Field of Circular Currents. 429 



also that rja is small compared with unity. From the con- 

 struction given in fig. 2 we see that the lines o£ force near a 

 circular filament are very approximately circles with their 

 centres on the circular axis of the filament. The field is 

 therefore similar to that round a straight cylindrical con- 

 ducting tube, in which the current flow is parallel to the axis. 

 Now it is well known * that the magnetic force outside a 

 straight cylindrical conducting tube can be calculated as if 

 the current were concentrated along the axis and that the 

 magnetic force inside the tube is zero. Let us suppose that 

 the circular conductor is built up of an infinite number of 

 infinitely thin ring-tubes, every point on any tube having the 

 same minimum distance from the circular axis, and make 

 the assumptions that the current in each ring-tube produces 

 no magnetic field inside it, and that the field outside can be 

 calculated as if the current were concentrated along the axis. 

 From the corresponding electrostatic problem of a charged 

 conducting ring we see, by the method of duality, that the first 

 assumption could be made rigorously true by assuming that 

 the density of the current distribution over the cross section 

 of the infinitely thin ring-tube was proportional to the 

 surface density of the electrostatic charge. In this case, 

 however, the magnetic force at points outside the tube is not 

 the same as if the current were concentrated along the circular 

 axis, and therefore the gain in rigour by making the first 

 assumption accurate is problematical. When r is very small 

 compared with a the surface density is approximately constant 

 over the ring f- In this case, therefore, when the current 

 density is uniform over the circular tube, the magnetic force 

 produced inside is negligible, and therefore our assumptions 

 are legitimate in obtaining an approximate solution. 



Let <!>! be the flux linked with the whole current when 

 unit current is flowing in the ring, and <3> 2 the flux linked 

 with part of the current only. By (21) we have 



(|y=87ra(F-E), 



where the modulus is (a—r)/a. 



By (17) , the force Z at a point in the plane of the circular 

 axis and at a distance a — % from its centre is given by 



„ 2i' , i' 8a 



Z= I + a l °Zj' 

 approximately, where i / = f 2 /r 2 , since the whole current in 

 the ring is unity. 



* A. Russell, ' Alternating Currents/ vol. i. p. 32. 



t F. W. Dyson, Phil. Trans, vol. clxxxiv. A, p. 67 (1893). 



