PROFESSOR CLERK MAXWELL OX THE ELECTROMAGNETIC FIELD. 
489 
Let C be the current in the conductor, and letjf , qj, r' be its components, then 
Xati= Calxx (^py - j g ^ , 
or X=[jjy(f — gjfir'. j 
Similarly, Y—pur' — (*y]?',\ (J) 
Z=^/3 p'—paq'. j 
These are the equations which determine the mechanical force acting on a conductor 
carrying a current. The force is perpendicular to the current and to the lines of force; 
and is measured by the area of the parallelogram formed by lines parallel to the current 
and lines of force, and proportional to their intensities. 
Mechanical Force on a Magnet. 
(77) In any part of the field not traversed by electric currents the distribution of 
magnetic intensity may be represented by the differential coefficients of a function 
which may be called the magnetic potential. When there are no currents in the field, 
this quantity has a single value for each point. When there are currents, the potential 
has a series of values at each point, but its differential coefficients have only one value, 
namely, 
d_l_ dp 
dy dz 
7- 
Substituting these values of a, (3, y in the expression (equation 38) for the intrinsic 
energy of the field, and integrating by parts, it becomes 
-2{^(t+f + t)|F- 
The expression 
2 (i“+w+^) dv=2 ^ v ( S9 > 
indicates the number of lines of magnetic force which have their origin within the 
space V. Now a magnetic pole is known to us only as the origin or termination of 
lines of magnetic force, and a unit pole is one which has 4x lines belonging to it, since 
it produces unit of magnetic intensity at unit of distance over a sphere whose surface 
is 4x. 
Hence if m is the amount of free positive magnetism in unit of volume, the above 
expression may be written 4xm, and the expression for the energy of the field becomes 
E=-X(i<pm)dV. (40) 
If there are two magnetic poles and m 2 producing potentials <£>, and <p 2 in the field , 
then if m 2 is moved a distance dx , and is urged in that direction by a force X, then the 
work done is Xdx, and the decrease of energy in the field is 
<7(i(Pi+?>2)(Wi+W 2 )), 
and these must be equal by the principle of Conservation of Energy. 
