Question

Consider the situation of the previous problem. Define displacement resistance Rd = V/id of the space between the plates, where V is the potential difference between the plates and id is the displacement current. Show that Rd varies with time as Rd=R(et/τ-1).

Answer 1

Explanation:

To Show that Rd varies with time as Rd=R(et/τ-1).

Strength of electric field for parallel frame capacitor = $E=\frac{Q}{A e_{0}}$  

Electric flux,

$\phi=E \cdot A=\frac{Q}{A e_{0}} \cdot A=\frac{Q}{\epsilon_{0}}$

Step 1:

Displacement of  current,

$\begin{aligned}i_{d}=\epsilon_{0} \frac{d \phi_{E}}{d t}=& \in_{0} \frac{d}{d t}\left(\frac{Q}{\epsilon_{0}}\right) \\i_{d} &=\left(\frac{d Q}{d t}\right)\end{aligned}$

Also, Q = CV  

$\begin{aligned}i_{d} &=\frac{d}{d t}\left(E_{0} C e^{-\frac{t}{R C}}\right) \\i_{d} &=E_{0} C\left(-\frac{1}{R C}\right) e^{-\frac{t}{R C}}\end{aligned}$

Step 2:

Displacement of resistance,  

$\begin{aligned}&R_{d}=\frac{E_{0}}{i_{d}}-R\\&R_{d}=\frac{E_{0}}{\frac{E_{0}}{R} e^{-\frac{t}{R C}}}-R |\\&R_{d}=R e^{\frac{t}{R C}}-R\\&R_{d}=R\left(e^{\frac{t}{R C}}-1\right)\end{aligned}$