RC time constant

You introduced 1 Coulomb worth of electrons into the inner volume of a dielectric material with ϵr=6. Thirty minutes later, you found that only 36.79% of the electrons were in the internal volume. Determine the conductivity σ of the dielectric material.

Correct answer:

s =  2.95E-14 S/m

Step-by-step explanation:

$$\begin{gathered} Q=1\text{C} \\ \epsilon =6 \\ {\epsilon }_0=8.85\cdot 10^{-12}=8.85\cdot 10^{-12}\text{F/m} \\ t=30\text{min}\to \text{s}=30\cdot 60\text{s}=1800\text{s} \\ p=36.79\% \\ T=t=1800\text{s} \\ T=R\cdot C \\ C=\epsilon \cdot {\epsilon }_0\cdot S/d=\epsilon \cdot K \\ T=R\cdot \epsilon \cdot {\epsilon }_0\cdot S/d \\ R=\rho \cdot d/S \\ T=\rho \cdot d/S\cdot \epsilon \cdot {\epsilon }_0\cdot S/d \\ T=\rho \cdot \epsilon \cdot {\epsilon }_0 \\ \rho =T/(\epsilon \cdot {\epsilon }_0)=1800/(6\cdot 8.85\cdot 10^{-12})=33898305084746\Omega \text{m} \\ s=1/\rho =1/33898305084746=2.95\cdot 10^{-14}\text{S/m} \end{gathered}$$

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