Question

Seawater at a frequency $f=9\times 10^{2}Hz$, has permittivity $\epsilon =80{\epsilon }_0$ and resistivity $\rho =0.25\Omega m$. Imagine a parallel plate capacitor is immersed in seawater and is driven by an alternatingv voltage source $V(t)=V_0$ sin $(2\pi ft)$. Then the conduction current density becomes $10^x$ times the displacement current density after time
$t=\frac{1}{800}s.$ The value of $x$ is ________
$($ Given $:\frac{1}{4\pi {\epsilon }_0}=9\times 10^{9}N{m}^2{C}^{-2})$

Answer 1

Answer: 6

$J_c=\frac{E}{\rho }=\frac{V}{\rho d}$
$J_d=\frac{1}{A}\frac{dq}{dt}$
$=\frac{C}{A}\frac{d{V}_c}{dt}$
$=\frac{ϵ}{d}\frac{d{V}_c}{dt}$
$\Rightarrow \frac{V_0\sin 2\pi ft}{\rho d}=10^x\times \frac{80{\epsilon }_0}{d}{V}_0(2\pi f)\cos 2\pi ft$
$\tan \left(2\pi \times \frac{900}{800}\right)=10^x\times \frac{40}{9\times 10^9}\times 900$
$=x=6$