Page 90 - Electromagnetics Handbook

Here we regard the quantities k E , k B ,... as base units for the discussion, while the
dimensionless quantities (2.266) serve to express the actual fields E, B,... in terms of
these base units. Of course, the time and space variables can also be scaled: we can write
t = tk t , l = lk l , (2.268)
if l is any length of interest. Again, the quantities t and l are dimensionless measure
numbers used to express the actual quantities t and l relative to the chosen base amounts
k t and k l . With (2.267) and (2.268), Maxwell’s curl equations become
k B k l ∂B k J k l k D k l ∂D
∇ × E =− , ∇ × H = J + (2.269)
k E k t ∂t k H k H k t ∂t
while the continuity equation becomes
∂ρ
k ρ k l
∇ · J =− , (2.270)
k J k t ∂t
where ∇ has been normalized by k l . These are examples of field equations cast into
dimensionless form — it is easily verified that the similarity parameters

k B k l k J k l k D k l k ρ k l
, , , , (2.271)
k E k t k H k H k t k J k t
are dimensionless. The idea behind electromagnetic similitude is that a given set of
normalized values E, B,... can satisfy equations (2.269) and (2.270) for many different
physical situations, provided that the numerical values of the coefficients (2.271) are all
fixed across those situations. Indeed, the differential equations would be identical.
To make this discussion a bit more concrete, let us assume a conducting linear medium
where

D =
E, B = µH, J = σE,
and use

=
k
, µ = µk µ , σ = σk σ ,
to express the material parameters in terms of dimensionless values
, µ, and σ. Then

k
k E k µ k H k σ k E
D =
E, B = µH, J = σE,
k D k B k J
and equations (2.269) become

k µ k l k H ∂H
∇ × E =− µ ,
k t k E ∂t

k E k
k l k E ∂E
∇ × H = k σ k l σE +
.
k H k t k H ∂t
Defining
k µ k l k H k E k
k l k E
α = , γ = k σ k l , β = ,
k t k E k H k t k H
we see that under the current assumptions similarity holds between two electromagnetics
problems only if αµ, γσ, and β
are numerically the same in both problems. A necessary
condition for similitude, then, is that the products

2 2
k l k l
(αµ)(β
) = k µ k
µ
, (αµ)(γ σ) = k µ k σ µσ,
k t k t

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