Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/366

 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, whose centre is the origin and whose plane is the plane of ; and the surrounding medium will be supposed to be free aether. The current may be taken to be of strength, so that the moment of the equivalent magnet is. Now in the older electrodynamics, the vector-potential due to a magnetic molecule of (vector) moment M at the origin is, where denotes distance from the origin. The vector-potential due to FitzGerald's magnetic oscillator would therefore be, where denotes a vector parallel to the axis of , and of magnitude. 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 to retarding the potential; so that the vector-potential a due to the oscillator is where  is still directed parallel to the axis of, and is of magnitude

The electric force at any point of space is $$-\mathbf\dot a$$, and the magnetic force  is : so that these quantities may be calculated without difficulty. The electric energy per unit volume is : 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 is

The part of this which is radiated is evidently that which is proportional to the inverse square of the distance, so the