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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 alternating 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 \varepsilon _{0}}=9\times 10^{9}Nm^{2}C^{-2})
Option: 1 6
Option: 2 10
Option: 3 11
Option: 4 13

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\begin{array}{l} 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 } \\ \\ \tan \left(2 \pi \times \frac{900}{800}\right)=10^{ x } \times \frac{40}{9 \times 10^{9}} \times 900 \\ \\ =\frac{ d c _{ 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 \\ \\ = x =6 \end{array}

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Deependra Verma

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