(15) 



426 Mr. A. Russell on the Magnetic 



Hence in bipolar coordinates 



Z=--Ecos£+-F, ^ 



r 2 r x I 



v 2i r-f + rJ -^ . . 2i 2rvr 2 -^ . , \ 

 r 2 TY — r^ r l rf — r 2 T ^ 



where cos < />=(r 1 2 + r 2 2 -4a 2 )/(2r^ 2 ) 7 and k 2 = l-r 2 2 /ri 2 . 



A simple and useful way of considering the force at any 

 point P due to a circular current is to consider that it is the 

 resultant (fig. 2) of two forces at right angles to AP and OP 

 respectively. The component at right angles to AP equals 

 (4w/?y 3 )E, and the component at right angles to OP equals 

 2z(F — - E) / (Vx cos 0), the modulus of E and F being 

 (l-r 2 2 /V) 1/2 . 



4. Applications of the Formulce. 

 i. Force in the plane of a circular current. 



From fig. 2 we see at once that the force at any point in 

 the plane of the circular current and inside the circle is 

 given by 



Z= 2! E + -F 



■>v> 



2i 2/ 

 ' E + -4^F (16) 



a — r a + r 



the modulus being 2(ar)^ 2 l(a + r). 



Again, at points in the plane of the circular current, but 

 outside the circle, we have 



« E+ % „ 



r — a r-ha 

 ; — r a + r 



Hence, whether the point be inside or outside the circle^ 

 formula (16) gives the magnetic force. When the point lies 

 within the circle the author at first used the formula * 



4ai TC/ 



2i 



a" — r 2 





E'+-^-E / , 

 a — r a + r 



where the modulus is r/a. By means of (9) we see that the 



* 'The Electrician/ vol. xxxi. p. 212 (1893) or 'Alternating Currents/ 

 vol. i. p. 30. 



