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 Incoming links: Dipole interaction correction to effective conductivity

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 80 / 2 / iii / Solution Created 2026-10-03 Updated 2026-10-14

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Set

E = e_{3}

and

β = 3 a_{0} (a_{1} - a_{0}) / (a_{1} + 2 a_{0})

. The incident field relevant to each sphere is its exciting field of a conductivity inclusion: the macroscopic field plus the regular field due to other inclusions and periodic images, excluding its own singular dipole. In the first interaction approximation write

e_{j}^{(1)} = - \frac{β}{p _{j}} Λ_{j} E .

(1)

The interior response from part (i) then gives the improved polarization

P^{(1)} (x) ≃ β \sum_{j = 1}^{N} χ_{B_{j}} (x) (E + e_{j}^{(1)}) .

(2)

Averaging, using

⟨ χ_{B_{j}} ⟩ = p_{j}

, gives

⟨ P^{(1)} ⟩ ≃ pβE + β \sum_{j} p_{j} e_{j}^{(1)} = pβE - β^{2} \sum_{j} Λ_{j} E .

(3)

Therefore the dipole interaction correction to effective conductivity is

a^{*} ≃ a_{0} I + pβ I - β^{2} j = 1 \sum N Λ_{j} .

(4)

The factors

p_{j}

cancel against the denominator in the given field correction. Linearity permits repeating the calculation for three independent incident gradients, recovering the tensor rather than just its action on

e_{3}

. This is a second-reflection estimate; further dipole reflections and higher multipoles have been omitted.

A literal pointwise reading of the field printed “on the surface” needs qualification. Even the isolated-sphere solution in part (i) has, on a concentric sphere of radius

R

e_{self} (R n) = - \frac{r _{1}^{3} ( a _{1} - a _{0} )}{( a _{1} + 2 a _{0} ) R ^{3}} (I - 3 n \otimes n) E .

(5)

For

E = e_{3}

, its values at

n = e_{3}

and

n = e_{1}

are respectively

2 k r_{1}^{3} E / R^{3}

and

- k r_{1}^{3} E / R^{3}

, where

k = (a_{1} - a_{0}) / (a_{1} + 2 a_{0})

. They differ whenever there is nonzero contrast. Thus a single constant matrix cannot give the total pointwise field everywhere on that sphere.

The consistent local-field interpretation is a spherical average, or equivalently the regular exciting field at the centre. Indeed

⟨ n \otimes n ⟩_{S} = I /3

, so the angular mean of the self-dipole is zero. The other-inclusion field is harmonic inside the surrounding sphere, and the mean value property gives its central value. With

Λ_{j}

encoding this average regular correction, the calculation above is justified. If “on the surface” is instead insisted upon as a constant total pointwise field, that premise is false and the isolated-sphere values above are a counterexample. The distinction between dipole fields and local exciting fields is also treated in sections 10.2–10.3 of www.math.utah.edu/~milton/TheoryCompositesNOPRINT.pdf .

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