## Monopoles

Electromagnetic units and the proof of the Biot-Savart integral using magnetic shells

### Introduction

In 1931, P. A. M. Dirac suggested that the existence of magnetic monopoles would explain the quantization of electric charge, the fact that it appears only in units of the electronic charge e (fractional charges appear in certain theories, but are not observed in nature). This article is not about these monopoles, so if the reader should refer to Jackson if interested in them, or to the article Energy Loss of Charged Particles, where the stopping of Dirac monopoles by matter is discussed. Here we will talk about magnetic charges, or isolated poles, or monopoles as an aid in understanding electricity and magnetism by analogy between electric and magnetic fields. We'll also discuss the systems of electromagnetic units, especially those things not appearing in current texts, and how it all hangs together.

The magnetic field of the earth does not exceed about 1 ørsted, which shows that there is no great amount of unbalanced magnetic charge in its makeup. In fact, isolated magnetic poles have never been observed, and are not accounted for in Maxwell's equations. Magnetic fields are all created by currents, either macroscopic ones or microscopic ones, just as electric fields are created by free or bound charges. Relativity indicates that E and B are the fundamental electromagnetic fields that describe the macroscopic interactions of electromagnetism and are associated in a field tensor that has different expressions in different inertial frames.

The original concept of electromagnetism was that of "action at a distance" evidenced by the mutual forces exerted by charges or currents at different locations on each other. Faraday's concept was that the influence was transmitted through a medium in which the charges or currents were immersed, and this concept was expressed mathematically by Maxwell, who based the theory on fields satisfying certain differential equations. A source charge created a field, which then acted upon the charge that was influenced. He searched in vain for a suitable medium, which became the famous "luminiferous ether" of pre-relativistic days, and for a suitable mechanism for the propagation of the influence. All attempts to detect the medium failed, and it was replaced by the concepts of relativity, in which there are still fields, but immaterial ones. This was at least a partial return to the earlier ideas, and the "reality" of electromagnetic fields can be a subject of discussion. It appears that the potentials are what travels at the speed of light, and the fields are derived from them. Whatever conclusion may be drawn, they do describe the physics very well indeed.

In the early days of Maxwell's theory, the equations D = κE and B = μH were thought of as relating a stress (E or H) to a strain (D or B). These four fields D, E, B and H were called displacement, electric intensity, induction and magnetic intensity, respectively. Their units were the familiar absolute electrostatic (esu) or absolute electromagnetic (emu) units. Practical workers, for their part, had arbitrarily defined units of electrical pressure and resistance, and from them derived the units of charge and current. The unit of electrical pressure was that of the Daniell cell, and that of resistance was a mercury column 1 sq. mm. in cross-section and 1 m long.

It was discovered, by an investigator named Giorgi, that by a certain choice of the units of mass, length and time quantities very close to the practical units could be brought into a consistent and absolute system of units. A consistent system of units is one in which the units are determined by the defining equations for the quantities, and not by arbitrary choice. An absolute system is one in which all units are derived from a set of basic units. The emu and esu were consistent absolute systems based on the centimeter-gram-second (cgs) mechanical units. Giorgi found that simply taking the meter-kilogram-second (mks) as fundamental units, the practical units were welcomed into the system, with slight adjustments. The units of pressure, resistance, current and charge were now called the volt (V), ohm (Ω), ampere (A) and coulomb (C), each honoring a worthy namesake. Each differed only by a factor of a power of 10 from the corresponding emu.

However, it did not stop there. It was decided to add the ampere as a fourth fundamental unit, which permitted the elimination of the speed of light from Maxwell's equations, and also removed fractional exponents in dimensional analysis. Also, the factors of 4π in the inherited relations D = E + 4πP and B = H + 4πM bothered some people, who would prefer D = E + P and B = H + M. This could be done by inserting factors of √4π at strategic places, a step called "rationalization" that was already popular in some circles. All of this, sanctioned by international agreement, established the Giorgi or MKSA system of units, now universally used by electrical engineers. The cost of this tidying-up was to load the system with strange constants and dimensions of no physical significance, and to thoroughly hide the basis of Maxwell's equations, except by appeal to a superseded mechanical theory. In MKSA D, E, B and H all have different dimensions, since instead of D = E + P, B = H + M we have D = εE + P and B = μH + M, which is worse than the original 4π! The factor μ, given a grandiose name, is just a numerical constant 4π x 10-7, but a constant with dimensions (H/m). The electric constant ε = 1/μc2 = 8.84 x 10-12, also has dimensions (F/m). Of course, HF/m2 = (m/s)-2, as it should. There is nothing physical in all this complication, and it confuses those who would try to explain it physically. However, MKSA units work well enough, and Giorgi was tickled by his success.

### Static Fields

It all begins with electric charge, labile and hard to manage in its free state, but an essential part of matter--it holds everything together. The force is proportional to the product of the algebraic amounts of charge and inversely proportional to the square of the distance between them, like Newton's gravity. This is all expressed by F = K(qq'/r2), Coulomb's Law, where K is a constant. (The vector nature is not expressed here for simplicity.) If F is in dyne (gm-cm/s2) and r is in cm, we can define the units of charge consistently by taking K = 1. The units of charge, written [q], are then [q] = gm1/2cm3/2s-1, called the esu of charge or statcoulomb. Coulomb measured charge absolutely in this way, using a torsion balance, which he invented. Later experiments by Cavendish and others showed that the inverse-square dependence was exact to a very great precision. That, and the superposition of electric forces, is the physical content of Coulomb's Law.

The field E is introduced through F = q'E. From this defining equation, [E] = gm1/2cm-1/2s-1, or dyne/esu. We also note that it is esu/cm2. For a point charge q, E = q/r2 = -q grad(1/r) (the vector properties appear in the last expression). This means that E = -grad φ, where φ = q/r is a scalar function called the scalar potential. Then [φ] = gm1/2cm1/2s-1 = erg/esu, as can be verified from the dimensions of erg = dyne-cm and esu. This is given the special name statvolt, which we shall soon relate to the familiar practical volt. Then, [E] = statvolt/cm as well as dyne/esu and esu/cm2. Each of these expressions suggests a different interpretation of E, and is well worth remembering.

If we generalize to a charge distribution specified by a charge density ρ, esu/cm3, then φ = ∫ρdV/R, where R is the distance from the location P(x,y,z) where φ is desired, called the field point, to the location of the charge density Q(x',y',z'), called the source point. R2 = (x - x')2 + (y - y')2 + (z - z')2, and grad R = -grad' R, where grad is taken with respect to P, and grad' with respect to Q. These relations will be much used below. Now E = - grad φ, so the determination of the field is reduced to differentiating the result of the integral. In vector calculus, this means that curl E = 0, and div E = 4πρ, first-order differential equations that are equivalent to Coulomb's Law.

Now we turn to magnetism. It originally involved only permanent and induced magnetism, which seemed like electricity, except for the absence of magnetic charge corresponding to electric charge. Coulomb performed experiments with long needle-shaped magnets that appeared to have equal and opposite poles at the two ends, and showed that their interaction was described by F = mm'/r2. These experiments were harder in the sense that the magnetic poles could not be isolated, but easier in that the magnetic poles didn't leak off or attract charges from the air. He measured m absolutely, and [m] = gm1/2cm3/2s-1, just like esu, and was called emu of magnetic charge. There is no name corresponding to coulomb, since magnetic poles are not included among MKSA units. We can proceed to define H by F = m'H, H = - grad ψ and ψ = m/r, just as with electricity. The unit of ψ is the gilbert, and the unit of H, unlike E, gets its own name, the ørsted, which is the same as gilbert/cm, emu/cm2 or dyne/emu. In cm, g and s, H has the same dimensions of E, of course, and ψ the same dimensions as φ. All of this is very easy to understand on the basis of the analogy with the electric field.

In vector calculus, the magnetic field satisfies div H = 4πρ', and curl H, where ρ' is the magnetic charge density. A permanent bar magnet can be considered to have a nonzero ρ' in the regions near its ends. This charge can be expressed in terms of the magnetization by ρ' = - div M, so that div (H + 4πM) = 0, since there is no other magnetic charge around. A new vector, B = H + 4πM, called the magnetic induction (or flux density) then has zero divergence, and this is one of Maxwell's equations, which thus explicitly provide that ρ' = 0: there is no magnetic free charge in nature.

### Gauss's Experiment

Gauss devised a simple experiment for measuring the magnetic field and the pole strengths of a bar magnet absolutely. It is worth doing this experiment at home, even though the results will not be accurate unless great care is taken. All you need is a small bar magnet, a compass (an ordinary lensatic compass is excellent), some cotton thread, a scale, a millimeter scale, and a watch with a second hand. The scale measures grams, the millimeter scale centimeters, and the watch seconds. These are the absolute units in terms of which we shall find the horizontal component of the earth's field, and the magnetic moment of the magnet. The design of the experiment is shown in the figure. The earth's field H exerts forces ±mH on the magnet's poles, which is a torque L = mHd sin θ = μH sin θ. The equation of motion is L = -Iθ", so for small oscillations θ" + (μH/I)θ" = 0, which gives harmonic oscillations of frequency f = (1/2π)√(μH/I). Hence μH = 4π2If2. The field on the axis of the magnet at a distance r from its center is Hr = 2μ/r3, directed along the axis. We arrange this to be perpendicular to the horizontal component of the earth's field H and note the angle α made by the resultant. Then, tan α = H/Hr, or H/μ = 2 tan α/r3. Since we know the product and the quotient of H and μ, we can find each separately.

The magnet I used was 0.4 x 0.5 x 2.5 cm in size, and weighed 5.00 grams (18 such magnets weighed 90 g). Its moment of inertia about an axis through the center, the way I suspended it, was I = (5.00/12)(2.52 + 0.42) = 2.67 gm-cm2. It was suspended by a long thread (about 1 m long) and its frequency of torsional oscillation measured by timing 25.5 oscillations in one minute, or f = 0.425 Hz. Next, the compass was opened and allowed to stabilize in the direction of the earth's field. The magnet, held perpendicular to the direction of the field, was then brought closer to the magnet until the needle deflected by a reasonable amount, here 35°. At this point the distance from the center of the magnet to the compass pivot was 11.0 cm. The product μH = 4π2If2 = 19.02 gm-cm2/s2, and the quotient H/μ = 2 tan α/r3 = 0.002146 cm-3. From these figures, H = 0.20 oe and μ = 95.1 emu-cm or erg/gauss. The value for the horizontal component of the field is remarkably close to the correct value (as measured with a tangent galvanometer), and the magnetic moment seems reasonable. From it we can calculate the magnetic field in the vicinity of the magnet. Incidentally, the local dip of the field is about 68°.

This is such a simple and enlightening experiment, depending only on easy theory, that I find it remarkable that I have never run across any recommendation of it to young learners, school pupils, or physics students at university. My magnet came from a "Things of Science" unit (No. 241, 1960, by Science Service, Inc.) that included 16 experiments, none of them Gauss's, or any of which involved quantitative measurements. Mathematics is avoided to keep the young in the dark until it is too late for them to learn it. The statement about magnetism in its instructions: "we are not able to understand what it is or how it operates" will certainly be true for those who restrict themselves to such limited investigations!

Gauss at Königsberg and Weber at Göttingen made ingenious and accurate measurements of electric and magnetic quantities, and the relations between them, creating the standards that were necessary for the success of Maxwell's theories and the discovery of electromagnetic waves. Many others helped, but we have no room to mention any details here.

### Connections between Electricity and Magnetism

Ørsted muddied the waters when a student of his convinced him in 1820 that an electric current was accompanied by an encircling magnetic field. He made the fact, and himself, known, without understanding much about it. This was the first definite connection between electricity and magnetism. Ampère, however, discovered that the interaction of current elements was much more puzzling than the simple interaction of charges was. He finally arrived at a suitable expression for this interaction, that is largely forgotten today, but is expressed by the vector relation curl H = 4πJ', where the magnitude of J' is current per unit volume, or current density. By Stokes's Theorem, we can put this in the form ∫H·ds = 4πI', where I' is the total current flowing through the area encircled by the path of integration. If we keep the area to the left as we go around the curve, I' is directed into the page. This is called Ampère's Law in his honor. Coulomb, Ampère and Faraday are well worth honoring. It is good that Joseph Henry was honored as well. Isolated in the woods at Albany and Princeton, he discovered the "inductive kick" of a coil whose current was interrupted at about the same time that Faraday was performing his experiments on induction, the only American to contribute anything useful to the field until Willard Gibbs.

We have used the symbol J' to represent the current density found from measuring the magnetic field absolutely. It is the emu of charge, q', per unit area and time, and this equation defines what we mean by the emu of charge, also called the abcoulomb. On the other hand, we already have an absolute unit of charge, the esu or statcoulomb that we can find independently of magnetic fields. If we let some measured current density J in esu per unit area and time create a magnetic field, and then measure that field absolutely, we find that if J and J' are numerically equal, the magnetic field produced by J is smaller by a factor of (1/c). That is, curl H = (4π/c)J. This is the way Ampère's Law appears in our Maxwell's Equations, and this way of handling the units is called the Gaussian system. The factors of c appear explicitly in the equations when it is necessary because we measure electrical quantities in esu, and magnetic quantities in emu. The relation between charge in esu and emu is q = cq', or q/q' = c. The ratio of esu to emu, then, is equal to the speed of light. Here, q and q' are the measures of the same charge in the two systems. Since the number in emu is smaller, its unit is larger. In the MKSA system, the coulomb was taken to be 1/10 the abcoulomb, or Q = 10q, I = 10i, where the capital letters are the measure in MKSA. We see that 1 C = (c/10) statcoulomb = 3 x 109 esu. The electronic charge, e = 1.602 x 10-19 C = 4.81 x 10-10 esu, once a familiar value.

Our conversions will all assume c = 3 x 1010 cm/s, and c2 = 9 x 1020, in place of the more accurate mantissas 2.9979 and 8.9874. An accurate conversion, when needed, can always be made by inserting the more precise figures instead of 3 or 9.

In emu, Coulomb's Law becomes F = qq'/c2r2, where q and q' are measured in emu. This is merely the use of a different constant K = 1/2, which serves to redefine the units of charge. Don't confuse this electric charge q' with the magnetic charge m, which also obeys Coulomb's Law. Now [q'] = gm1/2cm5/2s, so emu and esu have different dimensions, which should be obvious, since they differ by a factor c that has dimensions. The unit of the potential φ' = q'/r = q/cr, in erg/emu, is called the abvolt, and it is clear that the statvolt is c abvolts. The ergs are the same, and the charges differ by a factor c. Now, an actual volt is 1 J/1 C, or 107 ergs /0.1 abcoulomb = 108 abvolts. The abvolt is a tiny jolt, 1 x 10-8 V. Since c abvolts is one statvolt, then 300 V = 1 statvolt. Now we know what the measure of a charge, current or voltage in MKSA is in emu or esu. Don't try to convert the dimensions, however, since the MKSA dimensions are different from the emu and esu dimensions, and dimensional conversion is a real mess, fortunately an unnecessary one.

We still have to convert ohms, henries and farads, and the easy way to do this is to use the defining equations involving energy or power, which are the same in esu and emu. For example, P = RI2. Then, P = (c2R)(I/c)2. If the first is the definition in esu, then the second is the definition in emu (I/c is the current in emu), so that R' = c2R = 9 x 1020R, where R' is in abohms and R is in statohms. The abohm must be a very small unit compared to the statohm. We can find out about the ohm by 1 Ω = 1 V/1 A, and then expressing V and A in terms of emu or esu, which we have just done. For example, 1 Ω = (1/300) statvolt / (c/10) statampere = (1/30c) statohm, or 1 statohm = 9 x 1011 ohm. Similarly, 1 abohm = 10-9 ohm. The statohm is huge, the abohm minuscule. It is easy to see why engineers prefer the ohm.

For capacitance and inductance, the equations to use are U = Q2/2C and U = LI2/2, where C is the capacitance and L is the inductance. The work is left to the reader, who will find that 1 statfarad = c2 abfarad, and c2 stathenry = 1 abhenry. To relate them to MKSA, use Q = CV and V = L(dI/dt), just as we did V = RI in the preceding paragraph. Then, 1 H = 109 abhenry or 1/(9 x 1011) esu, and 1 F = 10-9 abfarad = 9 x 1011 statfarad. The statfarad has the dimensions of length, cm, and is a quite practical size, about 1 μF. The abhenry also has dimensions of length, and is rather small, about 0.001 μH. The stathenry and abfarad, both with dimensions s2/cm, are relatively unwieldy. The capacity of the earth, as a conducting sphere, is only 7 x 10-13 abfarad, or 0.7 picoabfarad.

We have now related all the electrical units between esu and emu, and between either system and MKSA. This was not such a big job after all, and we learned a lot along the way. It is easy to make mistakes and become confused (as I am very well aware) but if you have some idea of the relative sizes and check results for reasonableness and symmetry, it will all come out properly. Always ask yourself: "Why does this relation exist, and what does it mean?" and don't just take it for granted. Consider each of Maxwell's equations, and find as many examples and applications of each as you can.

We have got slightly ahead of ourselves in mentioning inductance above, but the reader is probably already quite familiar with it. Faraday discovered in 1830 that an electromotive force, the line integral of the electric field around a closed path, was induced in a conducting circuit when the magnetic flux linked with it changed. Magnetic flux Φ is the surface integral of H, when H is acting as the magnetic induction in free space. When it does so, its unit is called gauss instead of ørsted, but the units and dimensions are exactly the same. The amount of the flux is measured in maxwell or in lines. There aren't any actual lines of induction, but the term is colorful and traditional. Therefore, a gauss is one line per cm2. A change in flux of one line per second produces an emf ("voltage") of 1 abvolt in the circuit it links, and the defining equation is ∫ E·ds = -dΦ/dt. The minus sign results when the usual conventions of positive directions are observed. Recall that an induced current will move in the direction that opposes the change in flux (Lenz's Law) to determine the direction of the induced voltage in any practical case. Faraday conceived that this induced electric field existed in space, independently of any conductors, and so it does. In differential form, Faraday's Law is curl E' = -∂B'/∂t, where the primes remind us that each quantity is in emu. This equation gives us another connection between electricity and magnetism, and it must be consistent with the first, which was Ampère's Law. If we insist on measuring E in esu, then it is necessary to change this defining relation to curl E = -(1/c)∂B/∂t, where the speed of light appears once again. This appearance should not be mysterious, but expected. The factors of c in Maxwell's equations express the weakness of the magnetic interaction relative to the electric interaction.

The induction of a voltage in moving conductors is actually something different, a relativistic effect. The charges experience an electric field in their moving coordinate system, while the field in a nonmoving (with respect to the sources of the field) system is purely magnetic. This is of order v/c just like the usual magnetic phenomena. However, it must also be consistent with Faraday's Law, and can be brought into the same form by the fiction of "cutting lines of force" that is so popular in engineering. We get a voltage no matter how the flux linked with a circuit is changed, with sliding contacts as well as by distorting the circuit, or by changing the field. Since we restrict ourselves to nonmoving bodies here, we do not get into any trouble with such considerations.

Since 1 line per second corresponds to 10-8 V, then 108 lines per second will correspond to 1 V. This amount of flux is called a weber. Hence B can also be expressed as weber/m2, or tesla. A gauss provides 10,000 lines per square meter, so that 10,000 gauss = 1 tesla. In MKSA, then, Faraday's Law is curl E = -∂B/∂t, and we see why the factor c has disappeared. Actually, it is concealed in the relation between the units. We now have displayed all of Maxwell's equations, except for the term (1/c)∂D/∂t in Ampère's Law, which is Maxwell's celebrated displacement current that makes electromagnetic radiation possible. It was proposed theoretically long before it was actually measured, but it is as real as green apples. Here, E appears wearing its displacement hat, in esu/cm2, and receives the nickname of D. When there are material media present, the polarization P and magnetization M can be taken into account (approximately) by making them parts of D and B, respectively.

### Magnetic Shells and The Biot-Savart Law We have not mentioned yet any way to find the magnetic field of a current as convenient as taking the gradient of a scalar potential that is an integral over the sources of the field, which exists for the electrostatic and magnetostatic fields. Ampère's Law, a force-at-a-distance expression for the force exerted on one current element by another, was cast into the field picture as the Biot-Savart Law, shown in the diagram. Here, the field H at a point P was represented as an integral around a closed circuit, and then the force on a current element was expressed as the cross product of the vector current element and the magnetic field B = H. This split the interaction into two parts, each complicated enough because of the vector products. One cannot actually talk about the field due to an isolated current element, or the force on one, because the equation of continuity of charge is not satisfied. These formulas are guaranteed correct only when integrated over a closed circuit.

Most texts now introduce the Biot-Savart Law as fallen providentially from the heavens, or as the result of an experimental investigation. It is neither, but a good case can be made for it anyway, and, with the force expression, serves the same purpose as Coulomb's Law does in electrostatics. It can be derived, with some difficulty, from Ampère's original formula, but much more easily by a method coming from magnetostatics.

A dipole is an assembly of two equal and opposite charges, ±q, separated by a distance d directed from the negative charge to the positive one. The dipole moment is p = qd, with dimensions esu-cm or gm1/2cm5/2s-1. The potential of this dipole is found by adding the potentials due to each charge, which is approximately φ(P) = qd·grad'(1/R), becoming more and more exact as R becomes much greater than d. We can also write φ = -p·grad(1/R) = p·r/r3, where now r is the vector from the center of the dipole to the field point. For a magnetic dipole, ψ takes the same form, and pμ, where μ is the magnetic dipole moment, dimensions the same as those of the electric dipole moment. We note that these dimensions are the same as those of abampere-cm2. The familiar dipole fields can be found from ψ: Hr = 2μ cos θ/r3, Hθ = μ sin θ/r3, where θ is the angle between the radius vector and the axis of the dipole.

We can easily conceive of volume distributions of dipole moment, where P is the electric dipole moment per unit volume, and M is the magnetic dipole moment per unit volume. These vector fields have the same dimensions as E and H, respectively. In either case, they contribute a volume bound charge density of the negative divergence of either vector, which affects the fields in the same way as free charge, but is not free to move. We can also conceive of a two-dimensional sheet of dipoles, or shell, with a surface dipole moment density normal to the surface. In the magnetic case, if τ is the magnitude of the dipole moment per unit area, then τdS is the surface dipole density. The surface towards which dS points is called the positive surface, and the other is the negative surface. In a hydrodynamic analogy, such a surface sucks in flux uniformly on the (-) surface, and expels it uniformly on the (+) surface.

The potential due to such a shell can be calculated by integration: ψ = -τ∫ dS cos θ/r2, where θ is the angle between the positive normal to the surface and the radius vector from the observation point. The integral is the solid angle Ω subtended by the shell at the point P, taken as negative when we are looking at the (-) side of the shell. Hence, ψ = τΩ. The solid angle is determined only by the boundary of the shell. What it does otherwise is immaterial. We see that a closed shell gives ψ = 0, contributing nothing to the field, which proves the last statement. (Consider two shells with the same boundary.) Inside a closed shell, ψ is lower by 4πτ, if the positive side is facing outwards.

To find the field due to a shell, suppose the observation point P is given a virtual displacement δa. This changes ψ by -H·deltaa and Ω by δΩ: -H·δa = τδΩ. To find the change in Ω simply keep the field point fixed and displace the shell rigidly by -δa. To each arc element of the boundary curve ds corresponds an area increment dS = -δa x ds, so that the total change in solid angle is ∫ -δa· ds x r/r3, where we have interchanged dot and cross, as is allowable in a scalar triple product. The result is true for any δa, so H = τ∫ ds x r/r3, where the integral is extended around the boundary of the shell. This is the Biot-Savart integral, where the sign should be taken to be consistent with the direction of r and the positive side of the shell.

The field of a magnetic shell satisfies ∫ H·ds = 0 so long as the path of integration does not cross the shell. If we approach the shell from the (-) side, ψ → -2πτ, while on the other side ψ → 2πτ. Therefore, the value of ψ jumps by 4πτ when we cross the shell from the (-) side to the (+) side, making ∫ H·ds = 4πτ. This means that the field of the shell obeys exactly the same differential field equations as the field due to a current, and its sources, defined by curl and div, are exactly the same, provided τ = i' in emu, or i/c in Gaussian units. This means that the fields are the same. We can, therefore, replace a closed circuit in the form of a simple closed curve carrying a current i statamperes by a uniform magnetic shell of strength τ = i/c abamperes. This means that a circuit of area dS carrying a current i statamperes acts like a magnetic dipole of moment μ = (i/c)dS. Furthermore, the magnetic field H is given by the Biot-Savart integral. We have proved a real result using the imaginary concept of the magnetic shell, a process with definite beauty. Incidentally, electric shells are quite common: every metal is surrounded by one.

Consider H on the axis of a circular loop of radius a. The solid angle subtended by the loop at an axial distance z from its plane is Ω = ∫ 2πr2 sin θ dθ/r2 = 2π ∫(0,θ') sin θ dθ = 2π(1 - cos θ'), where θ' is the angle subtended by a radius of the loop at the observation point. Then, dΩ/dz = -2πa2(z2 + a2)-3/2, which gives the axial field at any point z on multiplying by -(i/c). At the center of the loop, z = 0 and H = 2πi/ca. At distances z >> a, H = 2(πa2i/c)z-3.

Consider two circular loops of radius a, placed coaxially and a distance a apart. At the point halfway between them, the first derivatives of H are equal and opposite, as indeed are all derivatives of odd order. The second derivative of H contains the factor (a2 - 4z2), so it is zero at this point, for both loops. Since the first nonvanishing derivative of the field is the fourth, there is a considerable region near the center of the loops where the axial field is practically constant. The field at the center is (4/5)3/2(4πi/ca) ≈ 10.829 (i/ac). If there are N turns in each coil, replace i by Ni. This arrangement is known as Helmholtz Coils, after their inventor.

We can go further with the Biot-Savart Law and cast it into the form B = curl A, where A(P) = ∫ J/R, an integral that looks very much like the one for the scalar potential. The vector field A is called the vector potential, and can be made to satisfy certain other conditions that do not affect the field derived from it but help in our algebra. More on this is mentioned in Electromagnetic Waves, but this is enough for the reader to work out the dimensions of A.

### References

This only lists my standard references, which will guide you to more extensive bibliographies.

M. Abraham and R. Becker, Classical Electricity and Magnetism 2nd ed.(New York: Hafner, 1949). This text contains a good introduction to vector calculus, using hydrodynamic analogies. A later edition has been reprinted by Dover.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (New York: John Wiley & Sons, 1975). Chapter 4. The third edition, I understand, uses Giorgi units, a regrettable concession to fashion. There is an excellent bibliography.

S. Ramo, J. R. Whinnery and T. van Duzer, Fields and Waves in Communication Electronics (New York: John Wiley & Sons, 1965). An unsurpassed engineering introduction to the subject, which, naturally, uses Giorgi units.

J. C. Maxwell, A Treatise on Electricty and Magnetism, 3rd ed., 2 Vols. (New York: Dover, 1954). Reprint of the 1891 edition.