Page:A Dynamical Theory of the Electromagnetic Field.pdf/53

Now if the current instead of being variable from the centre to the circumference of the section of the wire had been the same throughout, the value of F would have been

$F=T+\mu \gamma \left(1-{\frac {r^{2}}{r_{0}^{2}}}\right),$

where $\gamma$ is the current in the wire at any instant, and the total countercurrent would have been

$\int \limits _{0}^{\infty }\int \limits _{0}^{r}{\frac {1}{\rho }}{\frac {dF}{dt}}2\pi rdr={\frac {l}{R}}\left(T_{\infty }-T_{0}\right)-{\frac {3}{4}}\mu {\frac {l}{R}}C=-{\frac {L'C}{R}}$, say.

Hence

$L=L'-{\frac {1}{4}}\mu l$

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 ${\tfrac {1}{4}}\mu l$, where $l$ is the length of the wire, and $\mu$ is the coefficient of magnetic induction for the substance of the wire.

(116) The dimensions of the coil used by the Committee of the British Association in their experiments at King's College in 1864 were as follows:—

		metre.
Mean radius	$=a=$	.158194
Depth of each coil	$=b=$	.01608
Breadth of each coil	$=c=$	.01841
Distance between the coils	=	.02010
Number of windings	$n=$	313
Diameter of wire	=	.00126

The value of L derived from the first term of the expression is 437440 metres.

The correction depending on the radius not being infinitely great compared with the section of the coil as found from the second term is —7345 metres.

The correction depending on the diameter of the wire is per unit of length	+.44997
Correction of eight neighbouring wires	+.0236
For sixteen wires next to these	+.0008
Correction for variation of current in different parts of section	-.2500
Total correction per unit of length	.22437
Length	311.236 metres.
Sum of corrections of this kind	70 „
Final value of L by calculation	430165 „

This value of L was employed in reducing the observations, according to the method explained in the Report of the Committee^[1]. The correction depending on L varies as the square of the velocity. The results of sixteen experiments to which this correction had been applied, and in which the velocity varied from 100 revolutions in seventeen seconds to 100 in seventy-seven seconds, were compared by the method of

1. British Association Reports, 1863, p. 169.