Question

Consider the situation of the previous problem. Define displacement resistance R_d = V/i_dof 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 `R_d = R(e^(t"/"tau) - 1)` .

Answer 1

Electric field strength for a parallel plate capacitor = `E = Q/(∈_0A)`

Electric flux , `phi = E . A = Q/(∈_0A).A = Q/∈_0`

Displacment current , `i_d = ∈_0 (dphi_E)/dt = ∈_0 d/dt(Q/∈_0)`

`i_d = ((dQ)/dt)`

Also ,  `Q = CV`

`i_d = d/dt(E_0C_e^(-t"/"RC))`

`i_d = E_0C(-1/(RC))e^-t"/"RC`

Displacement resistance , `R_d = E_0/i_d - R`

⇒ `R_d = E_0/(E_0/Re^(-t"/"RC)) - R`

⇒ `R_d = Re^(t"/"RC) - R`

⇒ `R_d = R(e^(t"/"RC) - 1)`