A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 18 of 27

It appears from this, that in calculating the mutual induction of two coaxal solenoids, we must use in the expression (20) instead of the true length I the corrected length I — 2 c^ a, in which a portion equal to ac^ is supposed to be cut off at each end. When the solenoid is very long compared with its external radius,

(23) i \

679.] When a solenoid consists of a number of layers of wire of such a diameter that there are n layers in unit of length, the number of layers in the thickness dr is n dr, and we have

£

=4 Trfn*dr, and g = TT l\ n2 r2 dr. (24)

If the thickness of the wire is constant, and if the induction take place between an external coil whose outer and inner radii are x and y respectively, and an inner coil whose outer and inner radii are y and z, then, neglecting the effect of the ends,

Gg = $lnn(x-y)(y-z). (25)

68o.] INDUCTION COIL. 283

That this may be a maximum, x and z being given, and y variable, z* , „,

• = •?-;*• (2G)

J

This equation gives the best relation between the depths of the primary and secondary coil for an induction-machine without an iron core.

If there is an iron core of radius z, then G remains as before, but

g = TT ifn2 (r2 + 4 TT K z2) dr, (27)

-*)- (28)

If y is given, the value of z which gives the maximum value of g is

187TK ,„„,

z = 4 v - • I « " J

3yi87TK+l

When, as in the case of iron, K is a large number, z = f y, nearly.

If we now make x constant, and y and z variable, we obtain the maximum value of Gg when

x \y\ z : : 4 : 3 : 2. (30)

The coefficient of self-induction of a long solenoid whose outer and inner radii are x and y> and having a long iron core whose radius is z, is

L = %7T2ln*(v-y)2(x2 + 2xy + 3y2 + 24;TTKZ2). (31)

680.] We have hitherto supposed the wire to be of uniform thickness. We shall now determine the law according to which the thickness must vary in the different layers in order that, for a given value of the resistance of the primary or the secondary coil, the value of the coefficient of mutual induction may be a maximum.

Let the resistance of unit of length of a wire, such that n windings occupy unit of length of the solenoid, be p n2.

The resistance of the whole solenoid is

E = 2iilJ»*rdr. (32)

The condition that, with a given value of R, G may be a maximum

. dG ndR . „ . is -T- =C~r- , where C is some constant. *• _ dr l

This gives n2 proportional to - , or the diameter of the wire of

the exterior coil must be proportional to the square root of the radius.

In order that, for a given value of R, g may be a maximum

*.0, + lS£«.. (33)

284 CURRENT -SHEETS. [68 1.

Hence, if there is no iron core, the diameter of the wire of the interior coil should be inversely as the square root of the radius, but if there is a core of iron having a high capacity for magneti zation, the diameter of the wire should be more nearly directly proportional to the square root of the radius of the layer.

An Endless Solenoid.

681.] If a solid be generated by the revolution of a plane area A about an axis in its own plane, not cutting it, it will have the form of a ring. If this ring be coiled with wire, so that the windings of the coil are in planes passing through the axis of the ring, then, if n is the whole number of windings, the current-function of the

layer of wire is $ = — n y 0, where 6 is the angle of azimuth about

the axis of the ring.

If 12, is the magnetic potential inside the ring and 12' that out side, then 12-12' = 47T(£ + <?= 2ny0 + C. Outside the ring 12' must satisfy Laplace's equation, and must vanish at an infinite distance. From the nature of the problem it must be a function of 0 only. The only value of 12' which fulfils these conditions is zero. Hence

12' = 0, 12 = 2ny8+C.

The magnetic force at any point within the ring is perpendicular

to the plane passing through the axis, and is equal to 2ny-

where r is the distance from the axis. Outside the ring there is no magnetic force.

If the form of a closed curve be given by the coordinates z, r, and 0 of its tracing point as functions of s, its length from a fixed point, the magnetic induction through the closed curve is

[• z dr

2ny - -j- ds V0 r ds

taken round the curve, provided the curve is wholly inside the ring. If the curve lies wholly without the ring, but embraces it, the magnetic induction through it is

/"' z' dr _ , 2 n y / — -=-, ds = 2 n y a,

J Q T (IS

where the accented coordinates refer not to the closed curve, but to a single winding of the solenoid.

The magnetic induction through any closed curve embracing the

68 1.] ENDLESS SOLENOID. 285

ring1 is therefore the same, and equal to 2 n y a, where a is the linear

/*' zf dr' — -Tjds'. If the closed curve does not embrace the / ds

ring, the magnetic induction through it is zero.

Let a second wire be coiled in any manner round the ring, not necessarily in contact with it, so as to embrace it nf times. The induction through this wire is 2 n ri y a, and therefore M, the coefficient of induction of the one coil on the other, is M = 2 n ri a.

Since this is quite independent of the particular form or position of the second wire, the wires, if traversed by electric currents, will experience no mechanical force acting between them. By making the second wire coincide with the first, we obtain for the coefficient of self-induction of the ring-coil

L = 2 n2 a.

CHAPTER XIII.

PARALLEL CURRENTS.

Cylindrical Conductors.

682.] IN a very important class of electrical arrangements the current is conducted through round wires of nearly uniform section, and either straight, or such that the radius of curvature of the axis of the wire is very great compared with the radius of the transverse section of the wire. In order to be prepared to deal mathematically with such arrangements, we shall begin with the case in which the circuit consists of two very long parallel conductors, with two pieces joining their ends, and we shall confine our attention to a part of the circuit which is so far from the ends of the conductors that the fact of their not being infinitely long does not introduce any sensible change in the distribution of force.

We shall take the axis of z parallel to the direction of the con ductors, then, from the symmetry of the arrangements in the part of the field considered, everything will depend on //, the component of the vector-potential parallel to z.

The components of magnetic induction become, by equations (A),

m

dH

c — 0.

For the sake of generality we shall suppose the coefficient of magnetic induction to be p, so that a = /a a, b — /u, /3, where a and (3 are the components of the magnetic force.

The equations (E) of electric currents, Art. GO 7, give

u = 0, v = 0. 4 KW = — — -^ • (3)

dx dy

683.] STRAIGHT WIRE. 287

683.] If the current is a function of r, the distance from the axis of Zj and if we write

os = r cos 0, and y = r sin 0, (4)

and {3 for the magnetic force, in the direction in which 6 is measured perpendicular to the plane through the axis of z, we have

4™=f + 10 = 1*08,). (5)

dr r r dr ^

If C is the whole current flowing through a section bounded by a circle in the plane gey, whose centre is the origin and whose

radius is r, />

<?= / 2trrwdr = %(3r. (6) JQ It appears, therefore, that the magnetic force at a given point due to a current arranged in cylindrical strata, whose common axis is the axis of z, depends only on the total strength of the current flowing through the strata which lie between the given point and the axis, and not on the distribution of the current among the different cylindrical strata. For instance, let the conductor be a uniform wire of radius a, and let the total current through it be C, then, if the current is uniformly distributed through all parts of the section, w will be constant, and C=7rwa2'. (7) The current flowing through a circular section of radius r, r being less than a, is C'= -nwr2. Hence at any point within the wire, C Outside the wire 8 = 2 — . (9) f In the substance of the wire there is no magnetic potential, for within a conductor carrying an electric current the magnetic force does not fulfil the condition of having a potential. Outside the wire the magnetic potential is £l = 2C0. (10) Let us suppose that instead of a wire the conductor is a metal tube whose external and internal radii are a-j, and a2, then, if (7 is the current through the tubular conductor, C = 7Tw(al2-a.22). (11) The magnetic force within the tube is zero. In the metal of the tube, where ;• is between a-^ and a2, P= 2^-^--2r--2-2, (12) 288 PARALLEL CURRENTS. [684. and outside the tube, c /3=2-, (13) the same as when the current flows through a solid wire. 684.] The magnetic induction at any point is b = p (3, and since, by equation (2), fi - _ ^ (14) dr H^-jppdr. (15) The value of // outside the tube is A — 2iJL0Clogr, (16) where JUQ is the value of /x in the space outside the tube, and A is a constant, the value of which depends on the position of the return current. In the substance of the tube, a\ ~~ a-2 ai In the space within the tube H is constant, and #=^-2MoClog«1 + Me(l + -logr^). (18) U-^ — U>2 i*^ ' 685.] Let the circuit be completed by a return current, flowing in a tube or wire parallel to the first, the axes of the two currents being at a distance b. To determine the kinetic energy of the system we have to calculate the integral T = \ fjJHw dx cly dz. (19) If we confine our attention to that part of the system which lies between two planes perpendicular to the axes of the conductors, and distant I from each other, the expression becomes T= \l Hivdxdy. (20) If we distinguish by an accent the quantities belonging to the return current, we may write this ^-!-=jJHw'dx'dy'+jJH'wdxcly + jJHwdxdy+jJll'w'dx'dy'. (21) Since the action of the current on any point outside the tube is the same as if the same current had been concentrated at the axis of the tube, the mean value of H for the section of the return current is A — 2^C log I, and the mean value of H' for the section of the positive current is A — 2 /u0 GY/ log b. 687.] LONGITUDINAL TENSION. 289 Hence, in the expression for T, the first two terms may be written AC'-2n()CC'log6) and A'C-2 n0CC'logl>. Integrating the two latter terms in the ordinary way, and adding the results, remembering that C+ C' = 0, we obtain the value of the kinetic energy T. Writing this \LC2, where L is the co efficient of self-induction of the system of two conductors, we find as the value of L for unit of length of the system L If the conductors are solid wires, a.2 and a<£ are zero, and T /,2 (23) aiai It is only in the case of iron wires that we need take account of the magnetic induction in calculating their self-induction. In other cases we may make /x0, /LI, and // all equal to unity. The smaller the radii of the wires, and the greater the distance between them, the greater is the self-induction. To find the Repulsion, X, between the Two Portions of Wire. 686.] By Art. 580 we obtain for the force tending to increase b, *-»£<". = 2MO|C">, (24) which agrees with Ampere's formula, when JUQ = 1, as in air. 687.] If the length of the wires is great compared with the distance between them, we may use the coefficient of self-induction to determine the tension of the wires arising from the action of the current. If Z is this tension, In one of Ampere's experiments the parallel conductors consist of two troughs of mercury connected with each other by a floating bridge of wire. When a current is made to enter at the extremity of one of the troughs, to flow along it till it reaches one extremity VOL. II. U 290 PAEALLEL CURRENTS. [688. of the floating wire, to pass into the other trough through the floating bridge, and so to return along the second trough, the floating bridge moves along the troughs so as to lengthen the part of the mercury traversed by the current. Professor Tait has simplified the electrical conditions of this experiment by substituting for the wire a floating siphon of glass filled with mercury, so that the current flows in mercury through out its course. Fig. 40. This experiment is sometimes adduced to prove that two elements of a current in the same straight line repel one another, and thus to shew that Ampere's formula, which indicates such a repulsion of collinear elements, is more correct than that of Grassmann, which gives no action between two elements in the same straight line ; Art. 526. But it is manifest that since the formulae both of Ampere and of Grassmann give the same results for closed circuits, and since we have in the experiment only a closed circuit, no result of the experiment can favour one more than the other of these theories. In fact, both formulae lead to the very same value of the repulsion as that already given, in which it appears that b, the distance between the parallel conductors is an important element. When the length of the conductors is not very great compared with their distance apart, the form of the value of L becomes somewhat more complicated. 688.] As the distance between the conductors is diminished, the value of L diminishes. The limit to this diminution is when the wires are in contact, or when b = al + a2. In this case fiV (26) 689.] MINIMUM SELF-INDUCTION. 291 This is a minimum when a^ = a2t and then £ = 2 /(log 4 + 1), = 2^(1.8863), = 3.7726^. (27) This is the smallest value of the self-induction of a round wire doubled on itself, the whole length of the wire being 2 I. Since the two parts of the wire must be insulated from each other, the self-induction can never actually reach this limiting value. By using broad flat strips of metal instead of round wires the self-induction may be diminished indefinitely. On the Electromotive Force required to produce a Current of Varying Intensity along a Cylindrical Conductor. 689.] When the current in a wire is of varying intensity, the electromotive force arising from the induction of the current on itself is different in different parts of the section of the wire, being in general a function of the distance from the axis of the wire as well as of the time. If we suppose the cylindrical conductor to consist of a bundle of wires all forming part of the same circuit, so that the current is compelled to be of uniform strength in every part of the section of the bundle, the method of calculation which we have hitherto used would be strictly applicable. If, however, we consider the cylindrical conductor as a solid mass in which electric currents are free to flow in obedience to electromotive force, the intensity of the current will not be the same at different distances from the axis of the cylinder, and the electromotive forces themselves will depend on the distribution of the current in the different cylindric strata of the wire. The vector-potential //, the density of the current w, and the electromotive force at any point, must be considered as functions of the time and of the distance from the axis of the wire. The total current, C, through the section of the wire, and the total electromotive force, JE, acting round the circuit, are to be regarded as the variables, the relation between which we have to find. Let us assume as the value of H, H= S+To + T^+bc. + T.r**, (1) where S, T0, Tlf &c. are functions of the time. Then, from the equation d2H , 1 dH f . -J-H- H -=- = — 47TW, (2) dr2 r dr we find -TIW = Tl + &c + n*TnrZn~2. (3) U 2 292 PARALLEL CURRENTS. [690. If p denotes the specific resistance of the substance per unit of volume, the electromotive force at any point is p w, and this may be expressed in terms of the electric potential and the vector potential H by equations (B), Art. 598, dV dll ,A. <>w = -^-w d3> dS dTQ clT^ dTn -?w = T* + Tt+-W + -WT+^ + ^?T ' (5) Comparing the coefficients of like powers of r in equations <s)'nd(5)' Hence we may write -=- = — — , (9) T_,dT _£ 1 d'T J° 2' ^--pTt>- /B"?(i±FaF 690.] To find the total current (7, we must integrate w over the section of the wire whose radius is a, ra C=27T wrdr. (11) ^o Substituting the value of itw from equation (3), we obtain (12) The value of H at any point outside the wire depends only on the total current C, and not on .the mode in which it is distributed within the wire. Hence we may assume that the value of H at the surface of the wire is A C, where A is a constant to be determined by calculation from the general form of the circuit. Putting H=AC when r = a, we obtain 2n- (13) If we now write - = a, a is the value of the conductivity of P unit of length of the wire, and we have (15) 690.] VARIABLE CURRENT. 293 Eliminating T from these two equations, we find .dC dS, . dC . = o. (16) If I is the whole length of the circuit, R its resistance, and E the electromotive force due to other causes than the induction of the current on itself, dS E I Tl=-J' a = K' dC PcPC P fPC The first term, RC> of the right-hand member of this equation expresses the electromotive force required to overcome the resist ance according to Ohm's law. The second term, l(A + \)-;- , expresses the electromotive force dt which would be employed in increasing the electrokinetic momentum of the circuit, on the hypothesis that the current is of uniform strength at every point of the section of the wire. The remaining terms express the correction of this value, arising from the fact that the current is not of uniform strength at different distances from the axis of the wire. The actual system of currents has a greater degree of freedom than the hypothetical system, in which the current is constrained to be of uniform strength throughout the section. Hence the electromotive force required to produce a rapid change in the strength of the current is some what less than it would be on this hypothesis. The relation between the time-integral of the electromotive force and the time-integral of the current is (19) If the current before the beginning of the time has a constant value C0) and if during the time it rises to the value CL, and re mains constant at that value, then the terms involving the differ ential coefficients of C vanish at both limits, and ,\ (20) the same value of the electromotive impulse as if the current had been uniform throughout the wire. 294 PARALLEL CURRENTS. [691. On the Geometrical Mean Distance of Two Figures in a Plane.* 691.] In calculating the electromagnetic action of a current flowing in a straight conductor of any given section on the current in a parallel conductor whose section is also given, we have to find the integral where doc dy is an element of the area of the first section, dx'dy' an element of the second section, and r the distance between these elements, the integration being extended first over every element of the first section, and then over every element of the second. If we now determine a line R, such that this integral is equal to where A1 and A2 are the areas of the two sections, the length of R will be the same whatever unit of length we adopt, and whatever system of logarithms we use. If we suppose the sections divided into elements of equal size, then the logarithm of R, multiplied by the number of pairs of elements, will be equal to the sum of the logarithms of the distances of all the pairs of elements. Here R may be considered as the geometrical mean of all the distances between pairs of elements. It is evident that the value of R must be intermediate between the greatest and the least values of r. If RA and RB are the geometric mean distances of two figures, A and JB, from a third, C} and if RA+B is that of the sum of the two figures from C, then (A + B) log RA+B=A log RA + B log RB. By means of this relation we can determine R for a compound figure when we know R for the parts of the figure. 692.] EXAMPLES. (1) Let R be the mean distance from the point 0 to the line AB. Let OP be perpendicular to AB, then AB (log R + 1) = AP log OA + PB log OB+ OP AOB. i / Fig. 41. * Trans. R. S. Edin., 1871-2. 692.] GEOMETRIC MEAN DISTANCE. 295 (2) For two lines (Fig. 42) of lengths a and b drawn perpendicu lar to the extremities of a line of length c and on the same side of it. «£ (2 log 72 +3) = (c2 - (a-b}2) log+/c2 + (a- &)* + c2 log c 4- (a2 — c2) log \/a2 + c2 4- (b2 — c2) log \/b2 4- c2 / z\ * i a — ^ ~u - • -b — c(a — o) tan"1 — Fig. 42. (3) For two lines, PQ and RS (Fig. 43), whose directions inter sect at 0. PQ.RS(2logR+3) = logPR(20P.ORsin20-PR2cosO) + logQS(20Q.OSsin20-QS2cosO) - log PS (2 OP. OS sin2 0 - PS2 cos 0) -sinO {OP2. SPR- OQ2. S'QR+OR2. PltQ-OS2. PSQ}. Fig. 43. (4) For a point 0 and a rectangle ABCD (Fig. 44). Let OP, OQ, OR, OS, be perpendiculars on the sides, then AB.AD (2 log 72+ 3) = 2.0P.OQ log OA + 2 .OQ. OR log OB + 2. OR. OS log OC + 2.0S.OP logOD Fig. 44. 296 PARALLEL CURRENTS. [693. (5) It is not necessary that the two figures should be different, for we may find the geometric mean of the distances between every pair of points in the same figure. Thus, for a straight line of length 0, log 72 = log a—f, or E = ae~%, R = 0.223130. (6) For a rectangle whose sides are a and d, }0gR = logvV+^-iJiog /y/i + ^-^°g V1 + & + ietan-i*+i-tan-i£-«. o a a b When the rectangle is a square, whose side is 0, log 5 = Iog0 + i log 2 + | -ff, R = 0.447050. (7) The geometric mean distance of a point from a circular line is equal to the greater of the two quantities, its distance from the centre of the circle, and the radius of the circle. (8) Hence the geometric mean distance of any figure from a ring bounded by two concentric circles is equal to its geometric mean distance from the centre if it is entirely outside the ring, but if it is entirely within the ring al a2 where 0j and 02 are the outer and inner radii of the ring. R is in this case independent of the form of the figure within the ring. (9) The geometric mean distance of all pairs of points in the ring is found from the equation log R = ^0! — 2J^ log ^ 4- J *l ~®\ . For a circular area of radius 0, this becomes log R = Iog0-i, or R = ae~*, R = 0.77880. For a circular line it becomes 693.] In calculating the coefficient of self-induction of a coil of uniform section, the radius of curvature being great compared with 693-] SELF-INDUCTION OF A COIL. 297 the dimensions of the transverse section, we first determine the geometric mean of the distances of every pair of points of the section by the method already described, and then we calculate the coefficient of mutual induction between two linear conductors of the given form, placed at this distance apart. This will be the coefficient of self-induction when the total cur rent in the coil is unity, and the current is uniform at all points of the section. But if there are n windings in the coil we must multiply the coefficient already obtained by n2, and thus we shall obtain the coefficient of self-induction on the supposition that the windings of the conducting wire fill the whole section of the coil. But the wire is cylindric, and is covered with insulating material, so that the current, instead of being uniformly distributed over the section, is concentrated in certain parts of it, and this increases the coefficient of self-induction. Besides this, the currents in the neighbouring wires have not the same action on the current in a given wire as a uniformly distributed current. The corrections arising from these considerations may be de termined by the method of the geometric mean distance. They are proportional to the length of the whole wire of the coil, and may be expressed as numerical quantities, by which we must multiply the length of the wire in order to obtain the correction of the coefficient of self-induction. Let the diameter of the wire be d. It is covered with insulating material, and wound into a coil. We shall suppose that the sections of the wires are in square order, as in Fig. 45, and that the distance between the axis of each wire and that of the next is D, whether in the direction of the breadth or the depth of the coil. D is evidently greater than d. We have first to determine th ' excess of self-induction of unit of length of a cylindric wire of diameter d over that of unit of length of a square wire of side D, or , R for the square Og* R for the circle o o o o o o o o o Fig. 45. D = 2 (log-T + 0.1380606) 298 PARAkCEL CURRENTS. [693. The inductive action of the eight nearest round wires on the wire under consideration is less than that of the corresponding eight square wires on the square wire in the middle by 2x(. 01971). The corrections for the wires at a greater distance may be neg lected, and the total correction may be written 2(loge-=- + 0.11835). The final value of the self-induction is therefore L — n2M+ 2/(loge -j + 0.11835), where n is the number of windings, and I the length of the wire, M the mutual induction of two circuits of the form of the mean wire of the coil placed at a distance R from each other, where R is the mean geometric distance between pairs of points of the section. D is the distance between consecutive wires, and d the diameter of the wire. CHAPTER XIV. CIRCULAR CURRENTS. Magnetic Potential due to a Circular Current. 694.] THE magnetic potential at a given point, due to a circuit carrying a unit current, is numerically equal to the solid angle sub tended by the circuit at that point ; see Arts. 409, 485. When the circuit is circular, the solid angle is that of a cone of the second degree, which, when the given point is on the axis of the circle, becomes a right cone. When the point is not on the axis, the cone is an elliptic cone, and its solid angle is numerically equal to the area of the spherical ellipse which it traces on a sphere whose radius is unity. This area can be expressed in finite terms by means of elliptic integrals of the third kind. We shall find it more convenient to expand it in the form of an infinite series of spherical harmonics, for the facility with which mathematical operations may be performed on the general term of such a series z more than counterbalances the trouble of calculating a number of terms suffi cient to ensure practical accuracy. For the sake of generality we shall assume the origin at any point on the axis of the circle, that is to say, on the line through the centre perpen dicular to the plane of the circle. Let 0 (Fig. 46) be the centre of the circle, C the point on the axis which we assume as origin, H a point on the circle. Describe a sphere with C as centre, and CH as radius. The circle will lie on this sphere, and will form a small circle of the sphere of angular radius a. Fig. 46. 300 CIRCULAR CURRENTS. [694. Let CH = c, OC = b — c cos a, OH= a = c sin a. Let A be the pole of the sphere, and Z any point on the axis, and let CZ=z. Let R be any point in space, and let CR = r, and ACR = 6. Let P be the point when CR cuts the sphere. The magnetic potential due to the circular current is equal to that due to a magnetic shell of strength unity bounded by the current. As the form of the surface of the shell is indifferent, provided it is bounded by the circle, we may suppose it to coincide with the surface of the sphere. We have shewn in Art. 670 that if P is the potential due to a stratum of matter of surface-density unity, spread over the surface of the sphere within the small circle, the potential due to a mag netic shell of strength unity and bounded by the same circle is * = ii(rP). c dr ^ ' We have in the first place, therefore, to find P. Let the given point be on the axis of the circle at Z, then the part of the potential at Z due to an element dS of the spherical surface at P is $$ ~ZP' This may be expanded in one of the two series of spherical har monics, r], or ++&c. + <i + &c .j> the first series being convergent when z is less than c, and the second when z is greater than c. Writing dS = — c2 dp dfa and integrating with respect to <£ between the limits 0 and 2?r, and with respect to //, between the limits cos a and 1, we find or P=2vQ0dp + to>.+ rQidp. (O By the characteristic equation of Qi} 695-] SOLID ANGLE SUBTENDED BY A CIRCLE. 301 Hence ^=. (2) J^ ^ ^(^ + l) dp This expression fails when i = 0, but since Q0 = 1, As the function -~ occurs in every part of this investigation we d //. shall denote it by the abbreviated symbol Q/. The values of Q/ corresponding to several values of i are given in Art. 698. We are now able to write down the value of P for any point R, whether on the axis or not, by substituting r for z, and multiplying each term by the zonal harmonic of 6 of the same order. For P must be capable of expansion in a series of zonal harmonics of 0 with proper coefficients. When 0 = 0 each of the zonal harmonics becomes equal to unity, and the point E lies on the axis. Hence the coefficients are the terms of the expansion of P for a point on the axis. We thus obtain the two series (4) (4') 695.] We may now find o>, the magnetic potential of the circuit, by the method of Art. 670, from the equation We thus obtain the two series (6) !C2 i t? &'(«)& W + &c- + J+ The series (6) is convergent for all values of r less than c, and the series (6r) is convergent for all values of r greater than <?. At the surface of the sphere, where r — c, the two series give the same value for <o when Q is greater than a, that is, for points not occupied by the magnetic shell, but when 6 is less than a, that is, at points on the magnetic shell, 0/= CO+47T. (7) If we assume 0, the centre of the circle, as the origin of co ordinates, we must put a = - , and the series become 302 CIRCULAR CURRENTS. 1 0 (n a _ 1 - [696. . (8) where the orders of all the harmonics are odd *. 0# the Potential Energy of two Circular Currents. 696.] Let us begin by supposing the two magnetic shells which are equivalent to the currents to be portions of two concentric spheres, their radii being c^ and <?2, of which c^ is the greater (Fig. 47). Let us also suppose that the axes of the two shells coincide, and that QJ is the angle subtended by the radius of the first shell, and ez2 the angle subtended by the radius of the second shell at the centre C. Let o^ be the potential due to the first shell at any point within it, then the work required to carry the second shell to an infinite distance is the value of the surface-integral r/wco, JJ dr Hence Fig. 47. extended over the second shell. 4** sin* al(y< or, substituting the value of the integrals from equation (2), Art. 694, * The value of the solid angle subtended by a circle may be obtained in a more direct way as follows. — The solid angle subtended by the circle at the point Z in the axis is easily shewn i-* (a)) + &c. C Expanding this expression in spherical harmonics, we find (cos a-l) + (Q, ^cosa-Qo (a))- +&c. + (<& (a) coso- C for the expansions of cw for points on the axis for which z is less than c or greater than c respectively. Remembering the equations (42) and (43) of Art. 132 (vol. i. p. 165), the coefficients in these equations are evidently the same as those we have now obtained in a more convenient form for computation. 698.] POTENTIAL OF TWO CIRCLES. 303 697.] Let us next suppose that the axis of one of the shells is turned about C as a centre,, so that it now makes an angle 0 with the axis of the other shell (Fig. 48). We have only to introduce the zonal harmonics of 0 into this expression for M, and we find for the more general value of M, This is the value of the potential energy due to the mutual action of two circular currents of unit strength, placed so that the normals through the centres of the circles meet in a point C in an angle 0, the distances of the circumferences of the circles from the point C being <?x and c2 , of which c± is the greater. If any displacement dx alters the value of M, then the force acting in the direc tion of the displacement is X = -=— • For instance, if the axis of one of the shells is free to turn about the point C, so as to cause 0 to vary, then the moment of the force tending to increase & is 0, where _ dM Performing the differentiation, and remembering that dB where (j)/ has the same signification as in the former equations, 0 = — 4 7T2 sin2a1 sin2 a2 sin 0 c2 < J — $/(%) §/(a2) Qi(Q) + &c. ^ 1 698.] As t1 e values of Q{ occur frequently in these calculations the following table of values of the first six degrees may be useful. In this table /x stands for cos 0, and v for sin 6. 304 CIRCULAR CURRENTS. [699. 699.] It is sometimes convenient to express the series for M in terms of linear quantities as follows : — • Let a be the radius of the smaller circuit, I the distance of its plane from the origin, and c = \/a2-\-b2. Let A, B, and C be the corresponding- quantities for the larger circuit. The series for M may then be written, A2 M= 1.2.7T2^02COS0 C3 4- 2.3.7T2 -y=- a2b (cos2 6- i sin2(9) + 3.4.7T2 A2(£2-*A^ a2 (£2_1 ^2)(COS30_ 3 sin2 fl cog tf) -f &C.