332 Steady Currents in continuous Media [OH. x 



In considering the motion of material particles in general it is not usually true that the 

 motion of the particles is in the direction of the forces acting upon them. The velocity 

 of a particle at the end of any small interval of time is compounded of the velocity at 

 the beginning of the interval together with the velocity generated during the interval. 

 The latter velocity is in the direction of the forces acting on the particle, but is 

 generally insignificant in comparison with the original velocity of the particle. In the 

 particular case in which the original velocity of the particle was very small, the direction 

 of motion at the end of a small interval will be that of the force acting on the particle. 

 If the particle moves in a resisting medium, it may be that the velocity of the particle 

 is kept permanently very small by the resistance of the medium : in this case the 

 direction of motion of the particle at every instant, relatively to the medium, may be 

 that of the forces acting on it. 



On the modern view of electricity, a current of electricity is composed of electrons 

 which are driven through a conductor by the electric forces acting on them, and in 

 their motion experience frequent collisions with the molecules of the conductor. The 

 effect of these collisions is continually to check the velocity of the electrons, so that 

 their velocity is kept small just as if they were moving through a resisting medium 

 of the ordinary kind, and so it comes about that the velocity at all stages of the motion 

 is in the direction of the electric intensity. 



374. Let us select any tube of force of small cross-section inside a 

 conductor, and let P, Q be any two points on this tube of force, at which the 

 potentials are V P and VQ, the former being the greater. Let these points be 

 so near together that throughout the range PQ the cross-section of the tube 

 of force may be supposed to have a constant value o>, while the specific resist- 

 ance of the material of the conductor may be supposed to have a constant 

 value r. 



From what has been said in 373, it follows that the tube of force under 

 consideration is also a tube of flow. If C denotes the current, then the current 

 flowing through this tube of flow in the direction from P to Q will be Co). 

 This current may, within the range PQ, be regarded as flowing through a 

 conductor of cross-section o> and of specific resistance r. The resistance of 



PQ r 

 this conductor from P to Q is accordingly - , while the fall of potential 



is V P VQ. Thus by Ohm's Law 



so that ^P^ =(7T * 



If ;r- denotes differentiation along the tube of force, the fraction on the 



left of the foregoing equation reduces, when P and Q are made to coincide, 



dV 

 to -^- , so that the equation assumes the form 



- d -^=Cr .............. (309). 



