[SHAW] POTENTIAL ENERGIES 155 



§3. The Mutual Potential Energy of a Coil and a Small Magnet. 



It can be shown that if the point potential of a system of revolu- 

 tion, X, on its axis at a distance, x, from the origin is expressed in the 

 form 



flo,ai,a2.a3, 



where ûq, clu cli, etc., are constants, and if the point potential of a 

 system of revolution, F, on its axis near the origin is expressed in 

 the form 



hQ-\-hiX-\rh2X"-\-bzX^+ 



where 60, hi, b^, etc., are constants, then, if the axes of X and Y are 

 inclined at an angle 6, the origin being chosen at the point of inter- 

 section, the mutual potential energy of the two systems will be given by 



W=k(aoboPo+aibiPi+a2b2P2-\- ....) (2) 



where ^ is a constant. 



If X is our magnetic system we see from equation (1) that 



and equation (2) becomes 



W=k(aAPi+asbsP,+ ...) (3) 



where 



ai = M, a-i = ML\, ai= ML\, 

 and the series is so rapidly convergent in almost all practical cases 

 that further terms amount to magnitudes afïecting W by much less 

 than one part in a hundred thousand. 



If Y is the electrical system and consists for example of a plane 



circular coil of radius r, with its centre at a distance y from the origin 



and if N is the number of turns assumed to occupy a channel of, 



infinitesimal dimensions (corrected below), then it can be shown that 



è„ = 2x«(r2-f-/-)-''/2{cos <i> . P„(cos 0) -P„_ i(cos <^)} (4) 



where cos <i> = y{r'^-\-y')~^ , an if there are several coils we have 

 merely to consider each pair of reacting systems separately and then 

 superimpose the effects. From equation (4) we note the familiar 



6i=-2TiVrV(r2+3'2)-3/2 



^2= -4:TrNr\y''-ryi) / (r^+y-)^/^ 



/ 3 r*\ 



and ^5= —ÇfirNr'^i y^— — rV-j I /{r--\-y') 



\ 2 8/ 



11/2 



It will be seen that if the centre of Y is placed at a distance y = r/2 

 from the point of intersection of the axes, the value of 63 is reduced to 



