Field of Circular Currents. 427 



formulae give identical results. For points outside the circle, 

 however, we find by (16) and (9), or otherwise, 



Z=--i^E"+— F", 

 r — a- r 



the modulus now being a/r. It will be seen that it is best 

 to consider (16) as the standard formula, as it gives the force 

 at points in the plane of the filament, both outside and inside 

 the circle. 



It is worth noticing that if r be not greater than 0*7«, the 

 maximum inaccuracy of the formula 



H 



is less than 1 in 10,000. 



Let us now find the value of the magnetic force very close 

 to the filament. In this case, as the modulus of E and F 

 in (16) is nearly unity, we can use Legendre's formulae (1) 

 and (2) to find E and F. Hence 



a-r[_ 2\a + rJ { & a — r 2) J 



a + r L & a — r 4: \a + rj [ ° a-r J J 



2i 2i -. 4z(a + r) 



a — r a + r ° a — r 



when (a — r) 2 /( a + r J 2 * s negligibly small compared with unity. 

 The first term on the right-hand side of this equation is very 

 great compared with the second when r is nearly equal to a y 

 and. in this case, we may write 



Z= ——, very approximately. 



Similarly, when r is greater than a, we have 



. . (18) 

 r — a r + a ° r — a 7 



z= _JL + JL Iog 4(?±^ 



r — a r + a ° r—a 

 approximately. 



ii. The mutual inductance of tic o coaxial circular currents. 



Let the radii of the circles be a and b respectively, and 

 let c be the distance between their planes. We sball calculate 

 the flux M through the circle whose radius is b due to unit 

 current in the circle whose radius is a. Let Z be the 



