James Clerk Maxwell and Modern Physics

by Richard Glazebrook

We are dealing with a circuit with which lines of magnetic induction are linked, while the number of such lines linked with the circuit is varying. Now, let us suppose the variation to take place in consequence of the lines of induction moving outwards or inwards, as the case may be, so as to cut the circuit. Originally there are none linked with the circuit. As the magnetic field has grown to its present strength lines of magnetic induction have moved inwards. Each little element of the circuit has been cut by some, and the total number linked with the circuit can be found by adding together those cut by each element. Now, Professor Poynting’s statement of Maxwell’s fourth principle is that the electrical force in the direction of any element of the circuit is found by dividing by the length of the element the number of lines of magnetic induction which are cut in one second by it.

Moreover, the total number of lines of magnetic induction which have been cut by an element of unit length is defined as the component of the vector potential in the direction of the element; hence the electrical force in any direction is the rate of decrease of the component of the vector potential in that direction. We have thus a physical meaning for the vector potential, and shall find that in the dynamical theory this quantity is of great importance.

Professor Poynting has modified Maxwell’s third principle in a similar manner; he looks upon the variation in the electric displacement as due to the motion of tubes of electric induction,[63] and the magnetic force along any circuit is equal to the number of tubes of electric induction cutting or cut by unit length of the circuit per second, multiplied by 4π.

From the equations of the field, as found by Maxwell, it is possible to derive two sets of symmetrical equations. The one set connects the rate of change of the electric force with quantities depending on the magnetic force; the other set connects in a similar manner the rate of change of the magnetic force with quantities depending on the electric force. Several writers in recent years adopt these equations as the fundamental relations of the field, establishing them by the argument that they lead to consequences which are found to be in accordance with experiment.

We have endeavoured to give some account of Maxwell’s historical method, according to which the equations are deduced from the laws of electric currents and of electro-magnetic induction derived directly from experiment.

While the manner in which Maxwell obtained his equations is all his own, he was not alone in stating and discussing general equations of the electro-magnetic field. The next steps which we are about to consider are, however, in a special manner due to him. An electrical or magnetic system is the seat of energy; this energy is partly electrical, partly magnetic, and various expressions can be found for it. In Maxwell’s theory it is a fundamental assumption that energy has position. “The electric and magnetic energies of any electro-magnetic system,” says Professor Poynting, “reside, therefore, somewhere in the field.” It follows from this that they are present wherever electric and magnetic force can be shown to exist. Maxwell showed that all the electric energy is accounted for by supposing that in the neighbourhood of a point at which the electric force is R there is an amount of energy per unit of volume equal to KR²/8π, K being the inductive capacity of the medium, while in the neighbourhood of a point at which the magnetic force is H, the magnetic energy per unit of volume is μH²/8π, μ being the permeability. He supposes, then, that at each point of an electro-magnetic system energy is stored according to these laws. It follows, then, that the electro-magnetic field resembles a dynamical system in which energy is stored. Can we discover more of the mechanism by which the actions in the field are maintained? Now the motion of any point of a connected system depends on that of other points of the system; there are generally, in any machine, a certain number of points called driving-points, the motion of which controls the motion of all other parts of the machine; if the motion of the driving-points be known, that of any other point can be determined. Thus in a steam engine the motion of a point on the fly-wheel can be found if the motion of the piston and the connections between the piston and the wheel be known.

In order to determine the force which is acting on any part of the machine we must find its momentum, and then calculate the rate at which this momentum is being changed. This rate of change will give us the force. The method of calculation which it is necessary to employ was first given by Lagrange, and afterwards developed, with some modifications, by Hamilton. It is usually referred to as Hamilton’s principle; when the equations in the original form are used they are known as Lagrange’s equations.

Now Maxwell showed how these methods of calculation could be applied to the electro-magnetic field. The energy of a dynamical system is partly kinetic, partly potential. Maxwell supposes that the magnetic energy of the field is kinetic energy, the electric energy potential. When the kinetic energy of a system is known, the momentum of any part of the system can be calculated by recognised processes. Thus if we consider a circuit in an electro-magnetic field we can calculate the energy of the field, and hence obtain the momentum corresponding to this circuit. If we deal with a simple case in which the conducting circuits are fixed in position, and only the current in each circuit is allowed to vary, the rate of change of momentum corresponding to any circuit will give the force in that circuit. The momentum in question is electric momentum, and the force is electric force. Now we have already seen that the electric force at any point of a conducting circuit is given by the rate of change of the vector potential in the direction considered. Hence we are led to identify the vector potential with the electric momentum of our dynamical system; and, referring to the original definition of vector potential, we see that the electric momentum of a circuit is measured by the number of lines of magnetic induction which are interlinked with it.

Again, the kinetic energy of a dynamical system can be expressed in terms of the squares and products of the velocities of its several parts. It can also be expressed by multiplying the velocity of each driving-point by the momentum corresponding to that driving-point, and taking half the sum of the products. Suppose, now, we are dealing with a system consisting of a number of wire circuits in which currents are running, and let us suppose that we may represent the current in each wire as the velocity of a driving-point in our dynamical system. We can also express in terms of these currents the electric momentum of each wire circuit; let this be done, and let half the sum of the products of the corresponding velocities and momenta be formed.

In maintaining the currents in the wires energy is needed to supply the heat which is produced in each wire; but in starting the currents it is found that more energy is needed than is requisite for the supply of this heat. This excess of energy can be calculated, and when the calculation is made it is found that the excess is equal to half the sum of the products of the currents and corresponding momenta. Moreover, if this sum be expressed in terms of the magnetic force, it is found to be equal to μ H²/8 π, which is the magnetic energy of the field. Now, when a dynamical system is set in motion against known forces, more energy is supplied than is needed to do the work against the forces; this excess of energy measures the kinetic energy acquired by the system.