Page:A Dynamical Theory of the Electromagnetic Field.pdf/51

The value of L for this coil was found in the following way.

The value of L was calculated by the preceding formula for six different cases, in which the rectangular section considered has always the same breadth, while the depth was

A, B, C, A+B, B+C, A+B+C,

and $n=1$ in each case. Calling the results

L(A), L(B), L(C),&c,

we calculate the coefficient of mutual induction M(AC) of the two coils thus,

$2ACM(AC)=(A+B+C)^{2}L(A+B+C)-(A+B)^{2}L(A+B)-(B+C)^{2}L(B+C)+B^{2}L(B)$

Then if $n_{1}$ is the number of windings in the coil A and $n_{2}$ in the coil B, the coefficient of self-induction of the two coils together is

$L=n_{1}^{2}L(A)+2n_{1}n_{2}L(AC)+n_{2}^{2}L(B)$

(114) These values of L are calculated on the supposition that the windings of the wire are evenly distributed so as to fill up exactly the whole section. This, however, is not the case, as the wire is generally circular and covered with insulating material. Hence the current in the wire is more concentrated than it would have been if it had been distributed uniformly over the section, and the currents in the neighbouring wires do not act on it exactly as such a uniform current would do.

The corrections arising from these considerations may be expressed as numerical quantities, by which we must multiply the length of the wire, and they are the same whatever be the form of the coil.

Let the distance between each wire and the next, on the supposition that they are arranged in square order, be D, and let the diameter of the wire be d, then the correction for diameter of wire is

$+2\left(\log {\frac {D}{d}}+{\frac {4}{3}}\log 2+{\frac {\pi }{3}}-{\frac {11}{6}}\right)$

The correction for the eight nearest wires is

+0.0236.

For the sixteen in the next row

+0.00083.

These corrections being multiplied by the length of wire and added to the former result, give the true value of L, considered as the measure of the potential of the coil on itself for unit current in the wire when that current has been established for some time, and is uniformly distributed through the section of the wire.

(115) But at the commencement of a current and during its variation the current is not uniform throughout the section of the wire, because the inductive action between different portions of the current tends to make the current stronger at one part of the section than at another. When a uniform electromotive force P arising from any cause