CHAPTER X. 



STEADY CURRENTS IN CONTINUOUS MEDIA. 



Components of Current. 



370. IN the present chapter we shall consider steady currents of elec- 

 tricity flowing through continuous two- and three-dimensional conductors 

 instead of through systems of linear conductors. 



We can find the direction of flow at any point P in a conductor by 

 imagining that we take a small plane of area dS and turn it about at the 

 point P until we find the position in which the amount of electricity crossing 

 it per unit time is a maximum. The normal to the plane when in this 

 position will give the direction of the current at P, and if the total amount 

 of electricity crossing this plane per unit time when in this position is CdS, 

 then C may be defined to be the strength of the current at P. 



If I, m, n are the direction-cosines of the direction of the current at P, 

 then the current C may be treated as the superposition of three currents 

 1C, mC, nC parallel to the axes. To prove this we need only notice that the 

 flow across an area dS of which the normal makes an angle 6 with the direc- 

 tion of the currrent, and has direction-cosines l' t m', n, must be CdS cos 0, or 



CdS (IF + mm' + mi*). 



The first term of this expression may be regarded as the contribution from a 

 current 1C parallel to the axis Ox, and so on. The quantities 1C, mC, nC 

 are called the components of the current at the point P. 



Lines and Tubes of Flow. 



371. DEFINITION. A line of flow is a line drawn in a conductor such 

 that at every point its tangent is in the direction of the current at the point. 



DEFINITION. A tube of flow is a tubular region of infinitesimal cross- 

 section, bounded by lines of flow. 



