Page 194 - Electromagnetics

Figure 3.23: System of conductors used to derive Thomson’s theorem.

so that

2
δW e = ∇ ·∇(δ ) dV + |∇(δ )| dV ;
V 2 V
Thomson’s theorem will be proved if we can show that

∇ ·∇(δ ) dV = 0, (3.209)
V
because then we shall have
2
δW e = |∇(δ )| dV ≥ 0.
2 V
To establish (3.209), we use Green’s first identity

2
(∇u ·∇v + u∇ v) dV = u∇v · dS
V S
with u = δ and v = :

∇ ·∇(δ ) dV = δ ∇ · dS.
V S
Here S is composed of (1)the exterior surfaces S k (k = 1,..., n)of the n bodies, (2)
the surfaces S c of the “cuts” that are introduced in order to keep V a simply-connected
region (a condition for the validity of Green’s identity), and (3) the sphere S ∞ of very
large radius r.Thus
n

∇ ·∇(δ ) dV = δ ∇ · dS + δ ∇ · dS + δ ∇ · dS.
V k=1 S k S c S ∞
The first term on the right vanishes because δ = 0 on each S k . The second term
vanishes because the contributions from opposite sides of each cut cancel (note that ˆ n
occurs in pairs that are oppositely directed). The third term vanishes because ∼ 1/r,
2
2
∇ ∼ 1/r , and dS ∼ r where r →∞ for points on S ∞ .

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