373-375] Equation of Continuity 333 



Let I, m, n be the direction-cosines of the line of flow at P, and let u, v, w 

 be the components of current at P, so that u = 1C, etc. Then 



dV .97 



5- = I -5 = ICr ur, etc., 

 dx ds 



and we see that equation (309) is equivalent to the three equations 



18F 



u = ^~ 



T OX 



v = 



187 



r dy 



w = ^~- 



T OZ 



V (310). 



These equations express Ohm's Law in a form appropriate to flow through 

 a solid conductor. 



Equation of Continuity. 



375. Since we are supposing the currents to be steady, the amount of 

 current which flows into any closed region must be exactly equal to the 

 amount which flows out. This can be expressed by saying that the integral 

 algebraic flow into any closed region must be nil. 



Let any closed surface 8 be taken entirely inside a conductor. Let I, m, n 

 be the direction cosines of the inward normal to any element dS of this 

 surface, and let u, v, w be the components of current at this point. Then 

 the normal component of flow across the element dS is lu 4- mv + nw, and the 

 condition that the integral algebraic flow across the surface 8 shall be nil is 

 expressed by the equation 



([(lu + mv + nw) dS = 0. 



By Green's Theorem ( 176), this equation may be transformed into 



Mdu dv dw\ 

 ***+*r* 



and since this integral has to vanish, whatever the region through which it is 

 taken, each integrand must vanish separately. Hence at every point inside 

 the conductor, we must have 



This is the so-called " equation of continuity," expressing that no elec- 

 tricity is created or destroyed or allowed to accumulate during the passage of 

 a steady current through a conductor. 



