Page:PoyntingTransfer.djvu/2

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it is first recognisable as electric and magnetic to the parts where it is changed into heat or other forms?

The aim of this paper is to prove that there is a general law for the transfer of energy, according to which it moves at any point perpendicularly to the plane containing the lines of electric force and magnetic force, and that the amount crossing unit of area per second of this plane is equal to the product of the intensities of the two forces, multiplied by the sine of the angle between them, divided by 4\pi, while the direction of flow of energy is that in which a right-handed screw would move if turned round from the positive direction of the electromotive to the positive direction of the magnetic intensity. After the investigation of the general law several applications will be given to show how the energy moves in the neighbourhood of various current-bearing circuits.

The following is a general account of the method by which the law is obtained.

If we denote the electromotive intensity at a point (that is, the force per unit of positive electrification which would act upon a small charged body placed at the point) by \mathfrak{E}, and the specific inductive capacity of the medium at that point by K, the magnetic intensity (that is, the force per unit pole which would act on a small north-seeking pole placed at the point) by \mathfrak{H} and the magnetic permeability by \mu, Maxwell's expression for the electric and magnetic energies per unit volume of the field is

K\mathfrak{E}^{2}/8\pi+\mu\mathfrak{H}^{2}/8\pi (1)

If any change is going on in the supply or distribution of energy the change in this quantity per second will be

K\mathfrak{E}\frac{d\mathfrak{E}}{dt}/4\pi+\mu\mathfrak{H}\frac{d\mathfrak{H}}{dt}/4\pi (2)

According to Maxwell the true electric current is in general made up of two parts, one the conduction-current \mathfrak{K}, and the other due to change of electric displacement in the dielectric, this latter being called the displacement-current. Now, the displacement is proportional to the electromotive intensity, and is represented by K\mathfrak{E}/4\pi, so that when change of displacement takes place, due to change in the electromotive intensity, the rate of change, that is, the displacement-current, is K\tfrac{d\mathfrak{E}}{dt}/4\pi, and this is equal to the difference between the true current \mathfrak{E} and the conduction-current \mathfrak{K}. Multiplying this difference by the electromotive intensity \mathfrak{E} the first term in (2) becomes

\frac{K\mathfrak{E}}{4\pi}\frac{d\mathfrak{E}}{dt}=\mathfrak{EE-KE} (3)

The first term of the right side of (3) may be transformed by substituting for the components of the total current their values in terms of the components of the magnetic intensity, while the second term, the product of the conduction-current