214, 215] ELECTROMAGNETISM. 419 



The force in the direction of the axis is 

 L ^ 90^ 27nR 2 / 



At the center of the circle 



27T/ 



From this expression comes the definition often given of the 

 unit of current as that current which, flowing in a circle of unit 

 radius, produces the field 2?r at its center, or less correctly, the 

 current, which, flowing in an arc equal to the radius in a unit 

 circle, produces unit field at the center. 



The expression for the force is an example of the proposition 

 that similar geometrical circuits traversed by equal currents, 

 produce at corresponding points forces inversely proportional to 

 their linear dimensions. For at corresponding points the solid 

 angle, and therefore the potential is the same. In the circuit of 

 n times the dimensions, the potential changes by equal amounts 

 for displacements of n times the length, hence for equal displace- 

 ments the change is l/n as great, and the force is n times smaller. 



When the point is not on the axis of the circle, the cone, 

 having an oblique section circular, is elliptic, and we must cal- 

 culate the area of the spherical ellipse cut out by it from the unit 

 sphere. This involves an elliptic integral. 



We may however develop the result in an infinite series of 

 zonal spherical harmonics, as in the case of the potential of a disc, 

 in 102. Developing the above expression for o> at points on the 

 axis by the binomial theorem, we have 



272 



