SPECTRUM. This article will deal with the history of the initiation of the subject and the first stages of its development.
It is to Clerk Maxwell that we owe not only the origination of the idea that electrical and magnetic effects are propagated by waves, but also the establishment of equations giving a mathematical theory by which these effects can be cal culated. The equations known as Maxwell's equations are, if we accept the view that the constitution of matter is fundamentally electrical, the key to the solution of many of the most funda mental problems in physics. Maxwell's discovery was inspired by Faraday's view that the forces between electrified bodies were not due to direct action at a distance between the electric charges on these bodies, but to the presence in the region occupied by these bodies of lines of f orce, these lines started from bodies charged with electricity of one sign and ended on bodies charged with electricity of the opposite sign; these lines of force were supposed to be in a state of tension, and thus were striving to shorten while at the same time they repelled laterally the adjacent lines of force. These tensions and repulsions produced in Faraday's view the forces which electrified bodies exert on each other. Maxwell's first paper was "On Faraday's Lines of Electric Force" (Proc. Cambridge Philosophical Society, x. part i., 1855). In this he puts Faraday's conception into a form suitable for mathematical treat ment by the introduction of the idea of tubes of force, a tube of force being the tubular service formed by the lines of force which pass through a small closed curve drawn in the elec tric field. He shows that the electric force at any point along this tube is inversely proportional to the cross section of the tube at the point, so that the tubes are thinner when the force is large than when it is small, and the variations in the sizes of the tube will indicate the variation in the electric force. These tubes start from positively electrified bodies and end on negatively electrified ones. The charge enclosed by the tube on the surface from which it starts is equal to the charge it encloses on the surface where it ends, if the size of the tube is chosen so that this charge is unity. Thus a charge of electricity e will be the origin or terminus of e tubes of force. In the Philosophical Magazine, March, April, May, 1861, he works out a theory of magnetic lines of force, sup posing that a tube of magnetic force is a vortex whose axis of rotation coincides with the direction of the force, he shows that these vortexes would give rise to forces analogous to those ob served in the magnetic field. Inasmuch as contiguous positions of neighbouring vortexes must be moving in opposite directions he supposes that the vortexes are separated by particles which act like the idle wheels in a train of mechanism, so that each vortex has a tendency to make a neighbouring vortex rotate in the same direction as itself. The motion of these particles constituted in Maxwell's view the electric current. Though Maxwell in his later papers did not make much reference to this theory, it seems to have been the consideration of this hypothetical mechanism which suggested the conception which is the very keystone of his theory in the final form given in the paper "A Dynamical Theory of the Electromagnetic Field" (Phil. Trans., clv. 1864). This conception was that magnetic forces could be produced not only by the ordinary convective currents flowing through wires and electro lytes which had previously been supposed to be the sole source of magnetic force, but by another type of current to which he gave the name of displacement currents, and which can occur in insulators as well as conductors. The displacement current exists when the electric force is changing and the components of the intensity of this current, i.e., the current per unit area parallel to the axes x, y, z are where X, Y, Z, are the components of the electric force and K the specific induction capacity of the medium.
The difference between Maxwell's theory and the earlier ones is well illustrated by the consideration of the charging up of a condenser. Suppose the condenser consists of two parallel metal plates A and B and that it is charged by connecting A and B by wires with the terminals of a battery, if Q, —Q are the charges at a time t on the plates, then there is a convective current i along the wires equal to dQ/dt, this on the old view would be the only current that would have to be taken into ac count when calculating the magnetic forces; there would be no currents in the dielectric between the plates and the magnetic forces in that region would be derivable from a potential. On Maxwell's view there is a displacement current between the plates equal to K dX per unit area, where X is the electric force
dt between the plates and K the specific inductive capacity of the dielectric separating them. The density of the electricity on a plate is equal to Q/A where A is the area of a plate, hence by Coulomb's law so that where i is the current through the wires, this is the expression for the displacement current per unit area, the total current across the area A is equal to Ai/A or i. Thus the displacement current through the dielectric is the same as the current through the wire, so that the Maxwellian currents form a closed circuit; this can easily be seen to be true generally, so that on Maxwell's theory all current circuits are closed. The displacement as well as the con vective currents admit of simple representation in terms of the motion of tubes of electric force (J. J. Thomson, ELments of Elec tricity and Magnetism, chap. xiii.). The wave propagation of elec tric and magnetic force is an immediate consequence of Maxwell's generalization of the idea of an electric current. For let u, v, w be the components of the effective current, (X, Y, Z) (a, [3, y) the components of the electric and magnetic force respectively, K the specific inductive capacity of the medium, y its magnetic per meability. The equations which express Amperes law that the work done in taking a magnet pole round a closed circuit is equal to 4 times the current passing through the circuit are dy di3 da dy d f3 da4ru=---, 41rv=--- w 4tr=--- (A) dy dz dz dx ' dx dy while those which express Faraday's law that the electromotive force round a closed circuit is equal to the rate of diminution of the magnetic induction through the circuit are dad_Z_dY
_
_ dt dy dz ' µ dt dz
µ dt dx dy (B) Take now the case of a dielectric in which there are no con vective currents, then u= K dX , hence differentiating (1) of A 47r (It with respect to t we have K
d d
d d (3(C)
dy dt dz dt substituting the values of dy and
from 13, we get, since in the db dt dX dY dZ
+
— =o, µK = +
dx dy dz
dye
similar equations for Y, Z, a, (3, y. These equations are of the type of the wave equation I
=
i + i and rep jJ2
resent a disturbance propagated as a wave with the velocity V, in the electrical case is given by V = i/ Thus we see that on Maxwell's theory the components of the electric and magnetic force in a dielectric satisfy the wave equa tion, so that electric and magnetic forces travel as waves through the dielectric, the velocity of propagation being i/ -V µK. Now and K can be measured by purely electrical methods, the value of I/ µK for air being the ratio of the electrostatic to the electro magnetic unit of electricity which has been found by experi mental study to be very nearly 3 X io" , and thus equal to the velocity of light. Thus it follows from Maxwell's theory that electric and magnetic disturbances travel through dielectrics with the velocity ca light.
Let us consider the distribution of electric force in a plane electromagnetic wave. If X, Y, Z are the components of the electric force, 1, yn, n the direction cosines of the normal to the wave front and X the wave length we may put the second term on the left hand side of above equation is small compared with the first and may be neglected and the equation approximates to the wave equation, if on the other hand p is small compared with 47r/KO the first term is small compared with the second and the equation approximates to that which expresses the conduction of heat.
A case of great importance in the theory of the propagation of electrical waves through the atmosphere is when the conductivity of the medium is due to the presence of gaseous ions and the pressure of the gas is so low that the motion of the ions is not interfered with by collisions with the molecules of the gas. In this case, p, q, r the components of the convective current are given by the equations p = ZeE, q =
r = Zed' where t,
t are the components of the velocity of an ion, e its charge and the sum mation is to be extended to all the ions in unit volume. Thus d p = and since the motion of the ions is free mj = Xe, where dt m is the mass of an ion.
Thus if u is the total current du — _ K
+ ziet.
at 47r dt _ K
47r
+ m •X if there are n negative and n positive ions per unit volume 2 l e
m
where
are the masses of the negative and positive ions respectively, using this value of du/dt in equations (A) we get
d (dX d Y dZ) µK dt2 +4
p02X =
+ d 2 +
dx\ dx + d + dz / y y (equation E) for the electric forces and
d2a
Po2a=
-« (F) dt
dy dz for the magnetic. This equation differs from that for a non conducting dielectric by the presence of the term
on the left hand side, and this produces effects which differentiate sharply the behaviour of waves in a non-conducting and in an ionized medium.
These differences are (I) waves of any frequency can travel through the insulating medium while through the ionized one it is only waves whose frequency is greater than
which are able to do so; (2) the phase velocity of the waves in the insulating medium is independent of the wave length and equal to c the velocity of light; in the ionized medium, if V is the phase velocity and X the wave length V2=c2+ p°2X2 thus the phase velocity is 7-K always greater than the velocity of light and is infinite for in finitely long waves. (3) In the insulating medium the energy travels out with the velocity c, in the ionized medium it travels with the velocity V — X d V dX
since V is always greater than c, the velocity of the energy in the ionized medium is less than in the insulating one.
The velocity of the waves in the ionized medium depends upon the value of pot a quantity which is proportional to the number of ions per cubic centimetre in the medium. Thus if the number of ions in the upper regions of the atmosphere varies with the height above the surface of the earth the waves in these regions will be passing through a medium in which the refractive index varies from place to place, this, as in the analogous optical case of the mirage, will lead to a bending of the waves so that instead of continually travelling away from the earth they may be bent round so as to return to it. There are good reasons for believing that in the upper regions of the atmosphere there is a layer of ionized gas, known as the Heaviside side, and that this plays a most important part in long distance wireless. (See WIRELESS,
