[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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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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