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

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                                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 tend 
to make the current stronger at one part of the section than at another. 
When a uniform electromotive force P arising from any cause acts 
on a cylindrical wire of specific resistance [rho] we have 
p[rho] = P - dF/dt 
where F is got from the equation 
d<sup>2<\sup>F/dr<sup>2<\sup> + 1/r dF/dr = - 4[pi][mu]p 
r being the distance from the axis of the cylinder 
Let one term of the value of F be of the form Tr<sup>n<\sup> 
where T is a function of the time, then the term of p which 
produced it is of the form 
- 1/4[pi][mu] n<sup>2\sup> Tr<sup>n - 2<\sup> 
Hence if we write 
F  <s>S<\s>T + [mu][pi]/[rho] (-P + dT/dt)r<sup>2<\sup> + [mu][pi]/[rho])<sup>2<\sup> 1/<sup>2<\sup>.2<sup>2<\sup> d<sup>2<\sup>T/dt<sup>2<\sup> r<sup>4<\sup> + &c 
p[rho] = [theta]<s>1/[rho]<\s> [text?] + dT/dt) - [mu][pi]/[rho]) d<sup>2<\sup> T/dt<sup>2<\sup> r<sup>2<\sup> - [mu][pi]/[rho])<sup>2<\sup> 1/<sup>2<\sup>.2<sup>2<\sup> d<sup>3<\sup>T/dt<sup>3<\sup> r<sup>4<\sup> - &c 
The total counter current of self induction at any point is 
[integral](P/[rho] - p)dt = 1/[rho] T + [mu][pi]/[rho]<sup>2<\sup> dT/dt r<sup>2<\sup> + [mu][pi]/[rho])<sup>3<\sup>) 1/<sup>2<\sup>.2<sup>2<\sup> d<sup>2<\sup>T/dt<sup>2<\sup> r<sup>4<\sup> + &c
from t = 0 to t = [infinity] 
When t = 0 p = 0 [therefore] (dT/dt)<sub>0<\sub> = P (d<sup>2<\sup>T/dt<sup>2<\sup>)<sub>0<\sub> = 0 &c 
When t = [infinity] p = P/[rho] 
[therefore] (dT/dt)<sub>[infinity]<\sub> = 0 (d<sup>2<\sup>T/dt<sup>2<\sup>)<sub>[infinity]<\sub> = 0 &c
                            
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Manuscript details

Author
James Clerk Maxwell
Reference
PT/72/7
Series
PT
Date
1864
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J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7

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