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material are inclined to each other at an angle of 60 degrees, and the North pole of a magnet placed evenly between them, one would expect to get between the plates exactly the same distribution of field as if there were six poles, three norths and three souths, spaced alternately and evenly round the line of intersection of the plates. It may, however, be said to those who would care to try this experiment, that they must use a point pole of much greater strength than 40 units, or plates much more susceptible than iron if they hope to be at all successful.
The close connexion between the effect of an iron plate magnetically and that of a mirror optieally naturally makes us ask what would be the effect of a curved plate? Would it give magnified or diminished images like a concave or convex mirror, and what would be the position of these images with regard to the object?
If we take the case of spherical curvature, and consider the iron as being infinitely susceptible, the answer to these questions can be determined theoretically from the laws of magnetic potential. All the formulæ relating to electric images already known are applicable to magnetic images.
If we consider two points A and B at which are placed two poles respectively, of the strengths +m1 and m2, the Fig. 12.
magnetic potential at a point C due to m1 is
and that due to -m2 is
If we choose C at a point of zero potential, we have
but is a constant; therefore, if we take C at the points of
moves so that "remains constant it moves on the surface of
a sphere, therefore the equipotential surface of value zero is in the form of a spherical shell. If O be the centre of the sphere, it follows from a well-known property of a circle that
If, therefore, we are given m1 we can find from the equation
the strength of pole m, which when placed at B will give zero potential on a given spherical shell.
Now consider a magnet pole +m1 (fig. 13) brought up to
a point A near a sphere of very susceptible material whose radius is large as compared with the distance between A and its surface, so that we may neglect the potential of the sphere
due to m1 in dealing with the potential of points in the field we are considering, and so that we may take the surface of the sphere as an equipotential surface of practically zero value. The distribution of the field outside the sphere will be the same as if no sphere were there, but instead a pole of OC were placed at B. We may say that the sphere acting like a convex mirror has given a diminished image at B.
We can find the position of B by the following construction-Describe the arc OC (fig. 14) with A as a centre, and Fig. 14.
* [Note added after reading of Paper.-At this point our original paper contained the following remark as a footnote :-"If the magnetic object is large, or is far removed, then besides the image as above defined it is necessary to take into account the raising of the potential of the whole sphere, which would be represented by another image at the centre." some discussion took place on this point after the reading of this paper, it may be well to deal further with the matter. The case is then analogous to the case of an electric charge q brought up to an insulated conducting sphere having no previous charge. Lord Kelvin, in a paper dated July 7th, 1848, has shown that the effect on external points of the charge residing on the surface of such a sphere is the same as the effect of a charge at B (fig. 14), and a further charge of + at the centre รใ
O. His reasoning being applicable to the magnetic case, we see that the image of a North pole at A consists of a doublet having a South pole
situated at B of strength mand a North pole also of the strength m
situated at the centre O. Now any magnetic object such as a solenoid will have a South pole as well as a North pole, and if the object is small compared with the size of the sphere, both North and South may be taken as equidistant from the centre, and their images at the centre will therefore neutralize each other, and we have left the image that is considered in the text. If the object is large as compared with the size of the sphere, then in both the magnetic case and the optical case there is a confusion of images.]
then describe the arc OB with C as a centre, B being on the line AO. The condition that "1 shall be constant for all
points on the circumference is that
shall be equal to
and this is seen to be true from the similarity of the triangles ACO and COB.
If we had any number of magnetic points outside the sphere each would have its virtual image inside the sphere; thus any form of magnet, such as a solenoid carrying a current, would also have its image.
It is easy to see that the experiments above mentioned with a large iron plate can be explained in this way. For if we may consider the plate as part of an infinite sphere:
To assimilate to optical formulæ let us now express the relations in terms of the distance of object and image respectively from the pole E of the mirror, and write AÖ=d; BO=b; u=AE=d-r; v=BE=r-b (fig. 14). Now
The well-known optical mirror formula for a spherical
mirror differs from this in having instead of
In order that a magnetic image as at B in fig. 14 should be produced by a magnetic object at A, and with the relative strengths m, and m1, not only must the reflecting surface pass through the point E such that
but the curvature of the magnetic mirror is defined by the condition previously laid down that
= also. CO M2
Now the optical mirror which will give at B the image of A, and have its middle-point situated at E (as defined by the former of these conditions) will not have O for its centre.
In fact it will be a spherical surface having half the curvature or twice the radius. If (fig. 16) the dotted circle FEG be