Page:A Dynamical Theory of the Electromagnetic Field.pdf/48

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PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD.

PART VII. — CALCULATION OF THE COEFFICIENTS OF ELECTROMAGNETIC INDUCTION.

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 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, , and being the components of the current in A.

Now if we consider only a single element of A, we shall have

and the solution of the equation gives

where is the distance of any point from . Hence

where is the angle between the directions of the two elements , and 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 , are the components of magnetic induction due to unit current in A,