Question 07/01
CONDUCTING CHESSBOARD
Small square metal sheets are made from two different metals (green and yellow in the
picture) and combined into a single very large "chessboard". The two-dimensional
conductivity of one metal is S1 and of the other - is S2. Calculate
the effective two-dimensional conductivity of this "chessboard" by following steps:
(a) Show that the electrostatic potential inside every, say, green square is given
by the same harmonic function f1(x,y) (up to a constant which
needs to be added depending on the detailed position of the square).
Similar function g1(x,y) describes yellow squares.
(b) Functions f1(x,y) and g1(x,y) can be treated
as real parts of ANALYTIC functions
F(x,y)=f1(x,y)+if2(x,y)
and G(x,y)=g1(x,y)+ig2(x,y) of a complex
variable z=x+iy. Show that f2(x,y)
and g2(x,y) also solve (a different) chessboard conductivity problem.
(c) By comparing the solutions of the conductivity problems described in (b) find
the effective conductivity of this "chessboard".
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