ELECTRICAL METHODS 



469 



This equation gives the current which flows through the whole cross sec- 

 tion, i.e., through A sq. cm. The current through one square centimeter is 



1 



1 = • 



P 



dV 

 dL 



If the distance dL is not measured in the direction of current flow, 

 Equation 3 must be modified. In this case. 





where i is the current per square centimeter of area normal to L and 

 is the rate of change of potential in the direction of L. In particular, 



(2) 



dV 

 dL 



__)_'dV. • -_ldV. 



1 dV 



p d-r p dy p d3 



where 4, iy and 4 are the components of current density in the directions 



■p,T/ or/ 'p.T/ 



X, y, and z respectively, and ~ — , — — , and — — are the partial derivatives 



of V with respect to x, y and z respectively. 



Flow of Current in a Continuous Medium. — ^When a steady current 

 flows through a conductor, it behaves very much like an incompressible 

 fluid in that the total current which 

 flows into any closed surface 

 within the conductor is equal to the 

 total current which flows out of 

 that surface. In order to express 

 this fact mathematically, it will 

 be convenient to consider a small 

 cube within the conductor. (Figure 

 280.) 



X, y, and z are the coordinates 

 of the center of the cube and dx, 

 dy, dz are the edges of the cube. 



Fig. 280. — Steady state currents through a small 

 cube. 



Let iydxdz be the total current entering the cube at the face 1234 and 

 iy'dxdz be the total current leaving at the face 5678. Then the excess of 

 current entering at the face 1234 over that leaving the face 5678 is 



dxdz 

 \p dy p 'dy / 



dV dV'^ 



Aiy = iy dxdz — iy' dxdz = (iy — ij) dxdz = i 



, ^\ (-dV dV'\ 

 ' p \dy dy ) 



dxdz 



