TmK TliKATMKNI' OI' El.KCTHOKVNAMIfS. 3D 



Tlie ccfuatioiis may, of course, Ik- transfornieti, without ditticultv, 

 into the usual polar co-orfHuatos, or into the language of vector algebra. 



It is now convenient to deal with the magnetic action of a rect- 

 angular elementary circuit. Two principal cases arise according us the 

 field is required at a point lying in the axis of the small circuit or in 

 its plane. In the first case, let dy dz he the area of the circuit, anfl 

 the point considered be at a distance r along the axis of .*•. Then one 

 of the sides, of length (///, situated .', dz from the axis of y, will produce 



a field — - , of which the component in the axial direction is — x —. 



The transverse components c)f the opposite sides neutralise, lea\ ing an 

 axial resultant amounting to four times the above, or 



idy dz 2i c/A 



r- '1 r ?••' 



In the same way it may be shown that for a point in the plane of 



i dA. 

 the elementary circuit the field f/ H := — 7-- 



The product i dX may then be defined as the magnetic moment 

 of the elementar}' circuit. 



It is easy to show that, for a point in an}' direction, the area — or 

 magnetic moment — of the elementary circuit nuiy be resohed into 

 axial and equatorial components, and the effect of the two parts 

 calculated according to the preceding rules. We have, thus, a com- 

 plete picture of the magnetic field round an element of a circuit. 



The next step is to show that the field can be derived from a 

 potential. The necessary integration is gi\en in all treatises on 

 electricity, and shows that the magnetic potential in the axial direc- 

 tion has the value 



idA 



do = — —, 

 v 



while in the equatorial direction it vanishes; and that at any point 

 making an angle ;!^ with the axis of the elementary circuit the magnetic 



potential is rfO = — — con v. rsow the solid angle subtended bv the 

 ?•- ^ 



dA 



circuit at the point is cosy =^ ?r. Hence (/() =^ i du\ If, how- 



r- '^ 



ever, this relation is integrated, bearing in mind the way in which a 

 circuit may be supposed built up of small plane elements, the sides 

 of which cancel in their magnetic effects (as demonstrated by Ampere) 

 we Conclude that 



il = i v, 



that is, the magnetic potential produced by a current is etjual to the 



