XXXIX. Note on the Radiation from an Alternating Circular Electric Current. By Prof. W. McF. ORR, M.A.*

1. N this Journal of December 1903, p. 670, formulæ are given for the magnetic force at a great distance from a perfect conductor in the form of a very thin circular ring which carries an alternating current. I did not then notice that these formulæ can be expressed very simply in terms of Bessel's functions: I understand that the connexion had been noticed, though not definitely stated, by Dr. Pocklington (see 'Nature,' Mar. 26, 1903): it may, however, be worth while investigating more fully the expression which can thus be obtained for the rate of radiation of energy.

2. In the notation of the former paper, in case of a current of the type C1 cos (KS + KVt) + C2 cos (-KS+Vt), where s denotes distance measured along the ring from a fixed point, we have for the components of magnetic force at a point which is at a large distance R from the centre of the ring, and whose colatitude and longitude measured from the point s=0 are 0, 4, respectively, the equations

2

-(-)°C, sin (-ap+Vt-R)}F,(0) †, B÷-σR ̈1 cos 0 { С1 cos κ(ap+Vt−R)

-1

1

+(-)°C2 cos (-a$+Vt−R) } ƒ1(0),

yoR-1 sin {C1 cos (a + Vt-R)

where

F1(0):

=

+(-)°C2 cos (-ap+Vt-R)}fi(0),

cos cos(+ sin 0 sin )d,

(8)=1 sin sin σ(+ sin ✪ sin y)dy,

and κа=σ=any integer, a being the radius of the ring.

3. The relation Ka=σ=an integer holds, approximately, for some of the free periods ‡, which were more immediately referred to in the previous paper; if, however, the current is of the type C1 cos (op' +Vt) + C2 cos (− op' + xVt),

*Communicated by the Author.

+ The accidental omission of a factor from the expressions *(±ap+Vt−R) is here corrected.

The equation determining the free period appears, however, for every value of o to have an infinite number of roots; and there appear to be modes of free motion corresponding to the case σ=0, one of the corresponding values of being a pure imaginary.

K

o' denoting the longitude, where σ is still an integer but is unrestricted, it is readily seen that we obtain a=-—κаR-1 cos 0 { C1 sin (σ$ + Vt—R) — (−)°C, sin ( − σ$+«Vt−KR)}

S

π

cos cos (o+ka sin 0 sind,

-

with corresponding changes in the formulæ for ẞ and y.
Now

and

hence

cos (~ sin¥) =J。(x) + 2J2(μ) cos 2¥ + 2J ̧ (π) cos 4¥+...

sin (a sin )=2J, (x) sin ↓ + 2J3 (c) sin 34+ .....

cos (σ+æ sin¥) = cos σ¥{Jo(a)+2J1⁄2 (μ) cos 2¥ + 2J ̧(x) cos 4¥ + ..... } sin o{2J1(x) sin + 2J3(x) sin 3¥+.....}. Multiplying by cos and integrating, we obtain

cos cos (σ+ sin ↓)dỵ = ( − )°+1π{Jo_1(x)+Jo+1(x)}

and from the same equations this is readily seen to be

(−)°+12πJ'(x).

If now in the expression for the current we write C1=C2 =C/2, we see that the magnetic force due to the current C cos op cos Vt is given by the equations

a=2СπкаR-1 cos e sin op cos (Vt-R). σJ. (xa sin 0)/xa sin 0,

ка

if σ even;

oг ==—2СπêаR−1 cos 0 cos o sin ê(Vt—R). σJ ̧(xа sin 0) /κa sin 0,

into

7. being obtainable from 8 by changing the factor cos

- sin 0.

Hence the time-mean value of a2+ B2+y2 is
=2π2C2x2a2R2{cos2 0 sin2 op. o2J2(xa sin 0)/x2ɑ2 sin3 (
+cos2 op. J(ka sin 0)},

if σ is even, and an expression obtainable from this by inter-
changing sin op, cos op, ifo is odd. And the mean rate of
radiation across the sphere of radius R, being the mean
value of (4π)−1V√(a2+B2+y2)dS, accordingly becomes

**Ca3VS* sin {σ2J2(xa sin 0) cos2 0/x2a2 sin2 0+J (xa sin 0)}de,

0

σ

except that if o=0 the above must be doubled, since in that
case the mean value of cos op is 1 instead of . By means
of the relations

σx¬1J。(x)={{Jo_1(x) +Jo+1(x)}
J'(x) = {{J。-1(x)—J。+1(x)}

this may be written in the form

Ja(r)=(0)22-2020 F{o + 1; σ+1, 20+1; -a2} *, where F(a; P1, P2; y) denotes the hypergeometric series

α

1+

y+ Ριρο 1

a(a+1)

P1(P1+1) P2(P2+1)1.293 + ...

and integrating each term separately, we see that

S

sin 0.J2(v sin 0)d0=II (20+1)

F(o + 1; 0+3, 2σ+1; − a2)

0

The expression for the rate of radiation may thus be written in the form

+

(Kα) 20+2

II (20+3)

202(ka) 20-2

II (20+1).

F(0 + 3 ; 0+ 3, 2o +3; −x2α2),

F(o + ↓ ; o + §, 20 + 1; − x2a2)},

* Proc. Camb. Phil. Soc. May 1899.

or in the equivalent form

ka

¿π°C2xaV (* {J20-2(2x) +J20+2(2x) −20°x ̄°à ̄2J2(2x)}dæ,

the latter of which can be expressed also as

This expression agrees with that obtainable from Pocklington's paper by using the relation there obtained between the current and the electric force at the surface of the wire. (The solution there taken appears, however, to correspond to divergent, not convergent, waves.)

4. In case κα=σ the rate of radiation thus assumes the simple form

5. When σ=0 the expression obtained for the general case in § 3 has, as previously stated, to be doubled, and the result may thus be written

In case ka is small this becomes T2C2'a1V/3. The radiation in this case was investigated by FitzGerald †; his result is one-half of the above; he has, however, taken the mean value of sin2 (colatitude) over a sphere as instead of 3.

6. If xa is very great we may use approximate forms of the hypergeometric series for large values of the variable, or may proceed more simply as follows:-Except for small values of 0, the a component of the magnetic force at a great distance is now very small compared with the resultant of B, 7, and the first formula for the mean rate of radiation assumes the approximate form

2

sin OJ (ka sin 0)do.

Evidently in this integral we may use the approximate formula for J'(x) when a is large, viz.:

(2/a) sin {(2+1)π/4−x},

"Electrical Oscillations in Wires," Proc. Camb. Phil. Soc. 1897. + 'Scientific Writings,' p. 125; Trans. R. D. S. Nov. 18, 1883.

Thus the rate of radiation becomes 72C2V/2, but, as before, this expression is to be doubled in case σ=0.

7. Any periodically alternating current can be resolved into a number (in general doubly infinite) of elementary currents each of the simple harmonic type discussed above, and it is evident that the mean rate of radiation is the sum of the mean rates of radiation due to each of such constituents separately.

It appears from this and from the result of § 6 that if the current is a simple harmonic function of the time, and if the wave-length in free space is small compared with the radius, the radiation approximately depends only on the mean value of the square of the current averaged round the circle, and is otherwise independent of the law of variation from point to point.

8. Lord Rayleigh, in an investigation of the work done by given forces applied at given points of an elastic solid, has referred to the case when the forces act tangentially along a circle, in connexion with the subject of the present note. He states that "it would seem that (33) must lead to a more complicated expression for the energy radiated than that in Dr. Pocklington's investigation." As I understand it he considers in his expression (33) the applied forces F, F', to be the analogues of electromotive forces and proposes to replace them therein by expressions in terms of the conduction current obtained from Pocklington's resuits. The parallelism between the electromagnetic and the elastic-solid theories does not appear, however, to extend so far. A current of conduction presents itself in the electromagnetic equations as a discontinuity in the time-rate of change of the electric force t. The force F applied to the elastic solid is the analogue, not of an applied electromotive force, but of -du/dt, where u is an impressed conduction current; accordingly in Lord Rayleigh's expressions for the displace ment of the medium at a point P due to a force F applied at another point O, F is to be regarded as the equivalent of -du/dt. Yet again, in obtaining the rate of radiation we do not multiply F', or -du/dt, by the velocity of displacement *Phil. Mag. Oct. 1903.

+ Compare Macdonald, Electric Waves,' p. 16.