J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’

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                                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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