Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/319

683.] If the current is a function of $r$, the distance from the axis of $z$, and if we write

and $\beta$ for the magnetic force, in the direction in which $\theta$ is measured perpendicular to the plane through the axis of $z$, we have

If $C$ is the whole current flowing through a section bounded by a circle in the plane $xy$, whose centre is the origin and whose radius is $r$,

It appears, therefore, that the magnetic force at a given point due to a current arranged in cylindrical strata, whose common axis is the axis of $z$, depends only on the total strength of the current flowing through the strata which lie between the given point and the axis, and not on the distribution of the current among the different cylindrical strata.

For instance, let the conductor be a uniform wire of radius $a$, and let the total current through it be $C$, then, if the current is uniformly distributed through all parts of the section, $w$ will be

constant, and

The current flowing through a circular section of radius $r$, $r$ being less than $a$, is $C^{\prime }=\pi wr^{2}$. Hence at any point within the wire,

	⁠	$\beta ={\frac {2C^{\prime }}{r}}=2C{\frac {r}{a^{2}}}$.	(8)
Outside the wire		$\beta =2{\frac {C}{r}}$.	(9)

In the substance of the wire there is no magnetic potential, for within a conductor carrying an electric current the magnetic force does not fulfil the condition of having a potential.

Outside the wire the magnetic potential is

Let us suppose that instead of a wire the conductor is a metal tube whose external and internal radii are $a_{1}$ and $a_{2}$, then, if $C$ is the current through the tubular conductor,

The magnetic force within the tube is zero. In the metal of the tube, where $r$ is between $a_{1}$ and $a_{2}$,