Magnetic Monopoles

Equation (2.251) immediately suggests that

$$\nabla\cdot {\bf B} = 0,$$ (2.263)

because the divergence of a curl is identically zero. (See Section A.22.) In other words, the steady magnetic field generated by a pattern of steady circulating electric currents is divergence free. If we integrate the previous equation over a general volume $V$, bounded by a surface $S$, making use of the divergence theorem (see Section A.20), then we obtain

$$\oint_S {\bf B}\cdot d{\bf S} = 0.$$ (2.264)

We conclude that the flux of the magnetic field generated by a steady current pattern out of any closed surface is zero. This implies that the magnetic field-lines generated by a steady current pattern are solenoidal (see Section A.20) and, consequently, never begin or end.

Figure 2.21: Magnetic field-lines generated by a bar magnet.

What about magnetic fields generated by permanent magnets (the modern equivalent of loadstones)? Do they also never begin or end? We know that a conventional bar magnet has both a north and south magnetic pole (like the Earth). If we track the magnetic field-lines with a small compass then they all emanate from the north pole, spread out, and eventually re-converge on the south pole. See Figure 2.21. It appears likely (but we cannot prove it with a compass) that the field-lines inside the magnet connect from the south to the north pole so as to form closed loops that never begin or end.

Can we produce an isolated north or south magnetic pole; for instance, by snapping a bar magnet in two? A compass needle would always point toward an isolated south pole, so this would act like a negative magnetic charge. Likewise, a compass needle would always point away from an isolated north pole, so this would act like a positive magnetic charge. It is clear, from Figure 2.22, that if we take a closed surface $S$ containing an isolated magnetic pole, which is usually termed a magnetic monopole, then $\oint_S {\bf B}\cdot d{\bf S} \neq 0$. In fact, the flux will be positive for an isolated north pole, and negative for an isolated south pole. It follows from the divergence theorem (see Section A.20) that if $\oint_S {\bf B}\cdot d{\bf S} \neq 0$ then $\nabla\cdot{\bf B} \neq 0$. Thus, the statement that $\nabla\cdot{\bf B} = 0$ is equivalent to the statement that magnetic monopoles do not exist. It is actually quite possible to formulate electromagnetism so as to allow for magnetic monopoles. However, as far as we are aware, there are no magnetic monopoles in the universe. We know that if we try to make a magnetic monopole by snapping a bar magnet in two then we just end up with two smaller bar magnets. If we snap one of these smaller magnets in two then we end up with two even smaller bar magnets. We can continue this process down to the atomic level without ever producing a magnetic monopole. In fact, permanent magnetism is generated by electric currents circulating on the atomic scale, and so this type of magnetism is not fundamentally different to the magnetism generated by macroscopic currents.

Figure 2.22: Magnetic field-lines generated by magnetic monopoles.

In conclusion, all steady magnetic fields in the universe are generated by circulating electric currents of some description. Such fields are solenoidal; that is, they have field-lines that never begin or end, and also satisfy the field equation

$$\nabla \cdot {\bf B} = 0.$$ (2.265)

We have only proved that $\nabla\cdot{\bf B} = 0$ for steady magnetic fields, but, in fact, it turns out that this is also the case for time-dependent fields.