Page 201 - Electromagnetics

Figure 3.28: A system of current-carrying wires.

where the L mn are called the coefficients of inductance. Derive Neumann’s formula

µ 0 dl · dl
L mn = ,
4π |r − r |

n m
and thereby demonstrate the reciprocity relation L mn = L nm .
3.29 For the group of wires shown in Figure 3.28, show that the stored magnetic energy
is given by
N N
1
W = L mn I n I m .
2
m=1 n=1
3.30 Prove the minimum heat generation theorem: steady electric currents distribute
themselves in a conductor in such a way that the dissipated power is a minimum. Hint:
Let J be the actual distribution of current in a conducting body, and let the power it
dissipates be P. Let J = J + δJ be any other current distribution, and let the power it

dissipates be P = P + δP. Show that

1 1 2
δP = |δJ| dV ≥ 0.
2 V σ

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