Application to a Coil.

(112) To find the coefficient (M) of mutual induction between two circular linear conductors in parallel planes, the distance between the curves being everywhere the same, and small compared with the radius of either.

If r {\displaystyle r} be the distance between the curves, and a {\displaystyle a} the radius of either, then when r {\displaystyle r} is very small compared with a {\displaystyle a}, we find by the second method, as a first approximation,

To approximate more closely to the value of M, let a {\displaystyle a} and a 1 {\displaystyle a_{1}} be the radii of the circles, and b {\displaystyle b} the distance between their planes; then

We obtain M by considering the following conditions: –

1st. M must fulfil the differential equation

This equation being true for any magnetic field symmetrical with respect to the common axis of the circles, cannot of itself lead to the determination of M as a function of a , a 1 {\displaystyle a,a_{1}}, and b {\displaystyle b}. We therefore make use of other conditions.

2ndly. The value of M must remain the same when a {\displaystyle a} and a 1 {\displaystyle a_{1}} are exchanged.

3rdly. The first two terms of M must be the same as those given above.

M may thus be expanded in the following series:—

(113) We may apply this result to find the coefficient of self-induction (L) of a circular coil of wire whose section is small compared with the radius of the circle.

Let the section of the coil be a rectangle, the breadth in the plane of the circle being c {\displaystyle }, and the depth perpendicular to the plane of the circle being b {\displaystyle b}.

Let the mean radius of the coil be a {\displaystyle a}, and the number of windings n {\displaystyle n}; then we find, by integrating,

where M

{\displaystyle M(xy\ x’y’)} means the value of M for the two windings whose coordinates are x y {\displaystyle xy} and x ′ y ′ {\displaystyle x’y’} respectively; and the integration is performed first with respect to x {\displaystyle x} and y {\displaystyle y} over the rectangular section, and then with respect to x ′ {\displaystyle x’} and y ′ {\displaystyle y’} over the same space.

Here a

{\displaystyle a=} mean radius of the coil. „ r

{\displaystyle r=} diagonal of the rectangular section

b 2 + c 2 {\displaystyle ={\sqrt {b^{2}+c^{2}}}} „ θ

{\displaystyle \theta =} angle between r {\displaystyle r} and the plane of the circle. „ n

{\displaystyle =} number of windings. The logarithms are Napierian, and the angles are in circular measure.

In the experiments made by the Committee of the British Association for determining a standard of Electrical Resistance, a double coil was used, consisting of two nearly equal coils of rectangular section, placed parallel to each other, with a small interval between them.

PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD. 509 The value of L for this coil was found in the following way.

The value of L was calculated by the preceding formula for six different cases, in which the rectangular section considered has always the same breadth, while the depth was

A, B, C, A+B, B+C, A+B+C,

and n

1 {\displaystyle n=1} in each case. Calling the results

L(A), L(B), L(C),&c,

we calculate the coefficient of mutual induction M(AC) of the two coils thus,

Then if n 1 {\displaystyle n_{1}} is the number of windings in the coil A and n 2 {\displaystyle n_{2}} in the coil B, the coefficient of self-induction of the two coils together is

L

n 1 2 L ( A ) + 2 n 1 n 2 L ( A C ) + n 2 2 L ( B ) {\displaystyle L=n_{1}^{2}L(A)+2n_{1}n_{2}L(AC)+n_{2}^{2}L(B)}

(114) These values of L are calculated on the supposition that the windings of the wire are evenly distributed so as to fill up exactly the whole section. This, however, is not the case, as the wire is generally circular and covered with insulating material. Hence the current in the wire is more concentrated than it would have been if it had been distributed uniformly over the section, and the currents in the neighbouring wires do not act on it exactly as such a uniform current would do.

The corrections arising from these considerations may be expressed as numerical quantities, by which we must multiply the length of the wire, and they are the same whatever be the form of the coil.

Let the distance between each wire and the next, on the supposition that they are arranged in square order, be D, and let the diameter of the wire be d, then the correction for diameter of wire is

The correction for the eight nearest wires is

+0.0236.

For the sixteen in the next row

+0.00083.

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 tends to make the current stronger at one part of the section than at another. When a uniform electromotive force P arising from any cause

3 x 2 510 PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD. acts on a cylindrical wire of specific resistance ρ {\displaystyle \rho }, we have

p ρ

P − d F d t {\displaystyle p\rho =P-{\frac {dF}{dt}}}

where F is got from the equation

being the distance from the axis of the cylinder.

Let one term of the value of F be of the form T r n {\displaystyle Tr^{n}}, where T is a function of the time, then the term of p {\displaystyle p} which produced it is of the form

Hence if we write

The total counter current of self-induction at any point is

from \begin{array}{llc}\mathrm {When} \ t=0,\ p=0,&\therefore \left({\frac {dT}{dt}}\right){0}=P,&\left({\frac {d^{2}T}{dt^{2}}}\right){0}=0,\ \mathrm {etc.} \\\mathrm {When} \ t=\infty ,\ p={\frac {P}{\rho }},&\therefore \left({\frac {dT}{dt}}\right){\infty }=0,&\left({\frac {d^{2}T}{dt^{2}}}\right){\infty }=0,\ \mathrm {etc.} \end{array}}}

from t

0 {\displaystyle t=0} to

{When} \ t=0,\ p=0\ \mathrm {throughout\ the\ section} ,&\therefore \left({\frac {dT}{dt}}\right){0}=P,&\left({\frac {d^{2}T}{dt^{2}}}\right){0}=0,\ \mathrm {etc.} \\\mathrm {When} \ t=\infty ,\ p=0\ \mathrm {throughout} ,&\therefore \left({\frac {dT}{dt}}\right){\infty }=0,&\left({\frac {d^{2}T}{dt^{2}}}\right){\infty }=0,\ \mathrm {etc.} \end{array}}}

Also if l {\displaystyle l} be the length of the wire, and R its resistance,

R

ρ l π r 2 {\displaystyle R={\frac {\rho l}{\pi r^{2}}}}

and if C be the current when established in the wire,

The total counter current may be written (35).

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

where γ {\displaystyle \gamma } is the current in the wire at any instant, and the total countercurrent would have been

say.

Hence

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 μ l {\displaystyle {\tfrac {1}{4}}\mu l}, where l {\displaystyle l} is the length of the wire, and μ {\displaystyle \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

{\displaystyle =a=} .158194 Depth of each coil

b

{\displaystyle =b=} .01608 Breadth of each coil

c

{\displaystyle =c=} .01841 Distance between the coils = .02010 Number of windings n

{\displaystyle 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 least squares to determine what further correction depending on the square of the velocity should be applied to make the outstanding errors a minimum.

The result of this examination showed that the calculated value of L should be multiplied by 1.0618 to obtain the value of L, which would give the most consistent results."

We have therefore L by calculation 430165 metres. Probable value of L by method of least squares 456748 „ Result of rough experiment with the Electric Balance (see § 46) 410000 „ The value of L calculated from the dimensions of the coil is probably much more accurate than either of the other determinations.