Sources of magnetism

The previous lesson defined the magnetic field entirely by what it does to a moving charge, and never said what produces one.

Filling that gap needs an experimental input, exactly as Coulomb's law was an experimental input for electrostatics. It arrived within weeks of Oersted's demonstration: Jean-Baptiste Biot and Félix Savart measured the field around a long straight wire in October 1820 and found it fell off as the inverse of the distance, and Ampère worked out the general rule behind it over the following year. This lesson assumes the force law $F = q v B \sin θ$ and the right hand rule from the previous one.

The contribution of a current element

The rule Biot, Savart and Ampère arrived at splits the wire into short pieces and adds their contributions, in the same spirit as cutting a charge distribution into point charges. A piece of length $d l$ carrying current $I$ contributes, at a point a distance $r$ away,

d B = \frac{μ_{0}}{4 π} \frac{I d l \sin θ}{r^{2}}

where $θ$ is the angle between the current direction and the line to the point, and the direction of $d B$ is perpendicular to both, by the right hand rule. This is the Biot-Savart law.

Compare it with Coulomb's law term by term, because the differences are the whole character of magnetism. Both fall as the inverse square. Both have a constant out front. But Coulomb's field points along the line from the source to the point, while this one points at right angles to that line, and Coulomb's source is a scalar while this one is a piece of a vector, so the $\sin θ$ kills the contribution of any element pointing straight at the field point. The circulation that puzzled Oersted's audience is built into the cross product.

The constant $μ_{0}$ , the permeability of free space, was for seventy years exactly $4 π \times 10^{- 7}$ T m A⁻¹ by definition, because the ampere was defined by fixing it. Since the 2019 redefinition of SI the ampere is fixed through the elementary charge instead, so $μ_{0}$ is a measured quantity, currently $1.25663706212 \times 10^{- 6}$ , which differs from $4 π \times 10^{- 7}$ by about five parts in $10^{10}$ . For every calculation in this course the old exact value is used and the difference is invisible.

One caution: a current element on its own is not a physical object. Current has to flow in a complete circuit, so the Biot-Savart law is a rule for a term in an integral, never for a measurable field by itself.

The straight wire and the loop

Integrate the law along an infinite straight wire and the result, which is what Biot and Savart measured, is

B = \frac{μ_{0} I}{2 π r}

circling the wire, with the right hand rule giving the sense: thumb along the current, fingers curl the way the field goes. One ampere at one metre gives $2 \times 10^{- 7}$ T, which is a very small field, and shows how weak magnetism from ordinary currents is. Ten amperes at 2 cm, on the other hand, gives $1.0 \times 10^{- 4}$ T, twice the Earth's field, which is why a compass held near a household cable misbehaves.

Integrating around a circular loop of radius $R$ gives, at the centre,

B = \frac{μ_{0} I}{2 R}

and on the axis at distance $x$ , $B = μ_{0} I R^{2} / [2 (R^{2} + x^{2})^{3 / 2}]$ . Far away this becomes $μ_{0} μ / (2 π x^{3})$ with $μ = I A$ the magnetic moment of the previous lesson: an inverse cube, exactly like an electric dipole. A current loop seen from a distance is a magnetic dipole and nothing else, which is why a compass needle and a coil are interchangeable.

Fields from separate wires superpose, exactly as electric fields do, so a problem with several wires is a matter of getting each direction right and then adding.

Example. Two long parallel wires 10 cm apart carry 8.0 A and 12 A in the same direction. What is the field at the midpoint between them?

Each wire is 5.0 cm from the midpoint. From the first, $B_{1} = μ_{0} I / (2 π r) = (4 π \times 10^{- 7}) (8.0) / (2 π \times 0.050) = 3.2 \times 10^{- 5}$ T; from the second, $B_{2} = 4.8 \times 10^{- 5}$ T. Because the currents run the same way, their fields circle the two wires in the same rotational sense, so at a point between them the two fields point in opposite directions and subtract. The net field is $1.6 \times 10^{- 5}$ T, directed as the stronger wire dictates.

Now you. Two long parallel wires 8.0 cm apart carry 6.0 A and 10 A in opposite directions. What is the field at the midpoint?

Answer

Each wire is 4.0 cm away, giving $B_{1} = (4 π \times 10^{- 7}) (6.0) / (2 π \times 0.040) = 3.0 \times 10^{- 5}$ T and $B_{2} = 5.0 \times 10^{- 5}$ T. With the currents opposed the two fields point the same way between the wires, so they add: $8.0 \times 10^{- 5}$ T.

Two wires, and the old ampere

Now the experiment that forced this lesson. Take two long parallel wires a distance $d$ apart carrying $I_{1}$ and $I_{2}$ . The first produces a field $μ_{0} I_{1} / (2 π d)$ at the second, perpendicular to it, and the second, of length $L$ , feels $F = I_{2} L B$ from the previous lesson. So

\frac{F}{L} = \frac{μ_{0} I_{1} I_{2}}{2 π d}

Working through the directions with two right hand rules gives attraction for parallel currents and repulsion for antiparallel, which is the opposite of the intuition trained on magnets and charges, where like repels like.

Two wires a metre apart each carrying one ampere attract with $2 \times 10^{- 7}$ N per metre. That number is not a coincidence: from 1948 to 2019 it was the definition of the ampere, which is why $μ_{0}$ was exactly $4 π \times 10^{- 7}$ during that period. Defining a unit of current by a mechanical force is a striking choice, and it was made because force and length could be measured far more accurately than any property of a flowing charge. The modern definition fixes $e$ instead and lets the force be predicted.

Example. Two long parallel wires 25 cm apart carry 40 A and 60 A in the same direction. What force acts on a 3.0 m length of one of them?

$F / L = μ_{0} I_{1} I_{2} / (2 π d) = (4 π \times 10^{- 7}) (40) (60) / (2 π \times 0.25) = 1.92 \times 10^{- 3}$ N/m, so over 3.0 m the force is $5.76 \times 10^{- 3}$ N, attractive since the currents are parallel. Under six millinewtons from 40 and 60 amperes at a quarter of a metre: the magnetic force between currents is genuinely feeble, and everything practical about magnetism comes from stacking many turns or using iron.

Now you. Two long parallel wires 15 cm apart carry 30 A and 50 A in opposite directions. What force acts on a 2.0 m length of one of them?

Answer

$F / L = (4 π \times 10^{- 7}) (30) (50) / (2 π \times 0.15) = 2.0 \times 10^{- 3}$ N/m, so over 2.0 m the force is $4.0 \times 10^{- 3}$ N, repulsive because the currents oppose.

Ampère's law

The Biot-Savart law is to magnetism what the integral of Coulomb's law was to electrostatics: complete, and painful. The relief comes in the same shape as before, as a statement about a whole path rather than about individual sources.

Take the straight wire result and walk once round a circle of radius $r$ centred on the wire, in the direction the field points. The field has constant magnitude $μ_{0} I / (2 π r)$ everywhere on the circle and is everywhere tangent to it, so the product of field and path length is

B \times 2 π r = \frac{μ_{0} I}{2 π r} \times 2 π r = μ_{0} I

and the radius cancels, exactly as it did for flux through a sphere. Generalising, by the same style of argument that turned a sphere into any closed surface, gives Ampère's law:

\oint B \cos φ d l = μ_{0} I_{enc}

The sum of the field component along a closed path, all the way round, equals $μ_{0}$ times the current passing through any surface bounded by that path. Currents outside the loop contribute to $B$ at every point of it and contribute nothing to the total, exactly as external charges did for flux.

The quantity on the left is the circulation of the field, and it is the deep contrast with electrostatics. The circulation of the electrostatic field is zero round every closed path, which is the loop rule of the previous lesson and the reason a potential exists. The circulation of a magnetic field is not zero, so there is no magnetic potential. Electric field lines start and end on charges; magnetic field lines close on themselves.

Ampère's law, like Gauss's, is always true and useful only where symmetry lets $B$ come out of the integral. Two cases matter.

The solenoid. A long coil of $n$ turns per metre carrying $I$ has, inside it and far from the ends, a uniform field along the axis and almost nothing outside. Take a rectangular path with one side of length $L$ running along the axis inside, and the opposite side outside where the field is negligible; the two short sides are perpendicular to the field and contribute nothing. The path encloses $n L I$ of current, so $B L = μ_{0} n L I$ and

B = μ_{0} n I

with no reference to the coil's radius. A solenoid is the magnetic equivalent of the parallel plate capacitor: a way of making a uniform field in a defined region.

The toroid. Bend the solenoid into a doughnut and the field is entirely confined inside it, at $B = μ_{0} N I / (2 π r)$ for $N$ total turns. Nothing leaks, which is why transformer cores and fusion confinement devices are toroidal.

The solenoid formula also says why strong fields are hard. To reach 1 T needs $n I = 1 / μ_{0} = 7.96 \times 10^{5}$ ampere turns per metre, so a coil wound at 1000 turns per metre needs 796 A, and the $I^{2} R$ heating in ordinary copper at that current would destroy the coil in seconds. Every field above about 2 T is made either with superconducting wire, which has no resistance to heat it, or in a pulse short enough that the coil has not yet melted.

Example. A solenoid 40 cm long is wound with 800 turns and carries 2.5 A. What is the field inside it?

The turn density is $n = 800 / 0.40 = 2000$ turns per metre. Then $B = μ_{0} n I = (4 π \times 10^{- 7}) (2000) (2.5) = 6.28 \times 10^{- 3}$ T, about 6.3 mT, or 125 times the Earth's field.

Now you. A solenoid 25 cm long is wound with 1500 turns and carries 1.8 A. What is the field inside it?

Answer

$n = 1500 / 0.25 = 6000$ turns per metre, so $B = (4 π \times 10^{- 7}) (6000) (1.8) = 1.36 \times 10^{- 2}$ T, about 13.6 mT.

The law with nothing on the right

Gauss's law for the electric field said the flux out of a closed surface counts the charge inside. Ask the same question of the magnetic field and the answer is

\oint B \cos θ d A = 0

for every closed surface, always. This is the second of the four equations that will eventually be Maxwell's, and it is the only one with nothing on its right hand side.

What it asserts is that there is no magnetic charge. Cut a bar magnet in half and you do not get a north pole and a south pole; you get two shorter magnets, each with both. Every magnetic field line that enters a region leaves it, because field lines close on themselves rather than terminating.

There is no explanation for this within classical electromagnetism, which is worth saying plainly. Nothing in the theory forbids a magnetic monopole; the equations would accommodate one with a symmetrical extra term, and would arguably look better for it. Paul Dirac showed in 1931 that the existence of even a single monopole anywhere in the universe would force electric charge to be quantised, which is otherwise an unexplained fact. Searches have been thorough and unsuccessful: through moon rock, deep-sea sediment, accelerator debris and cosmic rays. Blas Cabrera's superconducting loop recorded exactly one candidate event, on 14 February 1982, of precisely the expected size, and in the decades of running since then, by that detector and much larger ones, nothing similar has ever appeared. The result is a bound rather than a discovery.

Where iron comes in

Ampère's proposal was that all magnetism is current, and it takes some believing when a bar magnet has no visible circuit in it. The currents are atomic: an electron has an intrinsic magnetic moment, close to one Bohr magneton, $μ_{B} = 9.274 \times 10^{- 24}$ J/T, and orbital motion contributes as well.

In most materials these moments point in random directions and cancel. Applying a field produces a small alignment, giving weak paramagnetism, or a small opposing response, giving weaker diamagnetism, and in both cases the effect disappears when the field is removed. In iron, cobalt, nickel and a few alloys, a quantum mechanical exchange interaction makes neighbouring moments prefer to be parallel, so they align spontaneously within regions called domains, and applying a modest field grows the favourably oriented domains at the expense of the others. That is ferromagnetism, and it multiplies an applied field by a factor of hundreds or thousands, which is why every motor and transformer has an iron core.

The magnitude is checkable. Iron has a density of 7874 kg/m³ and a molar mass of 55.845 g/mol, so it holds $8.49 \times 10^{28}$ atoms per cubic metre, and each contributes about 2.2 Bohr magnetons. If every one of them lined up, the magnetisation would be $M = (8.49 \times 10^{28}) (2.2) (9.274 \times 10^{- 24}) = 1.73 \times 10^{6}$ A/m, and the field it produces is $μ_{0} M = 2.18$ T. The measured saturation field of iron is 2.16 T. A count of atoms and a per-atom moment reproduce a bulk magnetic property to one per cent, which is about as direct a confirmation of Ampère's idea as could be asked for.

The limits are worth naming too. Heat a ferromagnet above its Curie temperature, 1043 K for iron, and thermal agitation destroys the alignment, leaving an ordinary paramagnet. And the response of iron is not linear: it saturates, it lags behind the applied field, and it retains magnetisation when the field is removed. That hysteresis is why permanent magnets exist and why transformer cores dissipate energy on every cycle.

The asymmetry that will not last

Two of the four field equations now exist: the flux of $E$ counts enclosed charge, and the flux of $B$ is always zero. Two more are needed, about circulation rather than flux, and one of them is already half written: the circulation of $E$ is zero, and the circulation of $B$ is $μ_{0} I_{enc}$ .

The set as it stands is a one-way street. Currents make magnetic fields; magnetic fields push currents around; and nothing at all connects a magnetic field back to an electric one. It is a natural thing to try, and several people tried it through the 1820s by putting a coil near a magnet and looking for a current, and every one of them found nothing. Faraday found out why in 1831, and the reason is the most consequential single fact in the subject.