# Confusion in the derivation of the potential of a magnetic shell

I am reading an old book on electromagnetism (THE MATHEMATICAL THEORY OF ELECTRICITY AND MAGNETISM) and I have some confusion in the following pages:

First let me clarify what a "magnetic shell" is:

A magnetic shell is a thin sheet of magnetic material of uniform thickness magnetized at every point in a direction perpendicular to the surface of sheet. It may be of any shape (plane, circular or curved). One face of the sheet exhibits north polarity and the other face exhibits south polarity. The magnetic shell may be considered as a polar sheet consisting of a large number of short magnetic or magnetic dipoles close to each other with their axes perpendicular to the face of the shell. The length of each magnetic dipole is the thickness of the shell. If the distribution of the magnetic dipoles over the surface of the shell is uniform, the shell is said to be uniform magnetic shell.

Now onto my main question:

On the first highlighted line it is written "$\phi dS$ is the pole strength of element $dS$"

Then on the next highlighted line its written "we regard any element $dS$ of the shell as a magnetic particle of moment $\phi dS$"

I say the moment should be "pole strength ($\phi dS$)" times "the infinitesimal length of the axis i.e. the thickness of the shell ($\tau$)".

We also cannot assume "the infinitesimal length of the axis i.e. the thickness of the shell ($\tau$)" to be unity since it is infinitesimal

Then how could the moment be $\phi dS$ instead of $\tau\phi dS$

In other words, how can the pole strength($=IdS$) become ($I\tau$)

• I think maybe $I\tau$ doesn't mean pole strength but instead it is some other quantity – N.G.Tyson Jun 20 '16 at 8:47

The first highlighted text refers to the strength of the magnet (aka. magnetic moment) not to the strength of a pole. The intensity of magnetisation $I$ refers to the strength of a pole per unit area. Thus we have a pole of strength $+I$ at one end and of strength $-I$ at the other end, separated by a distance $T$. This will give a magnetic moment per unit area of our shell of $I.T = \phi$. For an area $dS$ then the magnetic moment is $I.T.dS = \phi.dS$.