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16.3: The CGS Electromagnetic System

If you have been dismayed by the problems of CGS esu, you don't yet know what is in store for you with CGS emu. Wait for it:

Definition. One CGS emu of magnetic pole strength is that pole which, if placed 1 cm from a similar pole in vacuo, will repel it with a force of 1 dyne.

The system is based on the proposition that there exists a "pole" at each end of a magnet, and that point poles repel each other according to an inverse square law. Magnetic field strength \(\textbf{H}\) is defined as the force experienced by a unit pole situated in the field. Thus, if a pole of strength \(m\) emu is situated in a field of strength \(\textbf{H}\), it will experience a force \(\textbf{F}=m\textbf{H}\).

Definition. If a pole of strength 1 emu experiences a force of 1 dyne when it is situated in a magnetic field, the strength of the magnetic field is 1 oersted (Oe). It will probably be impossible for the reader at this stage to try to work out the conversion factor between Oe and \(\text{A m}^{-1}\), but, for the record

\[1 \text{ Oe} = \frac{250}{\pi} \text{A m}^{-1}.\]

Now hold on tight, for the definition of the unit of electric current.

Definition: One emu of current (1 abamp) is that steady current, which, flowing in the arc of a circle of length 1 cm and of radius 1 cm (i.e. subtending 1 radian at the centre of the circle) gives rise to a magnetic field of 1 oersted at the centre of the circle.

This will involve quite an effort of the imagination. First you have to imagine a current flowing in an arc of a circle. Then you have to imagine measuring the field at the centre of the circle by measuring the force on a unit magnetic pole that you place there.

It follows that, if a current \(I\) abamp flows in a circle of radius \(a\) cm, the field at the centre is of the circle is

\[H = \frac{2 \pi I}{a} \text{ Oe}.\]

The conversion between emu of current (abamp) and ampères is

1 emu = 10 \(\text{A}\).

The Biot-Savart law becomes

\[dH=\frac{I \ ds \ \sin \theta}{r^2}.\]

The field at a distance \(r\) in vacuo from a long straight current \(I\) is

\[H= \frac{2I}{r}.\]

Ampère's law says that the line integral of \(\textbf{H}\) around a closed plane curve is \(4 \pi\) times the enclosed current. The field inside a long solenoid of \(n\) turns per centimetre is

\[H=4 \pi n I.\]

So far, no mention of \(\textbf{B}\), but it is now time to introduce it. Let us imagine that we have a long solenoid of \(n\) turns per cm, carrying a current of \(I\) emu, so that the field inside it is \(4 \pi n I \) Oe. Suppose that the cross-sectional area of the solenoid is A. Let us wrap a single loop of wire tightly around the outside of the solenoid, and then change the current at a rate \(\dot I\) so that the field changes at a rate \(\dot H = 4 \pi n \dot I\). An EMF will be set up in the outside (secondary) coil of magnitude \(A\dot H\). If we now insert an iron core inside the solenoid and repeat the experiment, we find that the induced EMF is much larger. It is larger by a (supposed dimensionless) factor called the permeability of the iron. Although this factor is called the permeability and the symbols used is often \(\mu\), I am going to use the symbol \(\kappa\) for it. The induced EMF is now A times \(\kappa \dot H\). We denote the product of \(\mu\) and \(H\) with the symbol \(B\), so that \(B= \kappa H\). The magnitude of \(B\) inside the solenoid is

\[B = 4 \pi \kappa n I.\]

It will be evident from the familiar SI version \(B = \mu n I\) that the CGS emu definition of the permeability differs from the SI definition by a factor \(4 \pi\). The CGS emu definition is called an unrationalized definition; the SI definition is rationalized.  The relation between them is \(\mu = 4 \pi \kappa\).

In CGS emu, the permeability of free space has the value 1. Indeed the supposedly dimensionless unrationalized permeability is what, in SI parlance, would be the relative permeability.

The CGS unit of \(\text{G}\) is the gauss (\(\text{G}\)), and 1 \(\text{G} = 10^{-4} \text{T}\).

It is usually held that \(\kappa\) is a dimensionless number, so that \(B\) and \(H\) have the same dimensions, and, in free space, B and H are identical. They are identical not only numerically, but there is physically no distinction between them. Because of this, the unit oersted is rarely heard, and it is common to hear the unit gauss used haphazardly to describe either \(B\) or \(H\).

The scalar product of \(\textbf{B}\) and area is the magnetic flux, and its CGS unit, \(\text{G cm}^2\), bears the name the maxwell. The rate of change of flux in maxwells per second will give you the induced EMF in emus (abvolts). An abvolt is \(10^{-8}\) V.

The subject of magnetic moment has caused so much confusion in the literature that I shall devote an entire future chapter to it rather than try to do it here.

I end this section by giving the CGS emu version of magnetization. The familiar \(\textbf{B} = \mu_0 (\textbf{H} + \textbf{M})\) becomes, in its CGS emu guise, \(\textbf{B} = \textbf{H} + 4 \pi \textbf{M}\). The magnetic susceptibility \(\chi_m\) is defined by \(\textbf{M} = \chi_m \textbf{H}\). Together with \(\textbf{B} = \kappa \textbf{H}\), this results in \(\kappa = 1 + 4 \pi \chi_m\).

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