General Methods.

(109) The electromagnetic relations between two conducting circuits, A and B, depend upon a function M of their form and relative position, as has been already shown.

M may be calculated in several different ways, which must of course all lead to the same result.

First Method. M is the electromagnetic momentum of the circuit B when A carries a unit current, or

where F, G, H are the components of electromagnetic momentum due to a unit current in A, and

{\displaystyle ds’} is an element of length of B, and the integration is performed round the circuit of B.

To find F, G, H, we observe that by (B) and (C)

with corresponding equations for G and H,

{\displaystyle p’,q’}, and r ′ {\displaystyle r’} being the components of the current in A.

Now if we consider only a single element d s {\displaystyle ds} of A, we shall have

and the solution of the equation gives

where ρ {\displaystyle \rho } is the distance of any point from d s {\displaystyle ds}. Hence

where θ {\displaystyle \theta } is the angle between the directions of the two elements

{\displaystyle ds,ds’}, and ρ {\displaystyle \rho } is the distance between them, and the integration is performed round both circuits.

In this method we confine our attention during integration to the two linear circuits alone.

(110) Second Method. M is the number of lines of magnetic force which pass through the circuit B when A carries a unit current, or

where

{\displaystyle \mu \alpha ,\mu \beta ,\mu \gamma }, are the components of magnetic induction due to unit current in A, S’ is a surface bounded by the current B, and

{\displaystyle l,m,n} are the direction-cosines of the normal to the surface, the integration being extended over the surface.

We may express this in the form

where d {\displaystyle d}S’ is an element of the surface bounded by B, d s {\displaystyle ds} is an element of the circuit A, ρ {\displaystyle \rho } is the distance between them, θ {\displaystyle \theta } and θ ′ {\displaystyle \theta ‘} are the angles between ρ {\displaystyle \rho } and d s {\displaystyle ds} and between ρ {\displaystyle \rho } and the normal to d {\displaystyle d}S’ respectively, and φ {\displaystyle \varphi } is the angle between the planes in which θ {\displaystyle \theta } and θ ′ {\displaystyle \theta ‘} are measured. The integration is performed round the circuit A and over the surface bounded by B.

This method is most convenient in the case of circuits lying in one plane, in which case sin sin ⁡ θ

1 {\displaystyle \sin \theta =1}, and sin ⁡ φ

1 {\displaystyle \sin \varphi =1}.

111. Third Method. M is that part of the intrinsic magnetic energy of the whole field which depends on the product of the currents in the two circuits, each current being unity.

Let α , β , γ {\displaystyle \alpha ,\beta ,\gamma } be the components of magnetic intensity at any point due to the first circuit,

′ {\displaystyle \alpha ‘,\beta ‘,\gamma ‘} the same for the second circuit; then the intrinsic energy of the element of volume

The part which depends on the product of the currents is

Hence if we know the magnetic intensities I and I’ due to unit current in each circuit, we may obtain M by integrating

over all space, where θ {\displaystyle \theta } is the angle between the directions of I and I’.