Consider the situation of the previous problem. Define displaceme… - Vidyadip

Question

Consider the situation of the previous problem. Define displacement resistance $\text{R}_\text{d}=\frac{\text{V}}{\text{i}_\text{d}}$ of the space between the plates, where $V$ is the potential difference between the plates and $i_d$ is the displacement current. Show that $R_d$ varies with time as $\text{R}_\text{d}=\text{R}\big(\text{e}^{\text{t}/\tau}-1\big)$

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Answer

$\text{E}=\frac{\text{Q}}{\epsilon_0\text{A}} ($Electric field$)$
$\phi=\text{E.A.}=\frac{\text{Q}}{\epsilon_0\text{A}}\frac{\text{A}}{2}=\frac{\text{Q}}{\epsilon_02}$
$\text{i}_0=\epsilon_0\frac{\text{d}\phi_\text{E}}{\text{dt}}=\epsilon_0\frac{\text{d}}{\text{dt}}\Big(\frac{\text{Q}}{\epsilon_02}\Big)$
$=\frac{1}{2}\Big(\frac{\text{dQ}}{\text{dt}}\Big)$

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