Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/303

The component $X$ represents a retarding force acting on the pole in the direction opposite to that of its own motion. For a given value of $R$, $X$ is a maximum when ${\mathfrak {u}}=1.27\,R$.

When the sheet is a non-conductor, $R=\infty$ and $X=0$.

When the sheet is a perfect conductor, $R=0$ and $X=0$.

The component $Z$ represents a repulsion of the pole from the sheet. It increases as the velocity increases, and ultimately becomes ${\frac {1}{4c^{2}}}$ when the velocity is infinite. It has the same value when $R$ is zero.

666.] When the magnetic pole moves in a curve parallel to the sheet, the calculation becomes more complicated, but it is easy to see that the effect of the nearest portion of the trail of images is to produce a force acting on the pole in the direction opposite to that of its motion. The effect of the portion of the trail immediately behind this is of the same kind as that of a magnet with its axis parallel to the direction of motion of the pole at some time before. Since the nearest pole of this magnet is of the same name with the moving pole, the force will consist partly of a repulsion, and partly of a force parallel to the former direction of motion, but backwards. This may be resolved into a retarding force, and a force towards the concave side of the path of the moving pole.

667.] Our investigation does not enable us to solve the case in which the system of currents cannot be completely formed, on account of a discontinuity or boundary of the conducting sheet.

It is easy to see, however, that if the pole is moving parallel to the edge of the sheet, the currents on the side next the edge will be enfeebled. Hence the forces due to these currents will be less, and there will not only be a smaller retarding force, but, since the repulsive force is least on the side next the edge, the pole will be attracted towards the edge.

Theory of Arago's Rotating Disk.

668.] Arago discovered^[1] that a magnet placed near a rotating metallic disk experiences a force tending to make it follow the motion of the disk, although when the disk is at rest there is no action between it and the magnet.

This action of a rotating disk was attributed to a new kind

1. Annales de Chimie et de Physique, 1826.