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

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                                [integral]<sup>[infinity]<\sup><sub>0<\sub>[integral]<sup>r<\sup><sub>0<\sub> 2[pi] (P/[rho] - p)rdrdt = [mu]/[rho] t[pi]r<sup>2<\sup> + [half] [mu][pi]<sup>2<\sup>/[rho]<sup>2<\sup> dT/dt r<sup>4<\sup> + [mu]<sup>2<\sup>[pi]<sup>3<\sup>/[rho]<sup>3<\sup> 1/1<sup>2<\sup>.2<sup>2<\sup>.3 d<sup>2<\sup>T/dt<sup>2<\sup> r<sup>6<\sup> + &c 
from t = 0 to t = [infinity] 
When t = 0 p = 0 throughout the section [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] throughout [therefore](dT/dt)<sub>[infinity]<\sub> = 0 (d<sup>2<\sup>T/dt<sup>2<\sup>)<sub>[infinity]<\sub> = 0 &c 
Also if l be the length of the wire and R its resistance 
R = [rho]l/[pi]r<sup>2<\sup> 
and if C be the <s>total<\s> current when established in the wire C = [rho]l/R 
The total counter current may now be written 
l/R (T<sub>[infinity]<\sub> - T<sub>0<\\sub> - [half][mu] l/R C = - LC/R by 
Now if the current instead of being variable from the <s>[text?]<\s> centre  
to the circumference of the section of the wire had been the same 
throughout the value of F would have been 
F = T +[mu][gamma](1 - r<sup>2<\sup>/r<sup>2<\sup><sub>0<\sub>) 
where [gamma] is the current in the wire at any instant 
and the total counter current would have been 
[integral]<sup>[infinity]<\sup><sub>0<\sub>[integral]<sup>r<\sup><sub>0<\sub> 1/[rho] dF/dt 2[pi]rdr = l/R(T,sub>[infinity]<\sub> - t<sub>0<\sub>) - 3/4 [mu] l/R C = - L'C/R 
Hence L = L' - 1/4 [mu]l 
or the value of L which must be used in calculating the self 
induction of a wire; for variable currents is less than 
that which is deduced from the supposition of the current being 
constant throughout the section of the wire by 1/4 [mu]l 
where l is the length of the wire, and [mu] is the coefficient 
of magnetic induction for the substance of the wire 
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James Clerk Maxwell
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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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