172 SPHERE OF MAGNETIC MATERIAL IN A UNIFORM FIELD. 



The force at any point P (a, 0) just outside the sphere may be regarded as composed of a normal 

 (i.e. vertical) component 3/xF cos 0/0* + 2), and a tangential (i.e. horizontal) component 3F sin 0/(n + 2). 

 If \j/ be the inclination to OA of the resultant force R at P, then 



tan ^ = (/x- 1) sin 20-=-{(/i+l) + (^- 1) cos 20} (2), 



If p - 1 be small, then \j/ is small and R/F differs but little from unity, as of course is obvious a priori. 

 Over the surface, R obviously has its maximum value when 0=0, its minimum when = ?r/2, and 



R m ,/F= 3,1/0* + 2), I 



\ (4). 



R mln ./F= 3/0* + 2) J 

 Thus 



R m ./R m ,n. = / (5). 



The maximum occurs at the ends of the diameter = 0, where the force is entirely vertical, the 

 minimum at all points on the perpendicular great circle, where the force is entirely horizontal. 



Clearly the phenomena will depend mainly on whether p is large. If it were possible to suppose 

 p large, we should have a wide range of values of R, and the disturbing forces regarding F as 

 a disturbing field would vary widely in direction at different parts of the surface. But it is difficult 

 to suppose that p. is large. The Earth's own field is only of the order 0'5 C.G.S. units, and a disturbance 

 as large as O'Ol C.G.S. is exceptional. Even in the best magnetic steel p is low for such fields. Apart 

 from more theoretical considerations there is the fact that, according to the above solution, if p were 

 large, the horizontal (or tangential) component 3F sin 0/(p + 2) would tend to be negligible compared 

 to the vertical 3/xF cos 0/(p + 2), except at places where is nearly ir/2. Now the tendency is not for 

 the horizontal component of disturbances to be small compared to the vertical, but rather the opposite. 



A point calling for special remark is that R has the same value and the same absolute direction in space 

 for places on the same great circle through AA' whose angular co-ordinates are and TT + ; in other 

 words, the disturbing forces are equal and parallel at any two places diametrically situated with respect 

 to one another. 



88. As the Earth's crust is on the whole non-magnetic, the above simple problem is obviously an 

 imperfect representation of the facts. Thus it is worth glancing at the next simplest case, that presented 

 by an earth composed of a nucleus of radius a and permeability p, with a surrounding shell of per- 

 meability p and radius a', the medium external to a' having unit permeability. The imposed field 

 being F as before, I find for the potential external to the " earth " 



V = - Fr cos + F cos (/* + V) (X - 1) + 0* - /*') (V + *) ( a l a ') 3 ,K\ 



r 2 0* + V) 0*' + 2) + 2 (p - p) (p' - 1) (alaj 



In practice the only interesting case seems that in which p - 1 is small, i.e. in which the material of 

 the layer is only slightly magnetic. In this case, assuming a/a' no to be very small, the potential is 

 approximately given by 



V = - Fr cos + L F - a cos (7). 



p + 2 r 2 



This is identical in form with (1), the only difference being that in (1) a represents the radius of the 

 " earth," whilst in (7) it represents the radius of the magnetic nucleus. If the slightly magnetic crust 

 be thin, the phenomena are much the same as if the permeability were p throughout. If the thickness 

 of the crust be considerable, the variation in R over the surface is considerably reduced. Whether the 

 crust be thick or thin, the value and the direction of R are the same at diametral points. 



In this investigation the only assumption made as to the disturbing field F is that it may be regarded 

 as uniform so far as the Earth is concerned. Thus it need not be due to the Sun, but might represent any 

 stray field that happens to exist in any part of space traversed by the Earth, so long, of course, as the 

 hypothesis of uniformity in strength and direction is approximately satisfied. Some of the phenomena 



