the strength of the current is varied according to the simple periodic law. The circuit will be supposed to be a circle of small area S, whose centre is the origin and whose plane is the plane of xy; and the surrounding medium will be supposed to be free aether. The current may be taken to be of strength A cos (2πt/T), so that the moment of the equivalent magnet is SA cos (2πt/T). Now in the older electrodynamics, the vector-potential due to a magnetic molecule of (vector) moment M at the origin is (1/4π) curl (M/r), where r denotes distance from the origin. The vector-potential due to FitzGerald's magnetic oscillator would therefore be (1/4π) curl K, where Kdenotes a vector parallel to the axis of z, and of magnitude (1/r) SA cos (2πt/T). The change which is involved in replacing the assumptions of the older electrodynamics by those of Maxwell's theory is in the present case equivalent[1] to retarding the potential; so that the vector-potential a due to the oscillator is (1/4π) curl K where K is still directed parallel to the axis of z, and is of magnitude
.
The electric force E at any point of space is , and the magnetic force H is curl a: so that these quantities may be calculated without difficulty. The electric energy per unit volume is E2/8πc2: performing the calculations, it is found that the value of this quantity averaged over a period of the oscillation and also averaged over the surface of a sphere of radius r is
.
The part of this which is radiated is evidently that which is proportional to the inverse square of the distance,[2] so the
- ↑ Cf. pp. 298, 299.
- ↑ The other term, which is neglected, is very small compared to the term retained, at great distances from the origin; it is what would be obtained if the effects of induction of the displacement-currents were neglected: i.e. it is the energy of the forced displacement-currents which are produced directly by the variation of the primary current, and which originate the radiating displacement-currents.