ELECTROMAGNETIC FIELD ANALYSIS 505 



that the velocities of the charged particles are specified, and that the forces 

 which they exert on each other are completely neutralized by the forces 

 external to the field, in which case the convection current appears merely 

 as an ''impressed current". 



Except for the above restrictions, equations (10) form a complete set; 

 but for mathematical convenience two other equations are usually ad- 

 joined. These are 



II eEndS = q, 



(11) 



jj nHndS = 0, 



where the double integration is extended over a closed surface. The first 

 of these equations states that the total electric displacement through a 

 closed surface is equal to the net enclosed electric charge; the second denies 

 the physical existence of magnetic charge. These equations can be derived 

 from (10) and for this reason are not quite on the same footing with them. 



B Vbo = o D 



r 



— I 1 — 



A Vac=0 C 



Fig. 8 — A pair of parallel wires. 



Equation (10) tells us that, except when the field is static, we cannot 

 speak of the electromotive force or the voltage between two points without 

 specifying the path along which we add up the elementary voltages. In 

 fact, equation (10) gives us the difference between the voltages along two 

 different paths connecting the same pair of points. To illustrate, consider 

 a wave along a pair of perfectly conducting wires. Fig. 8. Voltages I ac 

 and Vbd along the wires are equal to zero; transverse voltages Vab and Vcd 

 are usually unequal; hence Vabd ^ Vacd • 



If two points are infinitely close, then we can define the voltage un- 

 ambiguously as the product Esds of the electric intensity and the distance 

 between the points. The difference between this voltage and the voltage 

 along any other infinitesimal path is an infinitesimal of the second order, 

 being dependent on the area enclosed by the two paths. In practice two 

 points are sufficiently close if the distance between them is small compared 

 with one quarter wavelength. 



Since, except in electrostatics, we cannot speak of the voltage between 

 two points without specifying the path, we cannot speak of the potential 



