Heterogeneous Medium Models (60 pts). a) In the notes on Heterogeneous Media, show that equation (*)...

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Heterogeneous Medium Models (60 pts). a) In the notes on Heterogeneous Media, show that equation (*) satisfies the appropriate boundary conditions, and perform the calculations needed to obtain the results (**) and (***). b) Given the Bruggeman effective medium approximation in 3D, - eff +20 +42 eff 02- eff 02 +20 = 0 eff and setting = 2 =1-0, solve for eff. You should get an expression involving square roots. Show that, for o < <1, (1+ko) and determine k by Taylor series. eff c) Show that when = 0, there is a percolation threshold, i.e. a minimum value of o needed for flux to exist through the heterogeneous medium. Determine that value. Heterogeneous Medium Models (60 pts). a) In the notes on Heterogeneous Media, show that equation (*) satisfies the appropriate boundary conditions, and perform the calculations needed to obtain the results (**) and (***). b) Given the Bruggeman effective medium approximation in 3D, - eff +20 +42 eff 02- eff 02 +20 = 0 eff and setting = 2 =1-0, solve for eff. You should get an expression involving square roots. Show that, for o < <1, (1+ko) and determine k by Taylor series. eff c) Show that when = 0, there is a percolation threshold, i.e. a minimum value of o needed for flux to exist through the heterogeneous medium. Determine that value.