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703.]

LINES OF MAGNETIC FORCE.

307

Second Expression for M.

An expression for M, which is sometimes more convenient, is got

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To draw the Lines of Magnetic Force for a Circular Current.

702.] The lines of magnetic force are evidently in planes passing through the axis of the circle, and in each of these lines the value of M is constant.

Calculate the value of K. =

sin 0
(Fsine-Esine)2

from Legendre's

tables for a sufficient number of values of 0.

Draw rectangular axes of a and 2 on the paper, and, with centre

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a (cosec 0-sin 0).

a (sin 0+ cosec ), draw a circle with radius.

0+cosec

For all points of this circle the value of c, will

be sin 0. Hence, for all points of this circle,

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Now A is the value of x for which the value of M was found. Hence, if we draw a line for which = A, it will cut the circle in two points having the given value of M.

Giving M a series of values in arithmetical progression, the values of 4 will be as a series of squares. Drawing therefore a series of lines parallel to z, for which has the values found for A, the points where these lines cut the circle will be the points where the corresponding lines of force cut the circle.

If we put m = 4πa, and M = nm, then

A = x = n2 K. a.

We may call n the index of the line of force.

The forms of these lines are given in Fig. XVIII at the end of this volume. They are copied from a drawing given by Sir W. Thomson in his paper on Vortex Motion *.'

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703.] If the position of a circle having a given axis is regarded as defined by b, the distance of its centre from a fixed point on the axis, and a, the radius of the circle, then M, the coefficient of induction of the circle with respect to any system whatever

* Trans. R. S., Edin., vol. xxv. p. 217 (1869).

of magnets or currents, is subject to the following equation

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To prove this, let us consider the number of lines of magnetic force cut by the circle when a or b is made to vary.

(1) Let a become a+da, b remaining constant. During this variation the circle, in expanding, sweeps over an annular surface in its own plane whose breadth is da.

If is the magnetic potential at any point, and if the axis of y be parallel to that of the circle, then the magnetic force perpen

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To find the magnetic induction through the annular surface we have to integrate

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dv αδα do, dy

where is the angular position of a point on the ring.

But this quantity represents the variation of M due to the variation of a, or

d M
da

δα. Hence

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(2) Let become b+db, a remaining constant. During this variation the circle sweeps over a cylindric surface of radius a and length db.

dV

dr

The magnetic force perpendicular to this surface at any point is

where is the distance from the axis.

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Hence

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Differentiating equation (2) with respect to a, and (3) with

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Transposing the last term we obtain equation (1).

(4)

(5)

(6)

704.]

TWO PARALLEL CIRCLES.

309

Coefficient of Induction of Two Parallel Circles when the Distance between the Arcs is Small compared with the Radius of either Circle.

704.] We might deduce the value of M in this case from the expansion of the elliptic integral already given when its modulus is nearly unity. The following method, however, is a more direct application of electrical principles.

First Approximation.

Let A and a be the radii of the circles, and 6 the distance between their planes, then the shortest distance

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the plane of a whose radius is a-c, c being a quantity small compared with a (Fig. 49).

Consider a small element ds of the circle a. At a point in the plane of the circle, distant p from the middle of ds, measured in a direction making an angle 0 with the direction of ds, the magnétic force due to ds is perpendicular to the plane, and equal to

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If we now calculate the surface-integral of this force over the space which lies within the circle a, but outside of a circle whose centre is ds and whose radius is c, we find it

с

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If c is small, the surface-integral for the part of the annular space outside the small circle e may be neglected.

We then find for the induction through the circle whose radius is a-c, by integrating with respect to ds,

Mac

= 4 a {log 8a-log c-2},

provided c is very small compared with a.

Since the magnetic force at any point, the distance of which from a curved wire is small compared with the radius of curvature,

is nearly the same as if the wire had been straight, we can calculate the difference between the induction through the circle whose radius is ac, and the circle A by the formula

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Hence we find the value of the induction between A and a to be Maa = 4πa (log 8 a-log r-2)

Мла

approximately, provided r is small compared with a.

705.] Since the mutual induction between two windings of the same coil is a very important quantity in the calculation of experimental results, I shall now describe a method by which the approximation to the value of M for this case can be carried to any required degree of accuracy.

We shall assume that the value of M is of the form

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3a2 + Az′

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and

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x2

y2

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+ B2

a

a

a2

+ &c., + Bay 2 a2 where a and a + are the radii of the circles, and y the distance

between their planes.

We have to determine the values of the coefficients A and B. It is manifest that only even powers of y can occur in these quantities, because, if the sign of y is reversed, the value of M must remain the same.

We get another set of conditions from the reciprocal property of the coefficient of induction, which remains the same whichever circle we take as the primary circuit. The value of M must therefore remain the same when we substitute a +x for a, and x for x in the above expression.

We thus find the following conditions of reciprocity by equating the coefficients of similar combinations of x and y,

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n (n − 1) A ̧ + (n + 1) n A„+1 + 1.2 A′„+1.2 A′n+1 = n An,

(n − 1) (n−2) Ẩ„+n (n−1) An+1+2.3 A′′n+2.3 A′′n+1 = (n − 2) A‚'i

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6 A3+342 = 2 B2+6 B2+2 B′3=6A3 + 3 A′ 2

(2 n − 1) A„+ (2 n + 2) An+1

2

= n (n − 2) B2+ (n + 1)n Bn+1+1 B' + 1.2 B'

n+1.

Solving these equations and substituting the values of the coefficients, the series for M becomes

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To find the form of a coil for which the coefficient of self-induction is a maximum, the total length and thickness of the wire being given.

706.] Omitting the corrections of Art. 705, we find by Art. 673

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where n is the number of windings of the wire, a is the mean radius of the coil, and R is the geometrical mean distance of the transverse section of the coil from itself. See Art. 690. If this section is always similar to itself, R is proportional to its linear dimensions, and n varies as R2.

Since the total length of the wire is 2 Tan, a varies inversely as n. Hence

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and we find the condition that I may be a maximum

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