Magnetic Flux

How much field passes through a loop

The last three chapters found the force a magnetic field puts on moving charges and currents, and where the field comes from. This chapter turns the question round. It asks when a magnetic field can push charges round a closed wire and drive a current, with no cell anywhere in the circuit. The answer depends on one quantity: how much field passes through the loop of wire.

Picture the field as lines, drawn close together where the field is strong. Hold a flat loop in a uniform field \(\vec B\). Face-on to the field, the loop catches the most lines. Turned edge-on, it catches none, because the lines slide past its plane. The count of lines through the loop is what the loop "feels". Physics calls it the magnetic flux.

To describe the tilt, give the flat surface an area vector \(\vec A\). Its size is the area \(A\), and it points along the normal, at right angles to the surface. Let \(\theta\) be the angle between \(\vec B\) and \(\vec A\). Only the part of the field along the normal, \(B\cos\theta\), crosses the surface. So the flux is \(B\cos\theta\) times the area.

In the sim, eight lines pierce the loop face-on and \(\Phi = 10\,\text{mWb}\). At \(60°\) only four lines pierce it and \(\Phi = 5\,\text{mWb}\), since \(\cos 60° = 0.5\). At \(90°\) no line crosses and the flux is zero. At \(180°\) eight lines cross again, but from the back, so the flux is \(-10\,\text{mWb}\). Flux is a scalar, and its sign depends on which way we choose the normal.

The SI unit of flux is the tesla square metre, named the weber (Wb). So the tesla is also a weber per square metre. A field is sometimes called a flux density for this reason: it is the flux through each square metre held face-on.

A coil of many turns

A coil of \(N\) turns, wound close together, has the same flux \(\Phi\) through every turn. Each turn will turn out to behave like a separate source in the circuit, and the turns add up. So what matters for a coil is \(N\Phi\), called the flux linkage. It is also measured in webers.

The field need not be uniform. Take a rectangle beside a long straight wire, its near side a distance \(a\) away and its far side at \(a + b\). The field \(\mu_0 I/2\pi x\) falls with the distance \(x\). So cut the rectangle into thin strips parallel to the wire, each of width \(dx\) and height \(c\). A strip carries the flux \(\frac{\mu_0 I}{2\pi x}\,c\,dx\), and the strips add to \(\Phi = \frac{\mu_0 Ic}{2\pi}\ln\frac{a+b}{a}\).

With \(I = 10\,\text{A}\), \(c = 0.2\,\text{m}\) and \(a = b = 0.1\,\text{m}\), the flux is \(2 \times 10^{-7} \times 10 \times 0.2 \times \ln 2\) \(= 2.77 \times 10^{-7}\,\text{Wb}\). The field of the wire comes from the chapter Sources of the Magnetic Field.

The flux through a solenoid

Inside a long solenoid the field is uniform and along the axis, \(B = \mu_0 nI\), where \(n\) is the number of turns per metre. Each turn is a flat loop face-on to that field. So the flux through one turn is \(\Phi = \mu_0 nIA\), with \(A\) the area of the cross-section.

A coil wound round the outside of the solenoid may have a larger area than the solenoid. It does not catch more flux. The field outside a long solenoid is nearly zero, so the only flux through the wide coil is the flux inside the solenoid's own cross-section. A coil of \(20\) turns of area \(5\times10^{-3}\,\text{m}^2\) round the solenoid above still links \(20 \times 2.51\times10^{-6}\) \(= 5.03\times10^{-5}\,\text{Wb}\).

Three ways to change a flux

The next lesson shows that a current is driven round a loop only while the flux through it changes. The formula \(\Phi = BA\cos\theta\) has three factors, so a flux can change in three ways. Change the field \(B\), for example by moving a magnet closer. Change the area \(A\) inside the field, by squeezing the loop or sliding it out of the field. Or change the angle \(\theta\), by turning the loop.

The three chips of Fig 1.1 change the flux by the same kind of amount. The field growing from \(0.2\) to \(0.5\,\text{T}\) raises it from \(0.02\) to \(0.05\,\text{Wb}\). Squeezing the loop from \(0.1\) to \(0.04\,\text{m}^2\) lowers it from \(0.05\) to \(0.02\,\text{Wb}\). Turning the loop through \(90°\) lowers it from \(0.05\,\text{Wb}\) to zero. As the loop turns it looks narrower, and its apparent width falls as \(\cos\theta\), exactly like the flux.

Real cases often combine two ways. A rod sliding on rails sweeps out area while the field itself may also grow. Then write the flux as a function of time first, \(\Phi(t) = B(t)\,l\,x(t)\), and change only afterwards. Moving a loop sideways inside a uniform field changes none of the three factors, so its flux stays the same.

Flux through a closed surface

Electric field lines start on positive charges and end on negative ones. Magnetic field lines have no ends. Round a straight wire they are circles. Through a loop or a bar magnet they are closed curves, running inside the magnet from its south pole to its north pole. No isolated north or south pole, a magnetic monopole, has ever been found. So a magnetic field line never starts or stops anywhere.

Now draw any closed surface, a sphere or a box, in a magnetic field. A line that enters the surface must leave it again somewhere, because it cannot end inside. So the inward flux and the outward flux cancel. Take the normal of a closed surface pointing outwards, as always. Then the net flux through the whole surface is exactly zero.

This is the magnetic partner of Gauss's law for the electric field, met in the chapter Electric Flux and Gauss's Law. There, the net flux out of a closed surface equals the enclosed charge over \(\epsilon_0\). Here, the enclosed "magnetic charge" is always zero. The chapter Magnetism and Matter returns to it: even a bar magnet cut in two gives two magnets, each with both poles.