660.]

DECAY OF CURRENTS IN THE SHEET.

267

differentiating with respect to a or y, the arbitrary function of z and t will disappear. We shall therefore leave it out of account.

a certain velocity, the equation between P and P' becomes

659.] Let us first suppose that there is no external magnetic system acting on the current sheet. We may therefore suppose P=0. The case then becomes that of a system of electric currents in the sheet left to themselves, but acting on one another by their mutual induction, and at the same time losing their energy on account of the resistance of the sheet. The result is expressed by the equation

the solution of which is

dP dP R =

dz dt

P = f (x, y, (z+Rt)).

(22)

(23)

Hence, the value of P on any point on the positive side of the sheet whose coordinates are x, y, z, and at a time t, is equal to the value of P at the point x, y, (z+Rt) at the instant when t=0.

If therefore a system of currents is excited in a uniform plane sheet of infinite extent and then left to itself, its magnetic effect at any point on the positive side of the sheet will be the same as if the system of currents had been maintained constant in the sheet, and the sheet moved in the direction of a normal from its negative side with the constant velocity R. The diminution of the electromagnetic forces, which arises from a decay of the currents in the real case, is accurately represented by the diminution of the force on account of the increasing distance in the imaginary case. 660.] Integrating equation (21) with respect to t, we obtain

If we suppose that at first P and P' are both zero, and that a magnet or electromagnet is suddenly magnetized or brought from an infinite distance, so as to change the value of P' suddenly from zero to P', then, since the time-integral in the second member of (24) vanishes with the time, we must have at the first instant

at the surface of the sheet.

P-P'

Hence, the system of currents excited in the sheet by the sudden

introduction of the system to which P' 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', as before, be the function from which the direct action of this system is to be deduced by the equations (3), (9), &c., dP

then dt will be the function corresponding to the system re

dt

664.]

presented by

dM
dt

MOVING TRAIL OF IMAGES.

269

St. This quantity, which is the increment of M

in the time dt, may be regarded as itself representing a magnetic

system.

If we suppose that at the time t a positive image of the system dM

dt is formed on the negative side of the sheet, the magnetic dt action at any point on the positive side of the sheet due to this image will be equivalent to that due to the currents in the sheet excited by the change in M during the first instant after the change, and the image will continue to be equivalent to the currents in the sheet, if, as soon as it is formed, it begins to move in the negative direction of z with the constant velocity R.

If we suppose that in every successive element of the time an image of this kind is formed, and that as soon as it is formed it begins to move away from the sheet with velocity R, we shall obtain the conception of a trail of images, the last of which is in process of formation, while all the rest are moving like a rigid body away from the sheet with velocity R.

663.] If P denotes any function whatever arising from the action of the magnetic system, we may find P, the corresponding function arising from the currents in the sheet, by the following process, which is merely the symbolical expression for the theory of the trail of images.

Let P, denote the value of P (the function arising from the currents in the sheet) at the point (x, y, z+RT), and at the time t-T, and let P, denote the value of P' (the function arising from the magnetic system) at the point (x, y, − (z+RT)), and at the time t-T. Then dP, dP

dP,

= R dr

and equation (21) becomes

(25)

T

(26)

and we obtain by integrating with respect to 7 from 70 to 7=∞,

as the value of the function P, whence we obtain all the properties of the current sheet by differentiation, as in equations (3), (9), &c.

664.] As an example of the process here indicated, let us take the case of a single magnetic pole of strength unity, moving with uniform velocity in a straight line.

Let the coordinates of the pole at the time t be

The coordinates of the image of the pole formed at the time

t-T are

¿ = u (t−7),

n = 0,

S=(c+w(t−7) +RT),

and if r is the distance of this image from the point (x, y, z),

To obtain the potential due to the trail of images we have to

calculate

If we write

dr 1

=

Q

log {Qr+u(x-ut)+(R−w) (z+c+wt)},

the value of r in this expression being found by making 7 = 0. Differentiating this expression with respect to t, and putting t = 0, we obtain the magnetic potential due to the trail of images, w (z+c)-ux

Q

-u2-w2+Rw

Qr+ux+(R-w) (z+c)

By differentiating this expression with respect to a or 2, we obtain the components parallel to x or z respectively of the magnetic force at any point, and by putting a = 0, z = c, and r = 2c in these expressions, we obtain the following values of the components of the force acting on the moving pole itself,

665.] In these expressions we must remember that the motion is supposed to have been going on for an infinite time before the time considered. Hence we must not take w a positive quantity, for in that case the pole must have passed through the sheet within a finite time.

If we make u = 0, and w negative, X = 0, and

or the pole as it approaches the sheet is repelled from it.

If we make w = 0, we find Q2 = 112 + R2,

668.]

FORCE ON MOVING POLE.

271

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 u 1.27 R.

When the sheet is a non-conductor, R∞ 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 1 when the velocity is infinite. It has the same value when

4c2

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 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 * Annales de Chimie et de Physique, 1826.