George Gabriel Stokes to Faraday   7 November 1855

69 Albert Street Regent’s Park | London Nov 7th 1855

My dear Sir,

I should not have ventured to write as I did, considering how deeply you have thought over the subject and how little I have attended to it, were it not that in our conversation I understood you to regard the so-supposed setting of a diamagnetic body in a uniform field across the lines of force as a mystery. However I did not without consideration; and having carefully considered the matter since I see no reason to change my opinion.

I do not question the accuracy of your experiment, Exp. R. 2812¹, nor object to the use of protosulphate of iron, but I do not see in the result anything more than the effect of the tendency of each particle of the phosphorus to go from places of stronger to places of weaker force.

I have obtained the following result mathematically.

Let a field of force be symmetrical about an axis; then if in travelling along the axis of force be a minimum or maximum at any point, it must be a maximum or minimum in travelling equatorially on arriving at the same point, and the variation of the square of the force in receding from that point axially is double the variation, of contrary sign, in receding from it equatorially. Thus, if C be the centre of a field which is symmetrical, not only round a horizontal axis, but also on opposite sides of a vertical plane perpendicular to that axis (i.e. when the north and south ends are alike) C must be a place of strongest or weakest force

in comparing any points along the axis, although it may be that the variation of force is but slight; and if A, B, C, D [sic] be 4 points near to C and equidistant from it, two A, B in the axial, and two D, E in the equatorial direction, the square of the force at D or E is half as much less or greater than at C as the square of the force at A or B is greater or less than at C. Thus the numbers might be suppose

If I recollect right Thomson at Liverpool² enunciated the result which I have mentioned, that if the force be at any point of the axis a maxm or minm going axially it must be a minm or maxm going equatorially. (See note B at end).

Now in your experiment the phosphorus as a whole went axially to the centre of the field; therefore the centre must have been a place of weakest force axially see note A at end; therefore it must have been a place of strongest force equatorially; therefore the observed setting of the phosphorus is what would result from the known law about going to places of weakest force; therefore we are not entitled from the result to draw any new conclusion.

I do not think the experiment need be repeated, but if you are not satisfied with the explanation I have given it might be worth while to remount the apparatus, to suspend a round piece of phosphorus by means of a lever and cacoon silk, so as to be free to move in the middle plane across the lines of force, and see whether when slightly displaced from the axis it would not tend to recede further from it.

By considering the mathematics of a body in a magnetic field, the body being composed of particles which severally tend to go from places of stronger to places of weaker force, but which do not sensibly influence one another, I arrived at the following conclusion.

If a small non-crystalline elongated body be suspended horizontally, so as to be free to turn round a vertical axis passing through its centre of gravity and be placed in an arbitrary magnetic field, there are two rectangular directions in one or other of which it will set, according as it is paramagnetic or diamagnetic. These directions have no immediate relation to the directions of the lines of force, though in ordinary fields they are, approximately at least, parallel and perpendicular to these lines. There may exist in the field what I will call dead points, that is, points where the setting force vanishes, and there may exist what I will call points of rest, that is, points where the whole force (or rather its square) is ultimately constant in passing to a consecutive point, so that the points of rest are positions of equilibrium of a small body free to move in all horizontal directions. But the points of rest (if any) are not in general dead points, and the dead points (if any) are not in general points of rest. Thus with pointed poles in the same horizontal plane the centre of the field would be a point of rest, but not a dead point. A dead point would be to the setting of the body something like what a magnetic pole of the earth is to the setting of the ordinary needle. In going once completely round it at a little distance, the direction of either set (i.e. para- or diamagnetic) would turn round through two right angles only. The lines of force would in general show nothing particular in the neighbourhood of a dead point, and thus we should have the phenomenon of a set taking place in all sorts of ways relating to the lines of force.

I don’t know whether there will appear to you anything paradoxical in this, or whether it will appear all natural. In the former case it might be worth while to investigate the thing further, so as to try to point out the mode of realising a dead point in experiment.

I have thought of a way in which the tendency of a bar, whether para- or diamagnetic, to set parallel to the lines of force, in consequence of mutual action, might perhaps be rendered sensible in experiment. Let two poles be prepared which I believe had best be, at the ends of the form of broad cylinders rendered concave. If the poles were placed at a distance from each other, the centre of the field would be a point of axially minimum and therefore equatorially maximum force, but when the poles were near together the centre of the field would be a point axially maximum and equatorially minimum force. In the former case a paramagnetic bar would set axially and a diamagnetic bar equatorially; in the latter case the settings would be reserved. Call the distance of the poles at which the transition takes place the critical distance, and suppose it determined experimentally. If over and above the set due to the tendency of each particle to go to places of stronger or weaker force in the otherwise undisturbed field (i.e. not deemed to be sensibly disturbed by all the other particles) there exists a cause of set (mutual disturbance) tending to make a bar whether para- or diamagnetic set along the lines of force, and therefore axially, it is plain that the axial setting is favoured thereby, and therefore the critical distance as determined by the use of a paramagnetic bar ought to come out somewhat less than as determined by the use of a diamagnetic bar, and less with a weakly than with a powerfully diamagnetic bar. Of course all the bars ought to be as nearly as possible of the same size and shape.

Although this method seems to me sound in principle, I doubt whether the forces would not be too small to allow the result to be sensible in experiment, except perhaps in the case of iron.

Note A. p.6 I presume at least the effect was due to this cause, and not any sensible reaction of the phosphorus on the iron poles[.]

Note B. On second thoughts I believe that what Thomson said had reference to a single flat pole and was not quite the same as this.

Believe me | Yours very truly | G.G. Stokes