1, we have completed the EQS column to the left. Starting from the top of the MQS column on the right, recall from Chap. 3 that the laws of primary interest are Ampère's law (with the displacement current density neglected) and the magnetic flux continuity law (Table 3.6.1).

These laws have associated with them continuity conditions at interfaces. If the interface carries a surface current density K, then the continuity condition associated with (1) is (1.4.16)

and the continuity condition associated with (2) is (1.7.6).

In the absence of magnetizable materials, these laws determine the magnetic field intensity H given its source, the current density J. By contrast with the electroquasistatic field intensity E, H is not everywhere irrotational. However, it is solenoidal everywhere.

The similarities and contrasts between the primary EQS and MQS laws are the topic of this and the next two chapters. The similarities will streamline the development, while the contrasts will deepen the understanding of both MQS and EQS systems. Ideas already developed in Chaps. 4 and 5 will also be applicable here. Thus, this chapter alone plays the role for MQS systems taken by these two earlier chapters for EQS systems.

Chapter 4 began by expressing the irrotational E in terms of a scalar potential. Here H is not generally irrotational, although it may be in certain source-free regions. On the other hand, even with the effects of magnetization that are introduced in Chap. 9, the generalization of the magnetic flux density _o H has no divergence anywhere. Therefore, Sec. 8.1 focuses on the solenoidal character of _o H and develops a vector form of Poisson's equation satisfied by the vector potential, from which the H field may be obtained.

In Chap. 4, where the electric potential was used to represent an irrotational electric field, we paused to develop insights into the nature of the scalar potential. Similarly, here we could delve into the way in which the vector potential represents the flux of a solenoidal field. For two reasons, we delay developing this interpretation of the vector potential for Sec. 8.6. First, as we see in Sec. 8.2, the superposition integral approach is often used to directly relate the source, the current density, to the magnetic field intensity without the intetermediary of a potential. Second, many situations of interest involving current-carrying coils can be idealized by representing the coil wires as surface currents. In this idealization, all of space is current free except for some surfaces within which surface currents flow. But, because H is irrotational everywhere except through these surfaces, this means that the H field may be expressed as the gradient of a scalar potential. Further, since the magnetic field is divergence free (at least as treated in this chapter, which does not deal with magnetizable materials), the scalar potential obeys Laplace's equation. Thus, most methods developed for EQS systems using solutions to Laplace's equation can be applied to the solution to MQS problems as well. In this way, we find "dual" situations to those solved already in earlier chapters. The method extends to time-varying quasistatic magnetic fields in the presence of perfect conductors in Sec. 8.4. Eventually, in Chap. 9, we shall extend the approach to problems involving piece-wise uniform and linear magnetizable materials.

Vector Field Uniquely Specified

A vector field is uniquely specified by its curl and divergence. This fact, used in the next sections, follows from a slight modification to the uniqueness theorem discussed in Sec. 5.2. Suppose that the vector and scalar functions C(r) and D(r) are given and represent the curl and divergence, respectively, of a vector function F.

The same arguments used in this earlier uniqueness proof then shows that F is uniquely specified provided the functions C (r) and D(r) are given everywhere and have distributions consistent with F going to zero at infinity. Suppose that F_a and F_b are two different solutions of (5) and (6). Then the difference solution F_d = F_a - F_b is both irrotational and solenoidal.

The difference solution is governed by the same equations as in Sec. 5.2. With F_d taken to be the gradient of a Laplacian potential, the remaining steps in the uniqueness argument are equally applicable here.

The uniqueness proof shows the importance played by the two differential vector operations, curl and divergence. Among the many possible combinations of the partial derivatives of the vector components of F, these two particular combinations have the remarkable property that their specification gives full information about F.

In Chap. 4, we determined a vector field F = E given that the vector source C = 0 and the scalar source D = /_o. In Secs. 8.1 we find the vector field F = H, given that the scalar source D = 0 and that the vector source is C = J.

The strategy in this chapter parallels that for Chaps. 4 and 5. We can again think of dividing the fields into two parts, a particular part due to the current density, and a homogeneous part that is needed to satisfy boundary conditions. Thus, with the understanding that the superposition principle makes it possible to take the fields as the sum of particular and homogeneous solutions, (1) and (2) become

In sections 8.1-8.3, it is presumed that the current density is given everywhere. The resulting vector and scalar superposition integrals provide solutions to (9) and (10) while (11) and (12) are not relevant. In Sec. 8.4, where the fields are found in free-space regions bounded by perfect conductors, (11) and (12) are solved and boundary conditions are met without the use of particular solutions. In Sec. 8.5, where currents are imposed but confined to surfaces, a boundary value approach is taken to find a particular solution. Finally, Sec. 8.6 concludes with an example in which the region of interest includes a volume current density (which gives rise to a particular field solution) bounded by a perfect conductor (in which surface currents are induced that introduce a homogeneous solution).

8.1 The Vector Potential and the Vector Poisson Equation

A general solution to (8.0.2) is

where A is the vector potential. Just as E = -grad is the "integral" of the EQS equation curl E = 0, so too is (1) the "integral" of (8.0.2). Remember that we could add an arbitrary constant to without affecting E. In the case of the vector potential, we can add the gradient of an arbitrary scalar function to A without affecting H. Indeed, because x (\nabla ) = 0, we can replace A by A^' = A + . The curl of A is the same as of A^'.

We can interpret (1) as the specification of A in terms of the assumedly known physical H field. But as pointed out in the introduction, to uniquely specify a vector field, both its curl and divergence must be given. In order to specify A uniquely, we must also give its divergence. Just what we specify here is a matter of convenience and will vary in accordance with the application. In MQS systems, we shall find it convenient to make the vector potential solenoidal

Specification of the potential in this way is sometimes called setting the gauge, and with (2) we have established the Coulomb gauge.

We turn now to the evaluation of A, and hence H, from the MQS Ampère's law and magnetic flux continuity law, (8.0.1) and (8.0.2). The latter is automatically satisfied by letting the magnetic flux density be represented in terms of the vector potential, (1). Substituting (1) into Ampère's law (8.0.1) then gives

The following identity holds.

The reason for defining A as solenoidal was to eliminate the A term in this expression and to reduce (3) to the vector Poisson's equation.

The vector Laplacian on the left in this expression is defined in Cartesian coordinates as having components that are the scalar Laplacian operating on the respective components of A. Thus, (5) is equivalent to three scalar Poisson's equations, one for each Cartesian component of the vector equation. For example, the z component is

With the identification of A_z and _o J_z /_o, this expression becomes the scalar Poisson's equation of Chap. 4, (4.2.2). The integral of this latter equation is the superposition integral, (4.5.3). Thus, identification of variables gives as the integral of (6)

and two similar equations for the other two components of A. Reconstructing the vector A by multiplying (7) by i_z and adding the corresponding x and y components, we obtain the superposition integral for the vector potential.

Remember, r^' is the coordinate of the current density source, while r is the coordinate of the point at which A is evaluated, the observer coordinate. Given the current density everywhere, this integration provides the vector potential. Hence, in principle, the flux density _o H is determined by carrying out the integration and then taking the curl in accordance with (1).

The theorem at the end of Sec. 8.0 makes it clear that the solution provided by (8) is indeed unique when the current density is given everywhere.

In order that x A be a physical flux density, J (r) cannot be an arbitrary vector field. Because div (curl) of any vector is identically equal to zero, the divergence of the quasistatic Ampère's law, (8.0.1), gives ( x H) = 0 = \nabla J and thus

The current distributions of magnetoquasistatics must be solenoidal.

Of course, we know from the discussion of uniqueness given in Sec. 8.0 that (9) does not uniquely specify the current distribution. In an Ohmic conductor, stationary current distributions satisfying (9) were determined in Secs. 7.1-7.5. Thus, any of these distributions can be used in (8). Even under dynamic conditions, (9) remains valid for MQS systems. However, in Secs. 8.4-8.6 and as will be discussed in detail in Chap. 10, if time rates of change become too rapid, Faraday's law demands a rotational electric field which plays a role in determining the distribution of current density. For now, we assume that the current distribution is that for steady Ohmic conduction.

\sectnonumTwo-Dimensional Current and Vector Potential Distributions

Suppose a current distribution J = i_z J_z(x, y) exists through all of space. Then the vector potential is z directed, according to (8), and its z component obeys the scalar Poisson equation

But this is formally the same expression, (4.5.3), as that of the scalar potential produced by a charge distribution (x^' , y^' ).

It was inconvenient to integrate the above equation directly. Instead, we determined the field of a line charge from symmetry and Gauss' law and integrated the resulting expression to obtain the potential (4.5.18)

where r is the distance from the line charge r = (x - x^' )² + (y - y^' )² and r_o is the reference radius. The scalar potential can thus be evaluated from the two-dimensional integral

The vector potential of a two-dimensional z-directed current distribution obeys the same equation and thus has a solution by analogy, after a proper interchange of parameters.

Two important consequences emerge from this derivation.

by a given charge distribution (x, y), has an MQS analog vector potential A_z(x, y) caused by a current density J_z(x, y) with the same spatial distribution as (x, y). The magnetic field follows from (1) and thus

Therefore the lines of magnetic flux density are perpendicular to the gradient of A_z. A plot of field lines and equipotential lines of the EQS problem is transformed into a plot of an MQS field problem by interpreting the equipotential lines as the lines of magnetic flux density. Lines of constant A_z are lines of magnetic flux.

(a)	Every two-dimensional EQS potential (x, y) produced
(b)	The vector potential of a line current of magnitude i along the z direction is given by analogy with (12), which is consistent with the magnetic field H = i (i/2 r) given by (1.4.10), if one makes use of the curl expression in polar coordinates,

The following illustrates the integration called for in (8). The fields associated with singular current distributions will be used in later sections and chapters.

Example 8.1.1. Field Associated with a Current Sheet

A z-directed current density is uniformly distributed over a strip located between x₂ and x₁ as shown in Fig. 8.1.1. The thickness of the sheet, , is very small compared to other dimensions of interest. So, the integration of (14) in the y direction amounts to a multiplication of the current density by . The vector potential is therefore determined by completing the integration on x^'

where K_o \equiv J_z .

This integral is carried out in Example 4.5.3, where the two dimensional electric potential of a charged strip was determined. Thus, with _o /_{o o} K_o, (4.5.24) becomes the desired vector potential.

The profiles of surfaces of constant A_z are shown in Fig. 8.1.1. Remember, these are also the lines of magnetic flux density, _o H.

Example 8.1.2. Two-Dimensional Magnetic Dipole Field

A pair of closely spaced conductors carrying oppositely directed currents of magnitude i is shown in Fig. 8.1.2. The currents extend to + and - infinity in the z direction, so the resulting fields are two-dimensional and can be represented by A_z. In polar coordinates, the distance from the right conductor, which is at a distance d from the z axis, to the observer location is essentially r - d cos . The A_z for each wire takes the form of (16), with r the distance from the wire to the point of observation. Thus, superposition of the vector potentials due to the two wires gives

In the limit d \ll r, this expression becomes

Thus, the surfaces of constant A_z have intersections with planes of constant z that are circular, as shown in Fig. 8.1.3. These are also the lines of magnetic flux density, which follow from (17).

If the line currents are replaced by line charges, the resulting equipotential lines (intersections of the equipotential surfaces with the x - y plane) coincide with the magnetic field lines shown in Fig. 8.1.3. Thus, the lines of electric field intensity for the electric dual of the magnetic configuration shown in Fig. 8.1.3 originate on the positive line charge on the right and terminate on the negative line charge at the left, following lines that are perpendicular to those shown.

8.2 The Biot-Savart Superposition Integral

Once the vector potential has been determined from the superposition integral of Sec. 8.1, the magnetic flux density follows from an evaluation of curl A. However, in certain field evaluations, it is best to have a superposition integral for the field itself. For example, in numerical calculations, numerical derivatives should be avoided.

The field superposition integral follows by operating on the vector potential as given by (8.1.8) before the integration has been carried out.

The integration is with respect to the source coordinates denoted by r^', while the curl operation involves taking derivatives with respect to the observer coordinates r. Thus, the curl operation can be carried out before the integral is completed, and (1) becomes

The curl operation required to evaluate the integrand in this expression can be carried out without regard for the particular dependence of the current density because the derivatives are with respect to r, not r^'. To make this evaluation, observe that the curl operates on the product of the vector J and the scalar = |r - r^'|^-1, and that operation obeys the vector identity

Because J is independent of r, the first term on the right is zero. Thus, (2) becomes

To evaluate the gradient in this expression, consider the special case when r^' is at the origin in a spherical coordinate system, as shown in Fig. 8.2.1. Then

where i_r is the unit vector directed from the source coordinate at the origin to the observer coordinate at (r, , ).

We now move the source coordinate from the origin to the arbitrary location r^'. Then the distance r in (5) is replaced by the distance |r - r^' |. To replace the unit vector i_r, the source-observer unit vector i_{r^' r} is defined as being directed from an arbitrary source coordinate to the observer coordinate P. In terms of this source-observer unit vector, illustrated in Fig. 8.2.2, (5) becomes

Substitution of this expression into (4) gives the Biot-Savart Law for the magnetic field intensity.

In evaluating the integrand, the cross-product is evaluated at the source coordinate r^'. The integrand represents the contribution of the current density at r^' to the field at r. The following examples illustrate the Biot-Savart law.

Example 8.2.1. On Axis Field of Circular Cylindrical Solenoid

The cross-section of an N-turn solenoid of axial length d and radius a is shown in Fig. 8.2.3. There are many turns, so the current i passing through each is essentially directed. To keep the integration simple, we confine ourselves to finding H on the z axis, which is the axis of symmetry.

In cylindrical coordinates, the source coordinate incremental volume element is dv^' = r^' dp dr^' dz^'. For many windings uniformly distributed over a thickness , the current density is essentially the total number of turns multiplied by the current per turn and divided by the area through which the current flows.

The superposition integral, (7), is carried out first on r^'. This extends from r^' = a to r^' = a + over the radial thickness of the winding. Because \ll a, the source-observer distance and direction remain essentially constant over this interval, and so the integration amounts to a multiplication by . The axial symmetry requires that H on the z axis be z directed. The integration over z^' and ^' is

In terms of the angle shown in Fig. 8.2.3 and its inset, the source-observer unit vector is

so that

The integrand in (9) is ^' independent, and the integration over ^' amounts to multiplication by 2.

With the substitution z^" = z^' - z, it follows that

In the limit where d/2a \ll 1, the solenoid becomes a circular coil with N turns concentrated at r = a in the plane z = 0. The field intensity at the center of this coil follows from (13) as the amp-turns divided by the loop diameter.

Thus, a 100-turn circular loop having a radius a = 5 cm (that is large compared to its axial length d) and carrying a current of i = 1 A would have a field intensity of 1000 A/m at its center. The flux density measured by a magnetometer would then be B_z = _o H_z = 4 x 10^-7 (1000) tesla = 4 gauss.

Further implications of this finding are discussed in the following demonstration.

Demonstration 8.2.1. Fields of a Circular Cylindrical Solenoid

The solenoid shown in Fig. 8.2.4 has N = 141 turns, an axial length d = 70.5 cm, and a radius a = 13.6 cm. A Hall-type magnetometer measures the magnitude and direction of H in and around the coil. The on-axis distribution of H_z predicted by (13) for the experimental length-to-diameter ratio d/2a = 2.58 is shown in Fig. 8.2.4. With i = 1 amp, the flux density at the center approaches 2.5 gauss. The accuracy with which theory and experiment agree is likely to be limited only by such matters as the care with which the probe can be mounted and the calibration of the magnetometer. Care must also be taken that there are no magnetizable materials, such as iron, in the vicinity of the coil. To avoid contributions from the earth's magnetic field (which is on the order of a gauss), ac fields should be used. If ac is used, there should be no large conducting objects near by in which eddy currents might be induced. (Magnetization and eddy currents, respectively, are taken up in the next two chapters.)

The infinitely long solenoid can be regarded as the analog for MQS systems of the "plane parallel plate capacitor." Just as the capacitor can be constructed to create a uniform electric field between the plates with zero field outside the region bounded by the plates, so too the long solenoid gives rise to a uniform magnetic field throughout the interior region and an exterior field that is zero. This can be seen by probing the field not only as a function of axial position but of radius as well. For the finite length solenoid, the on-axis interior field designated by H_ in Fig. 8.2.4 is given by (13) for locations on the z axis where d/2 \gg z.

In the limit where the solenoid is also very long compared to its radius, where d/2a \gg 1, this expression becomes

Probing of the field shows the field maintains the value and direction of (16) over the interior cross-section as well. It also shows that the magnetic field intensity just outside the windings at an axial location that is several radii a from the coil ends is relatively small.

Continuity of magnetic flux requires that the total flux passing through the solenoid in the z direction must be returned in the -z direction outside the solenoid. How, then, can the exterior field of a long solenoid be negligible compared to that inside? The outside flux returns in the -z direction through a much larger exterior area than the area a² through which the interior flux passes. In fact, as the coil becomes infinitely long, this return flux spreads out over an exterior area that stretches to infinity in the x and y directions. The field intensity just outside the winding tends to zero as the coil is made very long.

Stick Model for Computing Fields of Electromagnet

The Biot-Savart superposition integral can be completed analytically for relatively few configurations. Nevertheless, its evaluation amounts to no more than a summation of the field contributions from each of the current elements. Thus, on the computer, its evaluation is a straightforward matter.

Many practical current distributions are, or can be approximated by, connected straight-line current segments, or current "sticks." We will now use the Biot-Savart law to find the field at an arbitrary observer position r associated with a current stick having an arbitrary location. The result is a practical resource, because a numerical summation over differential volume current elements can then be replaced by one over the sticks.

The current stick, shown in Fig. 8.2.5, is represented by a vector a. Thus, the current is uniformly distributed between the base of this vector at r + b and the tip of the vector at r + c. The source coordinate r^' is located along the current stick. The objective in the following paragraphs is to carry out an integration over the length of the current stick and obtain an expression for H (r). Because the current stick does not represent a solenoidal current density at its ends, the field derived is of physical significance only if used in conjunction with other current sticks that together represent a continuous current distribution.

The detailed view of the current stick, Fig. 8.2.6, shows the source coordinate denoting the position along the stick. The origin of this coordinate is at the point on a line through the stick that is closest to the observer coordinate.

The projection of b onto a vector a is _b = a b /|a|. Thus, the current stick begins at this distance from = 0, as shown in Fig. 8.2.6, and terminates at _c, the projection of c onto the axis of a, as also shown.

The cross-product c x a/|a| is perpendicular to the plane of Fig. 8.2.6 and equal in magnitude to the projection of c onto a vector that is perpendicular to a and in the plane of Fig. 8.2.6. Thus, the shortest distance between the observer position and the axis of the current stick is r_o = |c x a|/|a|. It follows from this fact and the definition of the cross-product that

where ds is the differential along the line current and

Integration of the Biot-Savart law, (7), is first performed over the cross-section of the stick. The cross-sectional dimensions are small, so during this integration, the integrand remains essentially constant. Thus, the current density is replaced by the total current and the integral reduced to one on the axial coordinate of the stick.

In view of (17), this integral is expressed in terms of the source coordinate integration variable as

This integral is carried out to obtain

In evaluating this expression at the integration endpoints, note that by definition,

so that (20) becomes an expression for the field intensity at the observer location expressed in terms of vectors a , b, and c that serve to define the relative location of the current stick.

The following illustrates how this expression can be used repetitively to determine the field induced by currents represented in a piece-wise fashion by current sticks. Expressed in Cartesian coordinates, the vectors are a convenient way to specify the sticks making up a complex winding. On the computer, the evaluation of (22) is then conveniently carried out by a subroutine that is used many times.

Example 8.2.2. Axial Field of a Pair of Square Coils

Shown in Fig. 8.2.7 is a pair of coils, each having N turns carrying a current i in such a direction that the fields induced by each coil reinforce along the z axis. The four linear sections of the two coils comprise the sides of a cube, centered at the origin and with dimensions 2d.

We confine ourselves to finding H along the z axis where, by symmetry, it has only a z component. Thus, for an observer at (0, 0, z), the vectors specifying element (1) of the right-hand coil in Fig. 8.2.7 are

Evaluation of the z component of (22) then gives the part of H_z due to element (1). Because of the axial symmetry, the field induced by elements (2), (3), and (4) in the same coil are the same as already found for element (1). The field induced by element (5) in the second coil is similarly found starting from vectors that are the same as in (23), except that d -d in the z components of b and c. Here too, the other three elements each contribute the same field as already found. Thus, the axial field intensity, the sum of the contributions from the individual coils, is

This distribution is plotted on the inset to Fig. 8.2.8. Because the fields induced by the separate coils reinforce, the pair can be used to produce a relatively uniform field in the midregion.

Demonstration 8.2.2. Field of Square Pair of Coils

In the experiment of Fig. 8.2.8, the axial field is probed by means of a Hall magnetometer. The output is connected to the vertical trace of a high persistence scope. The probe is mounted on a carriage that is attached to a potentiometer in such a way that there is an output voltage proportional to the horizontal position of the probe. This is used to control the horizontal scope deflection. The result is a trace that follows the predicted contour. The plot is shown in terms of normalized coordinates that can be used to compare theory to experiment using any size of coils and any level of current.

8.3 The Scalar Magnetic Potential

The vector potential A describes magnetic fields that possess curl wherever there is a current density J (r). In the space free of current,

and thus H ought to be derivable there from the gradient of a potential.

Because

we further have

The potential obeys Laplace's equation.

Example 8.3.1. The Scalar Potential of a Line Current

A line current is a source singularity (at the origin of a polar coordinate system if it is placed along its z axis). From Ampère's integral law applied to the contour C of Fig. 1.4.4, we have

and thus

It follows that the potential \Psi that has H_ of (6) as the negative of its gradient is

Note that the potential is multiple valued as the origin is encircled more than once. This property reflects the fact that strictly, H is not curl free in all of space. As the origin is encircled, Ampère's integral law identifies J as the source of the curl of H.

Because \Psi is a solution to Laplace's equation, it must possess an EQS analog. The electroquasistatic potential

describes the fringing field of a capacitor of semi-infinite extent, extending from x = 0 to x = +, with a voltage V across the plates, in the limit as the spacing between the plates is negligible (Fig. 5.7.2 with V reversed in sign). It can also be interpreted as the field of a semi-infinite dipole layer with the dipole density _s = _s d = _o V defined by (4.5.27), where d is the spacing between the surface charge densities, \pm _s, on the outside surfaces of the semi-infinite plates (Fig. 5.7.2 with the signs of the charges reversed). We now have further opportunity to relate H fields of current-carrying wires to EQS analogs involving dipole layers.

The Scalar Potential of a Current Loop

A current loop carrying a current i has a magnetic field that is curl free everywhere except at the location of the wire. We shall now determine the scalar potential produced by the current loop. The line integral \oint H ds enclosing the current does not give zero, and hence paths that enclose the current in the loop are not allowed, if the potential is to be single valued. Suppose that we mount over the loop a surface S spanning the loop which is not crossed by any path of integration. The actual shape of the surface is arbitrary, but the contour C_l is defined by the wire which is its edge. The potential is then made single valued. The discontinuity of potential across the surface follows from Ampère's law

where the broken circle on the integral sign is to indicate a path as shown in Fig. 8.3.1 that goes from one side of the surface to a point on the opposite side. Thus, the potential \Psi of a current loop has the discontinuity

We have found in electroquasistatics that a uniform dipole layer of magnitude _s on a surface S produces a potential that experiences a constant potential jump _s/_o across the surface, (4.5.31). Its potential was (4.5.30)

where \Omega is the solid angle subtended by the rim of the surface as seen by an observer at the point r. Thus, we conclude that the scalar potential \Psi, a solution to Laplace's equation with a constant jump i across the surface S spanning the wire loop, must have a potential jump _s/_o i, and hence the solution

where again the solid angle is that subtended by the contour along the wire as seen by an observer at the point r as shown by Fig. 8.3.2. In the example of a dipole layer, the surface S specified the physical distribution of the dipole layer. In the present case, S is arbitrary as long as it spans the contour C of the wire. This is consistent with the fact that the solid angle \Omega is invariant with respect to changes of the surface S and depends only on the geometry of the rim.

Example 8.3.2. The H Field of Small Loop

Consider a small loop of area a at the origin of a spherical coordinate system with the normal to the surface parallel to the z axis. According to (12), the scalar potential of the loop is then

This is the potential of a dipole. The H field follows from using (2)

As far as its field around and far from the loop is concerned, the current loop can be viewed as if it were a "magnetic" dipole, consisting of two equal and opposite magnetic charges \pm q_m spaced a distance d apart (Fig. 4.4.1 with q q_m). The magnetic charges (monopoles) are sources of divergence of the magnetic flux _oH analogous to electric charges as sources of divergence of the displacement flux density _oE. Thus, if Maxwell's equations are modified to include the action of a magnetic charge density

in units of voltsec/m⁴, then the new magnetic Gauss' law must be

in analogy with

Now, magnetic monopoles have been postulated by Dirac, and recent searches for the existence of such monopoles have been apparently successful

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² Science Vol. 216, (June 4, 1982).

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. Because the search is so difficult, it is apparent that, if they exist at all, they are very rare in nature. Here the introduction of magnetic charge is a matter of convenience so that the field produced by a small current loop can be pictured as the field of a magnetic dipole. This can serve as a mnemonic for the reconstruction of the field. Thus, if it is remembered that the potential of the electric dipole is

the potential of a magnetic dipole can be easily recalled as

where

The magnetic dipole moment is defined as the product of the magnetic charge, q_m, and the separation, d, or by _o times the current times the area of the current loop. Another symbol is used commonly for the "dipole moment" of a current loop, m \equiv ia, the product of the current times the area of the loop without the factor _o. The reader must gather from the context whether the words dipole moment refer to p_m or m = p_m/_o. The magnetic field intensity H of a magnetic dipole at the origin, (14), is

Of course, the details of the field produced by the current loop and the magnetic charge-dipole differ in the near field. One has _o H \not= 0, and the other has a solenoidal H field.

8.4 Magnetoquasistatic Fields in the Presence of PerfectConductors

There are physical situations in which the current distribution is not prespecified but is given by some equivalent information. Thus, for example, a perfectly conducting body in a time-varying magnetic field supports surface currents that shield the H field from the interior of the body. The effect of the conductor on the magnetic field is reminiscent of the EQS situations of Sec. 4.6, where charges distributed themselves on the surface of a conductor in such a way as to shield the electric field out of the material.

We found in Chap. 7 that the EQS model of a perfect conductor described the low-frequency response of systems in the sinusoidal steady state, or the long-time response to a step function drive. We will find in Chap. 10 that the MQS model of a perfect conductor represents the high-frequency sinusoidal steady state response or the short-time response to a step drive.

Usually, we use the model of perfect conductivity to describe bodies of high but finite conductivity. The value of conductivity which justifies use of the perfect conductor model depends on the frequency (or time scale in the case of a transient) as well as the geometry and size, as will be seen in Chap. 10. When the material is cooled to the point where it becomes superconducting, a type I superconductor (for example lead) expels any mangetic field that might have originally been within its interior, while showing zero resistance to currrent flow. Thus, even for dc, the material acts on the magnetic field like a perfect conductor. However, type I materials also act to exclude the flux from the material, so they should be regarded as perfect conductors in which flux cannot be trapped. The newer "high temperature ceramic superconductors," such as Y_1Ba_2Cu_3O_7, show a type II regime. In this class of superconductors, there can be trapped flux if the material is cooled in a dc field. "High temperature superconductors" are those that show a zero resistance at temperatures above that of liquid nitrogen, 77 degrees Kelvin.

As for EQS systems, Faraday's continuity condition, (1.6.12), requires that the tangential E be continuous at a boundary between free space and a conductor. By definition, a stationary perfect conductor cannot have an electric field in its interior. Thus, in MQS as well as EQS systems, there can be no tangential E at the surface of a perfect conductor. But the primary laws determining H in the free space region, Ampère's law with J = 0 and the flux continuity condition, do not involve the electric field. Rather, they involve the magnetic field, or perhaps the vector or scalar potential. Thus, it is desirable to also state the boundary condition in terms of H or \Psi.

\sectnonumBoundary Conditions and Evaluation of Induced Surface Current Density To identify the boundary condition on the magnetic field at the surface of a perfect conductor, observe first that the magnetic flux continuity condition requires that if there is a time-varying flux density n _o H normal to the surface on the free space side, then there must be the same flux density on the conductor side. But this means that there is then a time-varying flux density in the volume of the perfect conductor. Faraday's law, in turn, requires that there be a curl of E in the conductor. For this to be true, E must be finite there, a contradiction of our definition of the perfect conductor. We conclude that there can be no normal component of a time-varying magnetic flux density at a perfectly conducting surface.

Correspondingly, if the H field is the gradient of the scalar potential \Psi, we find that

on the surface of a perfect conductor. This should be contrasted with the boundary condition for an EQS potential which must be constant on the surface of a perfect conductor. This boundary condition can be used to determine the magnetic field distribution in the neighborhood of a perfect conductor. Once this has been done, Ampère's continuity condition, (1.4.16), can be used to find the surface current density that has been induced by the time-varying magnetic field. With n directed from the perfect conductor into the region of free space,

Because there is no time-varying magnetic field in the conductor, only the tangential field intensity on the free space side of the surface is required in this evaluation of the surface current density.

Example 8.4.1. Perfectly Conducting Cylinder in a Uniform Magnetic Field

A perfectly conducting cylinder having radius R and extending to z = \pm is immersed in a uniform time-varying magnetic field. This field is y directed and has intensity H_o at infinity, as shown in Fig. 8.4.1. What is the distribution of H in the neighborhood of the cylinder?

In the free space region around the cylinder, there is no current density. Thus, the field can be written as the gradient of a scalar potential (in two dimensions)

The far field has the potential

The condition \Psi / n = 0 on the surface of the cylinder suggests that the boundary condition at r = R can be satisfied by adding to (5) a dipole solution proportional to sin /r. By inspection,

has the property \Psi / r = 0 at r = R. The magnetic field follows from (6) by taking its negative gradient

The current density induced on the surface of the cylinder, and responsible for generating the magnetic field that excludes the field from the interior of the cylinder, is found by evaluating (3) at r = R.

The field intensity of (7) and this surface current density are shown in Fig. 8.4.2. Note that the polarity of K is such that it gives rise to a magnetic dipole field that tends to buck out the imposed field. Comparison of (7) and the field of a two-dimensional dipole, (8.1.21), shows that the induced moment is id = 2 H_o R².

There is an analogy to steady conduction (H \leftrightarrow J) in the neighborhood of an insulating rod immersed in a conductor carrying a uniform current density. In Demonstration 7.5.2, an electric dipole field also bucked out an imposed uniform field (J) in such a way that there was no normal field on the surface of a cylinder.

Voltage at the Terminals of a Perfectly Conducting Coil

Faraday's law was the underlying reason for the vanishing of the flux density normal to a perfect conductor. By stating this boundary condition in terms of the magnetic field alone, we have been able to formulate the magnetic field of perfect conductors without explicitly solving for the distribution of electric field intensity. It would seem that for the determination of the voltage induced by a time-varying magnetic field at the terminals of the coil, knowledge of the E field would be necessary. In fact, as we now take care to define the circumstances required to make the terminal voltage of a coil a well-defined variable, we shall see that we can put off the detailed determination of E for Chap. 10.

The EMF at point (a) relative to that at point (b) was defined in Sec. 1.6 as the line integral of E ds from (a) to (b). In Sec. 4.1, where the electric field was irrotational, this integral was then defined as the voltage at point (a) relative to (b). We shall continue to use this terminology, which is consistent with that used in circuit theory.

If the voltage is to be a well-defined quantity, independent of the layout of the connecting wires, the terminals of the coil shown in Fig. 8.4.3 must be in a region where the magnetic induction is negligible compared to that in other regions and where, as a result, the electric field is irrotational. To determine the voltage, the integral form of Faraday's law, (1.6.1), is applied to the closed line integral C shown in Fig. 8.4.3.

The contour goes from the terminal at (a) to that at (b) along the coil wire and closes through a path outside the coil. However, we know that E is zero along the perfectly conducting wire. Hence, the entire contribution to the line integral comes from the short path between the terminals. Thus, the left side of (9) reduces to

It follows from Faraday's law, (9), that the terminal voltage is

where is the flux linkage

³ We drop the subscript f on the symbol for flux linkage where there is no chance to mistake it for line charge density.

By definition, the surface S spans the closed contour C. Thus, as shown in Fig. 8.4.3, it has as its edge the perfectly conducting coil, C_1, and the contour used to close the circuit in the region where the terminals are located, C_2. If the magnetic induction is negligible in the latter region, the electric field is irrotational. In that case, the specific contour, C_2, is arbitrary, and the EMF between the terminals becomes the voltage of circuit theory.

Our discussion has emphasized the importance of having the terminals in a region where the magnetic induction, _o H/ t, is negligible. If a time-varying magnetic field is significant in this region, then different arrangements of the leads connecting the terminals to the voltmeter will result in different voltmeter readings. (We will emphasize this point in Sec. 10.1, where we develop an appreciation for the electric field implied by Faraday's law throughout the free space region surrounding the perfect conductors.) However, there remains the task of identifying configurations in which the flux linkage is not appreciably affected by the layout of leads connected to the terminals. In the absence of magnetizable materials, this is generally realized by making coils with many turns that are connected to the outside world through leads arranged to link a minimum of flux. The inset to Fig. 8.4.3 shows an example. The large number of turns assures a magnetic field within the coil that is much larger than that associated with the wires that connect the coil to the terminals. By intertwining these wires, or at least having them close together, the terminal voltage becomes independent of the detailed wire layout.

Demonstration 8.4.1. Surface used to Define the Flux Linkage

The surface S used to define in (12) is often geometrically complex. It is helpful to picture the surface in terms of a model. Shown in Fig. 8.4.4 is a three-turn coil. The surface is filled in by stringing yarn between a vertical rod joining the terminals in the external region and points on the wire. The surface is filled in by connecting points of decreasing altitude on the rod to points of increasing distance along the wire. Note from Fig. 8.4.3 that da and ds are related by the right-hand rule, where the latter is directed along the contour from the positive terminal to the negative one.

Another way of demonstrating the relationship of the surface to the coil geometry takes advantage of the phenomenon familiar from blowing bubbles. A small coil, closed along the external segment between the terminals, can be dipped into materials like soap solution to form a continuous film having the wire as one continuous edge. In fact, if the film is formed from a material that hardens into a plastic sheet, a permanent model for the surface is obtained.

Inductance

When the flux linked by the perfectly conducting coil of Fig. 8.4.3 is due entirely to a current i in the coil itself, is proportional to i, = Li. Thus, the inductance L, defined as

becomes a parameter that is only a function of geometric variables and _o. In this case, the terminal voltage given by (11) assumes a form familiar from circuit theory.

The following example illustrates this rule.

Example 8.4.2. Inductance of a Long Solenoid

In Demonstration 8.2.1, we examined the field of a long N-turn solenoid and found that in the limit where the length d becomes very large, the field intensity along the axis is

where i is the current in each turn.

For an infinitely long solenoid this is not only the field on the axis of symmetry but everywhere inside the solenoid. To see this, observe that a uniform magnetic field intensity satisfies both Ampère's law and the flux continuity condition throughout the free space interior region. (A uniform field is irrotational and solenoidal.) Further, with the field given by (15) inside the coil and taken as zero outside, Ampère's continuity condition (1.4.16) is satisfied at the surface of the coil where K_ = Ni/d. The normal flux continuity condition is automatically satisfied, since there is no flux density normal to the coil surface.

Because the field is uniform over the circular cylindrical cross-section, the magnetic flux _

---

⁴ We use the symbol _ for the flux through one turn of a coil or a loop. passing through one turn of the solenoid is simply the cross-sectional area A of the solenoid multiplied by the flux density _o H.

The flux linkage, defined by (12), is obtained by summing the contributions of all the turns.

Thus, from (13),

For the circular cylindrical solenoid of radius a, A = a². The same arguments used to see that the interior field of a solenoid of circular cross-section is given by (15) show that the solenoid can have an arbitrary cross-sectional geometry and the field will still be given by (15) everywhere inside and be zero outside. Thus, (18) is applicable to a solenoid of arbitrary cross-section.

We have found that many interesting MQS cases can be treated by the use of the scalar potential obeying Laplace's equation. The vector potential, defined by (8.1.1), is necessary when analyzing fields with nonzero curl. There are other cases as well in which its use may be advantageous. The vector potential is the natural variable for evaluating the flux passing through a surface. In view of (8.1.1), integration of the flux density over the open surface S of Fig. 8.6.1 gives

and it follows from Stokes' theorem that this flux is equal to the line integral of A ds around the contour enclosing the surface.

In certain important cases, A has only one component and a vector field is again represented in terms of one scalar function. Two such cases are identified in the following subsections.

Vector Potential for Two-Dimensional Fields

Suppose that the flux density is parallel to the x - y plane and is independent of z. It can then be represented by a vector potential having only a z component.

Note that the divergence of this A is automatically zero and that in Cartesian coordinates, the components of the flux density are given in terms of A_z by

Consider now the evaluation of the net flux of magnetic flux density through a surface S that has length l in the z direction, as shown in Fig. 8.6.2. The points (a) and (b) denote the coordinates of the corners of the contour enclosing S. The contour consists of a pair of parallel straight segments of length l parallel to the z axis, one at the location (a) in the x - y plane and the other at (b), and contours joining (a) and (b) in x - y planes. Contributions to the contour integral, (2), from these latter segments of C are zero, because A is perpendicular to ds. Integration along the z-directed segments amounts to multiplication of A_z evaluated at (a) or (b) by the length of the segment. Thus, (1) becomes

The vector potential at (a) relative to (b) is the net magnetic flux per unit length passing through a surface of unit length in the z direction subtended between the two points and a corresponding pair at unity distance along the z axis. Note that the flux has a sign, relative to the direction of the contour integration, governed by the right-hand rule (Fig. 1.4.1).

\sectnonumVector Potential for Axisymmetric Fields in Spherical Coordinates

---

If the magnetic flux density is invariant with respect to rotation around the z axis, having components in the r and directions only, the vector potential again has a single component.

The net flux through the annular surface "spanned" over the contour shown in Fig. 8.6.3, having constant outer and inner radii denoted by (a) and (b), respectively, is given by the contributions to (2) of the azimuthal segments, A_ multiplied by the circumferences. The contour is closed by adjacent oppositely directed segments joining points (a) and (b) in a plane of constant . Thus, the contributions to the line integral of (2) from these segments cancel, even if A had components in the direction of ds on these segments. Thus, the net flux through the annulus is simply the axisymmetric stream function \Lambda at (a) relative to that at (b).

---

⁵ With A used to represent the velocity distribution of an incompressible fluid, \Lambda_s (or \Lambda_s /2) is called Stokes' stream function.

---

where

Lines of flux density are tangential to the axisymmetric surfaces of constant \Lambda_s. Just as A_z provides a ready visualization of the flux lines in two dimensions, \Lambda_s portrays the axisymmetric flux lines.

Boundary Value Solution by "Inspection"

In two-dimensional configurations, any surface of constant A_z can be replaced by the surface of a perfect conductor. Moreover, in the free space region between conductors, A_z satisfies Laplace's equation. Thus, any two-dimensional configuration from Chaps. 4 and 5 can be replaced by one where the potential lines are field lines. The equipotential (constant ) surfaces of the EQS perfect conductors become the perfectly conducting (constant A_z) surfaces of an MQS system.

Illustration. Field Trapped between Hyperbolic Perfect Conductors

The two-dimensional potential distribution of Example 4.1.1 suggests the vector potential A_z = \Lambda_o xy/a². The lines of magnetic field intensity, which are the surfaces of constant A_z, are shown in Fig. 8.6.4. Here, the surfaces A_z = \pm \Lambda_o are taken as being the surfaces of perfect conductors. Thus, the current density on the surfaces of these conductors are, given by using (4) to determine H and, in turn, (8.4.3) to find K_z. These currents shield the fields from the volume of the perfect conductors. The net flux per unit length passing downward between the upper pair of conductors is [in view of (7)] simply 2\Lambda_o.

This solution is the superposition of the fields of four line currents. Two directed in the +z direction are at infinity in the first and third quadrants, while two in the -z direction are in the second and fourth quadrants.

Example 8.6.1. Field and Inductance of Oppositely Directed Currents in Parallel Perfectly Conducting Cylinders

The cross-section of a pair of parallel perfectly conducting cylinders that extend to \pm in the z direction is shown in Fig. 8.6.5. The conductors have the same geometry as in the EQS case considered in Example 4.6.3. However, they should be regarded as shorted at one end and driven by a current source i at the other. Thus, current in the +z direction in the right conductor is returned in the left conductor. Although the net current in each conductor is given, its distribution on the surface of the conductors is to be determined.

Example 4.6.3 suggests our strategy. Instead of superimposing the potentials of a pair of line charges of opposite sign, we superimpose the A_z of oppositely directed line currents. With r_1 and r_2 the distances from the observer coordinate to the source coordinates, defined in Fig. 8.6.5, it follows from the vector potential for a line current given by (8.1.16) that

With the identification of variables

this expression is identical to that for the antidual EQS configuration, (4.6.18). We can conclude that the line currents should be located at a = (l² - R²)^1/2, and that the constant k used in that deduction (4.6.20) is identified using (10).

Here, the potential U in (4.6.20) is replaced by the flux per unit length \Lambda. Thus, the surfaces of constant A_z are circular cylinders and represent the field lines shown in Fig. 8.6.6.

The inductance per unit length L is now deduced from (11).

In the limit where the conductors represent wires that are thin compared to their spacing, the inductance per unit length of (12) is approximated using (4.6.28).

Once the vector potential has been determined, it is possible to evaluate the distribution of current density on the conductors. Note that the currents tend to concentrate on the inside surfaces of the conductors, where the magnetic field intensity is more intense.

We are one step short of a general relationship between the capacitance per unit length and inductance per unit length of a pair of parallel perfect conductors, regardless of the cross-sectional geometry. With and A_z defined as zero on one of the conductors, evaluated on the other conductor they represent the voltage and the flux linkage per unit length, respectively. Thus, with the understanding that and A_z are evaluated on the second conductor, L = A_z/i, and C = _l /, (4.6.5). Here, i and _l, respectively, are the line current and line charge density that give rise to the same fields as do those sources actually on the surfaces of the conductors. These quantitities are related by (10), so we can conclude that regardless of the cross-sectional geometry, the product of the inductance per unit length and the capacitance per unit length is

where c is the velocity of light (3.1.16).

Note that inductance per unit length of parallel circular conductors given by (12) and the capacitance per unit length for the same conductors under "open circuit" conditions (4.6.27) satisfy the general relation (14).

Method of Images

In the presence of a planar perfect conductor, the zero normal flux condition can be satisfied by symmetrically mounting source distributions on both sides of the plane. This approach is familiar from Sec. 4.7, where the boundary condition required a plane of symmetry on which the tangential electric field was zero. Here we require that the field intensity be tangential to the boundary. For two-dimensional configurations, the analogy between the electric potential and A_z makes the image method of Sec. 4.7 directly applicable here. In both cases, the symmetry plane is one of constant potential ( or A_z).

The most obvious example is an infinitely long line current at a distance d/2 from a perfectly conducting plane. If Fig. 4.7.1 were a picture of line charges rather than point charges, this would be the dual situation. The appropriate image is then an oppositely directed line current located at a distance d/2 to the other side of the perfectly conducting plane. By making a pair of symmetrically located line currents the image for this pair of currents, the boundary condition on yet another plane can be satisfied, the analog to the configuration of Fig. 4.7.3.

The following demonstration is intended to emphasize that the perfectly conducting symmetry plane carries a surface current that terminates the field in the region of interest.

\begindemo1Surface Currents Induced in Ground Plane by Overhead Conductor The metal cylinder mounted over a metal ground plane shown in Fig. 8.6.7 is familiar from Demonstration 4.7.1. Rather than being insulated from the ground plane and driven by a voltage source, this cylinder is shorted to the ground plane at one end and driven by a current source at the other. The height l is small compared to the length, so that the two-dimensional model describes the field distribution in the midregion.

A probe is used to measure the magnetic flux density tangential to the metal ground plane. The distribution of this field, and hence of the surface current density in the adjacent metal, can be determined by recognizing that the ground plane boundary condition of no normal flux density is met by symmetrically mounting a distribution of oppositely directed currents below the metal sheet. This is just what was done in determining the fields for the pair of cylindrical conductors, Fig. 8.6.5. Thus, (9) is the image solution for the region x \geq 0. In terms of x and y,

The flux density tangential to the ground plane at the location y = Y is

Normalized to H_o = i/ a, this distribution is shown as a function of the probe position, Y, in the inset to Fig. 8.6.7.

The role of the surface current density implied by this tangential field is demonstrated by the same probe measurement of the magnetic flux density normal to the conducting sheet. Provided that the frequency is high enough so that the sheet does indeed behave as a perfect conductor, this flux density is small compared to that tangential to the sheet. This is also true at the surface of the cylindrical conductor.

To appreciate the physical origins of this distribution, a dc current source is used in place of the ac source. The distribution of current in the sheet is then dictated by the rules of steady conduction, as enunciated in the first half of Chap. 7. If the sheet is long enough compared to its width, the current is uniformly distributed over the sheet and over the cross-section of the cylinder. By contrast with the high-frequency ac case, where the field is terminated by surface currents in the sheet, the magnetic field now extends below the sheet.

The method of images is not restricted to the two-dimensional situations where there is a convenient analogy between and A_z. In the following example, involving a three-dimensional field, the symmetry conditions are viewed without the aid of the vector potential.

Example 8.6.2. Current Loop above a Perfectly Conducting Plane

A current loop with time-varying current i is mounted a distance h above a perfectly conducting plane, as shown in Fig. 8.6.8. Its axis is inclined at an angle with respect to the normal to the plane. What is the net field produced by the current loop and the currents it induces in the plane?

To satisfy the boundary condition in the plane of the perfectly conducting sheet, an image loop is mounted as shown in Fig. 8.6.9. For each current segment in the actual loop, there is a segment in the image loop giving rise to an oppositely directed vertical component of H. Thus, the net normal flux density in the plane of the perfect conductor is zero.

Two-Dimensional Boundary Value Problems

The vector potential of a two-dimensional field parallel to the x - y plane is z directed and thus only one scalar function describes fully the associated field, as already pointed out earlier. In problems in which currents are confined to the boundaries, the scalar potential can be used as effectively as the vector potential. The lines of steepest descent of the scalar potential are the lines of constant height of the vector potential. When the region of interest contains current distributions, then use of the vector potential is required. We shall consider both situations in the examples to follow.

Example 8.6.3. Inductive Attenuator

The cross-section of two conducting electrodes that extend to infinity in the \pm z directions is shown in Fig. 8.6.10. The time-varying current in the +z direction in the electrode at y = b is returned in the -z direction through the \sqcup-shaped electrode. This current is so rapidly varying that the electrodes behave as though they were perfectly conducting. The gaps of width insulating the electrodes from each other are small compared to the other dimensions of interest. The magnetic flux (per unit length in the z direction) passing through these gaps in the directions shown is defined as \Lambda (t).

The magnetic fields are two dimensional and there are no sources in the region of interest. Thus, _oH can be represented in terms of A_z, which satisfies

The walls are perfectly conducting in the sense that they are modeled as having no normal _oH. This means that A_z is constant on these walls. We define A_z to be zero on the vertical and bottom walls. Thus, A_z must be equal to \Lambda on the upper electrode, so that the flux per unit length in the z direction through the gaps is \Lambda.

The boundary value problem is now formally identical to the EQS capacitive attenuator that was the theme of Sec. 5.5, with the identification of variables

Thus, it follows from (5.5.9) that

The lines of magnetic flux density are the lines of constant A_z. They are the equipotential "lines" of Fig. 5.5.3, shown in Fig. 8.6.10 with arrows added to indicate the field direction. Remember, there is a z-directed surface current density that is proportional to the tangential field intensity. For the flux lines shown, K_z is out of the page in the upper electrode and returned into the page on the side walls and (to an extent determined by b relative to a) on the bottom wall as well.

From the cross-sectional view given by Fig. 8.6.10, the provision for the current through the driven plate at the top to recirculate through the side and bottom plates is not shown. The following demonstration emphasizes the implied current paths at the ends of the configuration.

Demonstration 8.6.2. Inductive Attenuator

One configuration described by Example 8.6.3 is shown in Fig. 8.6.11. Here the upper plate is shorted to the adjacent walls at the near end and driven at the far end through a step-down transformer by a 20 kHz oscillator. The driving voltage v(t) at the far end of the upper plate is measured by means of an oscilloscope. The lower plate is shorted to the side walls at the far end and also connected to these walls at the near end, but in such a way that the induced current i(t) can be measured by means of a current probe.

The walls and upper and lower plates are made from brass or copper. To insure that the resistances of the plate terminations are negligible, they are made from heavy copper wire with the connections soldered. (To make it possible to adjust the spacing b, braided wire is used for the shorts on the lower electrode.)

If the length w of the plates in the z direction is large compared to a and b, H within the volume follows from (20). The surface current density K_z in the lower plate then follows from evaluation of the tangential H on its surface. In turn, the total current follows from integration of K_z over the width, a, of the plate.

With the objective of relating this current to the driving voltage, note that (8.4.11) gives

so that with the driving voltage a sinusoid of magnitude V,

Thus, in terms of the driving voltage, the output current is i_o sin ( t), where it follows from (21) and (23) that

We have found that the output current, normalized to I, has the dependence on spacing between upper and lower plates shown by the inset to Fig. 8.6.11. With the spacing b small compared to a, almost all of the current through the upper plate is returned in the lower one, and the field between is essentially uniform. As the spacing b becomes comparable to the distance a between the side walls, most of the current through the upper electrode is returned in these side walls. Thus, for large b/a, the normalized output current of Fig. 8.6.11 reflects the exponential decay in the -y direction of the field.

Value is added to this demonstration if it is compared to its EQS antidual, Demonstration 5.5.1. For the EQS configuration, the lower plate was properly constrained to essentially the same potential as the walls by connecting it to these side walls through a resistance (which was then used to measure the induced current). Up to frequencies above 100 Hz in the EQS case, this resistance could be as high as that of the oscilloscope (say 1 M\Omega) and still constrain the lower plate to essentially the same zero potential as the walls. In the MQS case, we did not use a resistance to connect the lower plate to the side walls (and hence provide a means of measuring the output current), because that resistance would have had to be extremely low, even at 20 kHz, to prevent flux from leaking through the gaps between the lower plate and the side walls. We used the current probe instead. The effects of finite conductivity in MQS systems are the subject of Chap. 10.

In a final example, we exemplify how the particular and homogeneous solutions are combined to satisfy boundary conditions while also illustrating how the inductance of a distributed winding is determined.

Example 8.6.4. Field and Inductance of Distributed Winding Bounded by Perfect Conductor

The cross-section of a distributed winding of radius a is shown in Fig. 8.6.12. It consists of turns carrying current i in the +z direction at a location (r, ) and returning the current at (r, - ) in the -z direction. The density of turns, each carrying the current i in the +z direction for 0 \leq and in the -z direction for < < 2, is

The total number of wires N in the left-hand half of the coil is

so that the current density is

The windings are very long in the z direction so that effects of the end turns are ignored and the fields taken as independent of z.

The coil is bounded at r = a by a perfect conductor. With the following steps we determine the field distribution throughout the winding and finally, its inductance.

The vector potential is z independent and must satisfy Poisson's equation (8.1.6). In polar coordinates,

First we look for a particular solution. If it is to take a product form, inspection shows that sin is the appropriate dependence. Substitution of an r dependence rⁿ shows that the equation can be satisfied if n = 2. Thus, we have "guessed" a particular solution.

The magnetic flux density normal to the perfectly conducting surface at r = a must be zero, so the total vector potential must be constant there. It follows that one must add a vector potential with no associated current density in the region r < a, a homogeneous solution A_zh. At r = a, the homogeneous solution, A_zh, must be the negative of the particular solution, A_zp.

A linear combination of the two solutions to Laplace's equation that have the same dependence as this condition is

The coefficient D must be zero so that the solution is finite at the origin. The coefficient C is then adjusted to make (31) satisfy the condition of (30). Hence, the sum of the particular and homogeneous solutions is

A graphical representation of what has been accomplished is given in Fig. 8.6.13, where the surfaces of constant A_z (and hence the lines of field intensity) are shown for the particular, homogeneous, and total solutions.

Each turn of the coil links a different magnetic flux. Thus, to determine the total flux linked by the distribution of turns, it is necessary to carry out an integration. To do this, first observe that the flux linked by the turns with their right legs within the area rd dr in the neighborhood of (r, ) and their left legs within a similar area in the neighborhood of (r,- ) is

Here, l is the length of the system in the z direction.

The total flux linked by all of the turns is obtained by integrating over all of the turns.

Substitution for A_z from (32) and use of (26) then gives

where L will be recognized as the inductance.

8.7 Summary

Just as Chap. 4 was initiated with the representation of an irrotational vector field E, this chapter began by focusing on the solenoidal character of the magnetic flux density. Thus, _o H was portrayed as the curl of another vector, the vector potential A.

The determination of the magnetic field intensity, given the current density everywhere, was pursued first using the vector potential. The integration of the vector Poisson's equation for A was the first of many exploitations of analogies between EQS and MQS descriptions. In Cartesian coordinates, the superposition integral for A, (8.1.8) in Table 8.7.1, has components that are analogous to the scalar potential superposition integral, (4.5.3), from Table 4.9.1. Similarly, the two-dimensional superposition integral, (8.1.14), has as its analog (4.5.20) from Table 4.9.l.

Especially if a computer is to be used, it is often most practical to work directly with the magnetic field intensity. The Biot-Savart law, (8.2.7) in Table 8.7.1, gives H directly as an integration over the given distribution of current density.

In many applications, the current distribution can be approximated by piece-wise continuous straight-line segments. In this case, the total field is conveniently represented by the superposition of contributions given by (8.2.22) in Table 8.7.1 due to the individual "sticks."

In regions free of current density, H is not only solenoidal, but also irrotational. Thus, like the electric field intensity of Chap. 4, it can be represented by a scalar potential \Psi , H = - \Psi. The magnetic scalar potential is, in general, discontinuous across a surface carrying a surface current density. It is its normal derivative that is continuous. The scalar potential provides an elegant representation of the fields in free space regions surrounding current loops. The superposition integral, (8.3.12) in Table 8.7.1, is written in terms of the solid angle \Omega.

Through the combined effects of Faraday's law, flux continuity, and Ohm's law, currents are induced in a conductor by a time-varying magnetic field. In a perfect conductor, these currents are on the surface, distributed in such a way as to shield the magnetic field out of the conductor. As a result, the normal component of the magnetic flux density must be zero on the surface of a perfect conductor.

Although useful for representing any solenoidal field, the vector potential is especially useful in the situations summarized by Table 8.7.2. It is especially convenient for describing systems with perfectly conducting boundaries. In two dimensions, the boundary condition on a perfect conductor is satisfied by making the vector potential constant on the boundary. The approaches of Chaps. 4 and 5 apply equally well to solving MQS boundary value problems involving perfect conductors. In fact, the two-dimensional EQS and MQS configurations of perfect conductors in free space, exemplified by the configurations of Figs. 4.7.2 and 8.6.7, were found to be duals. Formally, the solution for H follows from that for E by identifying A_z, /_o _o J_z. However, while the electric field intensity E is perpendicular to the surfaces of constant , H is tangential to the surfaces of constant A_z.

The boundary conditions obeyed by the vector potential at surfaces of discontinuity (containing surface currents) reflect the discontinuity in tangential H field and the continuity of the normal flux density. The vector potential itself must be continuous (a discontinuity of A would imply an infinite H in the surface)

where Ampère's continuity condition

requires that curl A have discontinuous tangential components. The condition that A be continuous, (1), guarantees the continuity of the normal flux density. [According to (1), the integral of A ds around an incremental closed contour lying on one side of the surface is equal to that on the other. Thus, the normal flux which each of these integrals represents, is the same as well.]

In fluid mechanics, the scalar A_z would be called a "stream-function", because in two dimensions, lines of constant vector potential constitute the flux lines. In axisymmetric configurations, the flux lines are lines of constant \Lambda_s, as defined in Table 8.7.2. Of course, a similar representation can be used for any solenoidal vector. For example, an expression for the two-dimensional lines of electric field intensity in a region free of charge density could be obtained by finding a vector potential representation of E. Thus, in these special cases, the vector potential is convenient for plotting any solenoidal field.

The electric potential of EQS systems, evaluated on the surface of a perfectly conducting capacitor electrode, can be used to evaluate the terminal voltage. The vector potential is similarly related to the terminal characteristics of a lumped parameter element, this time an inductor. Indeed, we found in Sec. 8.6 that the flux per unit length linked by a pair of conductors in two dimensions was simply the difference of vector potentials evaluated on the two conductors. In Sec. 8.4, we found that the terminal voltage is the time rate of change of this flux linkage.

The division of the field into particular and homogeneous parts makes possible a number of different approaches to obtaining the total field. The particular part can be obtained using the vector potential, using the Biot-Savart law, or by superimposing the fields of thin coils represented in terms the scalar magnetic potential. The homogeneous solution is both irrotational and solenoidal, so it is possible to use either the vector or the scalar potential to represent this part of the field everywhere. The vector potential helps determine the net flux, as required for calculating the inductance, but is of limited usefulness for three-dimensional configurations. The scalar potential does not directly portray the net flux, but does generally apply to three-dimensional configurations.