530-533] General Equations 465 



532. If the medium is a conducting medium, the presence of the electric 

 forces sets up currents, and the components u, v, w of the current at any 

 point are, as in 374, connected with the currents by the equations 



X = TU, Y = TV, Z = TW, 



these equations being the expression of Ohm's Law, where r is the specific 

 resistance of the conductor at the point. 



On substituting these values for X, Y, Z in equations (464) (466), we 

 obtain a system of equations connecting the currents in the conductor 

 with the changes in the magnetic field. 



533. There is, however, a further system of equations expressing rela- 

 tions between the currents and the magnetic field. We have seen ( 480) 

 that a current sets up a magnetic field of known intensity, and since the 

 whole magnetic field must arise either from currents or from permanent 

 magnets, this fact gives rise to a second system of equations. 



In a field arising solely from permanent magnetism, we can take a unit 

 pole round any closed path in the field, and the total work done will be nil. 

 Hence on taking a unit pole round a closed circuit in the most general 

 magnetic field, the work done will be the same as if there were no perma- 

 nent magnetism, and the whole field were due to the currents present. The 

 amount of this work, as we have seen, is 47rSi, where ^i is the sum of all the 

 currents which flow through the circuit round which the pole is taken. If 

 u, v, w are the components of current at any point, we have 



1 1 



mv -f nw) dS, 



the integration being over any area which has the closed path as boundary. 

 Hence our experimental fact leads to the equation 



Transforming the line integral into a surface integral by Stokes' Theorem 

 ( 438), we obtain the equation in the form 



7/87 8/3 \ /8 87 \ fdp da \) , 



l( ^- ~ 4)7TU + m ^- 4:7TV H" n I 5 o ^ 7rw; ( ^ = 0. 



\9y dz J \dz dx I \dx dy J) 



As with the integral of 529, each integrand must vanish for all values 

 of I, m, n, so that we must have 



87 8/9 /^ihr-kx 



^-^ (473), 



dy dz 



da. 87 



^- -5* (474), 



8-z cte 



P-^ (475). 



dx dy 



J. 30 



