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

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                                (113) We may aply this result to find the oefficient of self induction (L) of 
a circular coil of wire whose section is small compared with the radius of the 
circle
Let the section of the coil be a rectangle the breadth in the plane 
of the circle being c, and the depth perpendicular to the plane of 
the circle being b 
Let the mean radius of the coil be a and the number of windings n 
then we find by integrating 
L = n<sup>2<\sup>/b<sup>2<\sup>c<sup>2<\sup> [integral][integral][integral][integral] M(x y x' y') dxdydx'dy' 
where M(x y x' y') means the value of M for two windings whose 
coordinates are xy and x'y' respectively and the integration is 
performed first with respect to x and y over the rectangular section 
and then with respect to x' and y' over the same space 
L = 4[pi]n<sup>2<\sup>a{log 8a/r + 1/12 - 4/3([theta] - [pi]/4)cot2[theta] - [pi]/3cosec2[theta] - 1/6 cot<sup>2<\sup>[theta]log cos[theta] - 1/6 tan<sup>2<sup>[theta] log sin[theta]} 
+ [pi]n<sup>2<\sup>r<sup>2<\sup>/24a{log 8a/r(2sin<sup>2<\sup>[theta] + !) + 3.45 + 27.475 cos<sup>2<\sup>[theta] - 3.2 ([pi]/2 - [theta]) sin<sup>3<\sup>/cos[theta] 
+ 1/5 cos<sup>4<\sup>[theta]/sin<sup>2<\sup>[theta] log cos[theta] + 13/3 sin<sup>4<sup>/cos<sup>2<\sup>[theta] log sin[theta]} &c 
Here a = mean radius of the coil 
r = diagonal of the rectangular section = [square root]b<sup>2<\sup> + c<sup>2<\sup> 
[theta] = angle between r and the plane of the circle 
n = number of windings 
The logarithms are Napierian and the angles are in circular measure 
108 In the experiments made by the Committee of the British Association 
for determining a standard of Electrical Resistance, a double coil was 
used, consisting of two nearly equal coils of rectangular section placed 
parallel to each other with a small interval between them 
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 
                            
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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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