Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/172

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ELECTROMAGNETIC FORCE.

[493.

of two magnetic shells whose edges are the closed curves $s_{1}$ and $s_{2}$ .

If we make

M=\int _{0}^{s_{2}}\int _{0}^{s_{1}}{\frac {\cos \epsilon }{r}}\,ds_{1}\,ds_{2}

,

where $\epsilon$ is the angle between the directions of the elements $ds_{1}$ and $ds_{2}$ , and $r$ is the distance between them, the integration being extended once round $s_{2}$ and once round $s_{1}$ , and if we call $M$ the potential of the two closed curves $s_{1}$ and $s_{2}$ , then the potential energy due to the mutual action of two magnetic shells whose strengths are $i_{1}$ and $i_{2}$ bounded by the two circuits is

-i_{1}i_{2}M

,

and the force $X$ , which aids any displacement $\delta x$ , is

i_{1}i_{2}{\frac {\delta M}{\delta x}}

.

The whole theory of the force acting on any portion of an electric circuit due to the action of another electric circuit may be deduced from this result.

493.] The method which we have followed in this chapter is that of Faraday. Instead of beginning, as we shall do, following Ampère, in the next chapter, with the direct action of a portion of one circuit on a portion of another, we shew, first, that a circuit produces the same effect on a magnet as a magnetic shell, or, in other words, we determine the nature of the magnetic field due to the circuit. We shew, secondly, that a circuit when placed in any magnetic field experiences the same force as a magnetic shell. We thus determine the force acting on the circuit placed in any magnetic field. Lastly, by supposing the magnetic field to be due to a second electric circuit we determine the action of one circuit on the whole or any portion of the other.

494.] Let us apply this method to the case of a straight current of infinite length acting on a portion of a parallel straight conductor.

Let us suppose that a current $i$ in the first conductor is flowing vertically downwards. In this case the end of a magnet which points north will point to the right-hand of a man looking at it from the axis of the current.

The lines of magnetic induction are therefore horizontal circles, having their centres in the axis of the current, and their positive direction is north, east, south, west.

Let another descending vertical current be placed due west of the first. The lines of magnetic induction due to the first current