Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/318

 CHAPTER XIII. PARALLEL CURRENTS.

682.] a very important class of electrical arrangements the current is conducted through round wires of nearly uniform section, and either straight, or such that the radius of curvature of the axis of the wire is very great compared with the radius of the transverse section of the wire. In order to be prepared to deal mathematically with such arrangements, we shall begin with the case in which the circuit consists of two very long parallel conductors, with two pieces joining their ends, and we shall confine our attention to a part of the circuit which is so far from the ends of the conductors that the fact of their not being infinitely long does not introduce any sensible change in the distribution of force.

We shall take the axis of $$z$$ parallel to the direction of the conductors, then, from the symmetry of the arrangements in the part of the field considered, everything will depend on $$H$$, the component of the vector-potential parallel to $$z$$.

The components of magnetic induction become, by equations (A),

For the sake of generality we shall suppose the coefficient of magnetic induction to be $$\mu$$, so that $$a = \mu \alpha$$, $$b = \mu \beta$$, where $$\alpha$$ and $$\beta$$ are the components of the magnetic force.

The equations (E) of electric currents, Art, 607, give Rh