1. Tardigrade
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  3. Physics
  4. Seawater at a frequency f =9 × 102 Hz, has permittivity ε=80 ε0 and resistivity ρ=0.25 Ω m. Imagine a parallel plate capacitor is immersed in seawater and is driven by an alternatingv voltage source V(t)=V0 sin (2 π f t). Then the conduction current density becomes 10 x times the displacement current density after time t =(1/800) s . The value of x is (. Given .: (1/4 π ε0)=9 × 109 Nm 2 C -2)

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Q. Seawater at a frequency $f =9 \times 10^{2} Hz$, has permittivity $\varepsilon=80 \varepsilon_{0}$ 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 $(2 \pi f t)$. 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 ________
$\left(\right.$ Given $\left.: \frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} Nm ^{2} C ^{-2}\right)$

JEE MainJEE Main 2021Current Electricity

Solution:

$J _{ c }=\frac{ E }{\rho}=\frac{ V }{\rho d }$
$J _{ d }=\frac{1}{ A } \frac{ d q }{ dt }$
$=\frac{C}{A} \frac{d V_{c}}{d t}$
$=\frac{\epsilon}{ d } \frac{ d V _{ c }}{ dt }$
$\Rightarrow \frac{ V _{0} \sin 2 \pi ft }{\rho d }=10^{ x } \times \frac{80 \varepsilon_{0}}{ d } V _{0}(2 \pi f ) \cos 2 \pi ft$
$\tan \left(2 \pi \times \frac{900}{800}\right)=10^{ x } \times \frac{40}{9 \times 10^{9}} \times 900$
$= x =6$