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

```                                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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## Cite as

J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7