1. Given data

• Frequency: $f = 9\times 10^2 = 900\,\text{Hz}$
• Permittivity of seawater: $\varepsilon = 80\varepsilon_0$
• Resistivity: $\rho = 0.25\,\Omega\,\text{m}$
• Applied voltage: $V(t)=V_0\sin(2\pi f t)$
• Time: $t=\dfrac{1}{800}\,\text{s}$

We need the ratio of conduction current density to displacement current density at this instant.

2. Current densities in dielectric medium

For a parallel plate capacitor, electric field varies as $E(t)=E_0\sin(2\pi f t).$

Conduction current density

$J_c = \sigma E = \frac{1}{\rho}E$ So, $J_c = \frac{1}{\rho}E_0\sin(2\pi f t).$

Displacement current density

$J_d = \varepsilon \frac{dE}{dt}$ Thus, $J_d = \varepsilon E_0 (2\pi f)\cos(2\pi f t).$

Hence, $\frac{J_c}{J_d} = \frac{\frac{1}{\rho}E_0\sin(2\pi f t)}{\varepsilon E_0 (2\pi f)\cos(2\pi f t)}$ $\Rightarrow \frac{J_c}{J_d} = \frac{\tan(2\pi f t)}{\rho\varepsilon 2\pi f}.$

3. Evaluate $\tan(2\pi f t)$

$2\pi f t = 2\pi\times 900\times \frac{1}{800} = 2\pi\times \frac{9}{8} = \frac{9\pi}{4}.$

Now, $\tan\left(\frac{9\pi}{4}\right)=\tan\left(2\pi+\frac{\pi}{4}\right)=\tan\left(\frac{\pi}{4}\right)=1.$

So, $\frac{J_c}{J_d}=\frac{1}{\rho\varepsilon 2\pi f}.$

4. Substitute material constants

Given, $\rho=0.25, \qquad \varepsilon=80\varepsilon_0, \qquad f=900.$

Therefore, $\frac{J_c}{J_d}=\frac{1}{0.25\times 80\varepsilon_0\times 2\pi\times 900}.$

Since $0.25\times 80 = 20$, $\frac{J_c}{J_d}=\frac{1}{20\varepsilon_0\times 1800\pi} = \frac{1}{36000\pi\varepsilon_0}.$

5. Use the given relation

Given, $\frac{1}{4\pi\varepsilon_0}=9\times 10^9$ So, $\frac{1}{\pi\varepsilon_0}=4\times 9\times 10^9 = 36\times 10^9.$

Hence, $\frac{J_c}{J_d}=\frac{1}{36000}\cdot \frac{1}{\pi\varepsilon_0} = \frac{36\times 10^9}{36000}.$

Now, $\frac{36\times 10^9}{36000} = 10^6.$

Thus, $\frac{J_c}{J_d}=10^6 = 10x.$

So, $x=10^5.$

6. Final answer

$\boxed{x=100000}$

7. Comparison with stored answer

Stored correct answer is $6$.

My derived answer is $100000$, so I do not agree with the stored answer.

It appears the intended interpretation may have been that the ratio is $10^x$ times, not $10x$ times. In that case, $\frac{J_c}{J_d}=10^6 \Rightarrow x=6,$ which matches the stored answer.