548 



Dr. A. Russell on the Effective Resistance 



V. The Density of the Current in the Inner 

 and Outer Conductors. 



1. With low frequency currents. 



For the inner conductor we find from equation (65), that 



i=i \X(mr)} 1/2 cos (»* + e), . . . (107) 



where tan e = bei mr/hev mr. 



It is known * that X(?nr) always increases asm? 1 increases. 

 Hence the amplitude of the current density is always greatest 

 at the surface of the inner conductor and least along the 

 axis. When the sixth and higher powers of mr can be 

 neglected (L07) becomes 



i = 2 (l + wiV/64) cos (art + e), . . . (108) 



where tan e = mV 2 /4. 



If mr=l, (107) gives 



t = 1-015 io cos (»* + 14° 150, 



and (108) gives t= 1-016 1 cos (»* + 14 c 2'). 



Hence when mr is not greater than 1, (108) may be used. 

 Even when mr = 2 the inaccuracy in the value of the ampli- 

 tude of the current density given by (108) is less than 

 2 per cent. In this case the amplitude of the current- 

 density at the surface of the conductor is about 23 per cent, 

 greater than at its axis, and the current along the axis lags 

 by about 52° behind the surface current. 

 " Similarly when me is small, we find by (78) that 



7=m 



i- 



loo- 



mr 



} 



cos cot. 



(109), 



Hence the amplitude of the current-density in the outer 

 conductor diminishes as r increases. 



3. With high frequency currents. 



In this case, by (20) and (21), we get for the inner con- 

 ductor 



£mr 



, V: 



" cos( t 

 r V 



mr ttx 



r* + 7i-8-> • • (110 > 



i \/ ZTrmr^^V" 1 ^ \/2 8, 

 * Russell's ' Alternating Currents/ vol. i. p. 373. 



