556 BELL SYSTEM TECHNICAL JOURNAL 



To get an insight into the significance of this equation, let us con- 

 sider a conductor which is part of a single-mesh circuit, and extend our 

 integrals over the region occupied by this conductor. Then the first 

 integral on the right of (79) represents twice the power dissipated in 

 heat in the conductor, while fxf S S{HH*)dv is four times the average 

 amount of magnetic energy stored in it. 



On the other hand, when we look at the conductor from the stand- 

 point of circuit theory, these two quantities are respectively RP and 

 LP; R and L being by definition the "resistance" and "inductance" 

 of the conductor. Hence we have the equation, 



// 



lEH*']ndS = iR + iooL)P = ZP, (80) 



from which the impedance Z can be computed when the field intensities 

 are known at the surface of the conductor. 



If, on the other hand, the conductor is part of a two-mesh circuit 

 and /i and I2 are the amplitudes of the currents in meshes 1 and 2 

 respectively, the average amount of energy dissipated in heat per 

 second can be regarded as made up of three parts, two of which are 

 proportional to the squares of these amplitudes, while the third is 

 proportional to their product. The first two of these parts being 

 dependent on the magnitude of the current flowing in one mesh only 

 are attributed to the self-resistance of the conductor to the corre- 

 sponding current; the third part is attributed to the mutual resistance 

 of the conductor. Designating the self-resistances by Rn and i?22 and 

 the mutual resistance by i?i2, we represent the energy dissipated in 

 heat in the form l/liRiJi^ + 2R12I1I2 + R^^H)- Similarly, the 

 average amount of energy stored in the conductor can be represented 

 in the form l/4(Lii7i2 + 2L 12/1/2 + Li^I^), where Lu and L22 are 

 called respectively s el j -inductances and L12 mutual inductance. In this 

 case, equation (79) can be written as follows: 



// 



[E//*]n^5 = Zn/x2 + 2Z12/1/2 + Z22/2^ (81) 



where the quantities Zw, Z22 and Z12 are respectively the self-im- 

 pedances and the mutual impedance of the conductor. 



In general, if the conductor is part of a ife-mesh circuit, we can 

 obtain all its self-and mutual impedances by evaluating the integral 

 X X[EH*\dS over its surface, and picking out the coefficients of 

 various combinations of /'s. 



We shall have an occasion to apply these results in computing the 

 eff^ect of eccentricity upon the resistance of parallel cylindrical con- 

 ductors. 



