Question $3$

a$)$ A certain carbon electrode used in electrical arcing applications is $47$ percent porous. The resistivity of graphite

$($in polycrystalline form$)$ at room temperature is about $9\mathrm{.}1m,$ while the air pores can be assumed to be

perfect insulators.

Estimate the effective resistivity of the carbon electrode using the appropriate Reynolds and Hough rule

$,$ see section $5,$ Resistivity of Mixtures & Porous Materials $-$ Heterogeneous Mixtures,

simplified to either ${}_{\text{e}\text{f}\text{f}}={}_c\frac{(1+\frac{1}{2}{x}_d)}{(1-x_d)}$ or ${}_{\text{e}\text{f}\text{f}}={}_c\frac{(1-x_d)}{(1+2x_d)},$ same section$)$

and also use the simple conductivity mixture rule ${}_{\text{e}\text{f}\text{f}}={}_a+{}_{}.$

Compare your estimates with the measured value of $18m.$

b$)$ Silver particles are dispersed in a graphite paste to increase the effective conductivity of the paste. If the volume

fraction of dispersed silver is $30$ percent, what is the effective conductivity of this paste?

Note: Look up the values in Table $2($see Question $2$ above$)$ needed to calculate the room temperature $(20C)$

resistivity of silver.