Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/300

introduction of the system to which $P^{\prime }$ is due is such that at the surface of the sheet it exactly neutralizes the magnetic effect of this system.

At the surface of the sheet, therefore, and consequently at all points on the negative side of it, the initial system of currents produces an effect exactly equal and opposite to that of the magnetic system on the positive side. We may express this by saying that the effect of the currents is equivalent to that of an image of the magnetic system, coinciding in position with that system, but opposite as regards the direction of its magnetization and of its electric currents. Such an image is called a negative image.

The effect of the currents in the sheet on a point on the positive side of it is equivalent to that of a positive image of the magnetic system on the negative side of the sheet, the lines joining corresponding points being bisected at right angles by the sheet.

The action at a point on either side of the sheet, due to the currents in the sheet, may therefore be regarded as due to an image of the magnetic system on the side of the sheet opposite to the point, this image being a positive or a negative image according as the point is on the positive or the negative side of the sheet.

661.] If the sheet is of infinite conductivity, $R=0$, and the second term of (24) is zero, so that the image will represent the effect of the currents in the sheet at any time.

In the case of a real sheet, the resistance $R$ has some finite value. The image just described will therefore represent the effect of the currents only during the first instant after the sudden introduction of the magnetic system. The currents will immediately begin to decay, and the effect of this decay will be accurately represented if we suppose the two images to move from their original positions, in the direction of normals drawn from the sheet, with the constant velocity $R$.

662.] We are now prepared to investigate the system of currents induced in the sheet by any system, $M$, of magnets or electromagnets on the positive side of the sheet, the position and strength of which vary in any manner.

Let $P^{\prime }$, as before, be the function from which the direct action of this system is to be deduced by the equations (3), (9), &c., then ${\frac {dP^{\prime }}{dt}}\delta t$ will be the function corresponding to the system re-