Question

A leaky parallel plate capacitor is filled completely with a material having dielectric constant $K=5$ and electrical conductivity $\sigma =7.4\times {10}^{-12}{\Omega }^{-1}{m}^{-1}$. If the charge on the plate at the instant $t=0$ is $q=8.85\mu C$, then calculate the leakage current at the instant $t=12$s.

• A. $0.30$ $\mu$A.
• B. $0.20$ $\mu$A.
• C. $0.40$ $\mu$A.
• D. $0.50$ $\mu$A.

Answer 1

Answer: (B)

$0.20$ $\mu$A.

$\text{Given}:$ K = 5, $\sigma =7.4\times {10}^{-12}{\Omega }^{-1}{m}^{-1}$, q(t = 0) = 8.85$\mu C$

$\text{Solution}:$

${\tau }_C=CR=\frac{K{\epsilon }_{0}A}{d}\frac{\rho d}{A}=K{\epsilon }_0\rho$

$5\times 8.86\times \frac{{10}^{-12}}{7.4\times {10}^{-12}}\approx 6s$

Initial current

$i_0=\frac{\frac{q_0}{C}}{r}=\frac{q_0}{CR}=\frac{q_0}{{\tau }_C}=\frac{8.55}{6}=1.425\mu A$

Now, current as function of i

$i=i_0{e}^{\frac{-t}{{\tau }_C}}$

$i=\left(1.425\right)e^{\frac{-12}{6}}=0.193\mu A\approx 0.2\mu A$

$\text{Hence B is the correct option}$