Application to a Coil. (112) To find the coefficient (M) of mutual induction between two circular linear conductors the distance between the curves being everywhere the same and small compared with the radius of either. If r be the distance between the curves and a the radius of either then when r is very small compared with a we find by the second method, as a first approximation M = 4[pi]a (log<sub>e<\sub> 8a/r  2) To approximate more closely to the value of M let a and a<sub>1<\sub> be the radii of the circles and b the distance between their planes, then r<sup>2<\sup> = (a  a<sub>1<\sub>)<sup>2<\sup> + b<sup>2<\sup> We obtain M by considering the following conditions 1<sup>st<\sup> M must fulfil the differential equation d<sup>2<\sup>M/da<sup>2<\sup> + d<sup>2<\sup>M/db<sup>2<\sup> + 1/a dM/da = 0 This equation being true for any magnetic field symmetrical with respect to the common axis of the circles, cannot of itself lead to the determination of M as a function of a a<sub>1<\sub> & b. We therefore make use of <s>an<\s>other conditions 2<sup>nd<\sup> The value of M must remain the same when a and a<sub>1<\sub> are exchanged 3<sup>rd<\sup> The first two terms of M must be the same as those given above M may thus be expanded in the following series M = 4[pi]a log 8a/r {1 + [half] [(]a  a<sub>1<\sub>[)]/a + 1/16 [(]3b<sup>2<\sup> + (a<sub>1<\sub>)<sup>2<\sup>/a<sup>2<\sup>/a<sup>2<\sup>  1/32(3b<sup>2<\sup> + (a  a<sub>1<\sub>)<sup>2<\sup>(a  a<sub>1<\sub>)/a<sup>3<\sup> + &c}  4[pi]a {2 + [half] [(]a  a<sub>1<\sub>[)]/a + 1/16 [(]b<sup>2<\sup>  3(a  a<sub>1<\sub>)<sup>2<\sup>/a<sup>2<\sup>/a<sup>2<\sup>  1/48(6b<sup>2<\sup>  (a  a<sub>1<\sub>)<sup>2<\sup>(a  a<sub>1<\sub>)/a<sup>3<\sup> + &c}
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Manuscript details
 Author
 James Clerk Maxwell
 Reference
 PT/72/7
 Series
 PT
 Date
 1864
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Cite as
J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7
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