## Extract

1. *Energy of a Charged Sphere in Steady Motion*: *Heaviside's Method*.— When the velocity of a system of electric charges is changed in any manner, waves of disturbance travel out from the system in all directions with velocity *v*, the speed of light, and carry both energy and momentum to the distant parts of the electro-magnetic field. Dr. Oliver Heaviside was the first to notice that these waves might be made to yield important information as to energy and momentum of the moving system. Thus, when a charged sphere is in steady motion the electro-magnetic field possesses electric energy U, magnetic energy T and momentum M. If, now, the sphere be suddenly brought to rest, the forces which hold it at rest do no work, since their points of application do not move, and hence the total energy in the field is the same after the sphere is brought to rest as it was before. When the sphere is stopped, a pulse with depth equal to the diameter of the sphere travels outwards. The electric and magnetic forces in this pulse ultimately vary inversely as the distance and hence the energy and the momentum in the pulse tend to constant values as the pulse travels out to infinity. Outside the pulse the electric and magnetic forces are the same as if the sphere had continued in steady motion and therefore are, after an infinite time, inversely proportional to the square of the distance from the point which the centre of the sphere would have reached had its motion continued. Hence the energy and momentum outside the pulse ultimately vanish. On the other hand, the field within the spherical region bounded by the pulse is the same as that due to the sphere at rest and hence the energy in this part of the field is ultimately U_{s}, the electrostatic energy of the sphere at rest. Thus, if W be the limiting value of the energy in the pulse, the energy in the field is ultimately U_{s} + W. But, before the sphere was stopped, the energy was U + T.

## Footnotes

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- Received June 13, 1907.

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