NEET-XII-Physics
40: Electromagnetic Waves
- #4Consider 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).Ans : Electric field strength for a parallel plate capacitor = `` E=\frac{Q}{{\in }_{0}A}``
`` \,\mathrm{\,Electric\,}\,\mathrm{\,flux\,},\varphi \mathit{=}E\mathit{.}A\mathit{=}\frac{Q}{{\mathit{\in }}_{0}A}\mathit{.}A=\frac{Q}{{\in }_{0}}``
`` \,\mathrm{\,Displacement\,}\,\mathrm{\,current\,},{i}_{\,\mathrm{\,d\,}}={\in }_{0}\frac{d{\varphi }_{\,\mathrm{\,E\,}}}{dt}={\in }_{0}\frac{d}{dt}\left(\frac{\,\mathrm{\,Q\,}}{{\in }_{0}}\right)``
`` {i}_{\,\mathrm{\,d\,}}=\left(\frac{dQ}{dt}\right)``
`` \,\mathrm{\,Also\,},Q\mathit{=}CV``
`` {i}_{\,\mathrm{\,d\,}}=\frac{d}{dt}\left({E}_{\mathit{0}}{{C}_{e}}^{\mathit{-}t\mathit{/}RC}\right)``
`` {i}_{\,\mathrm{\,d\,}}={E}_{0}C\left(\mathit{-}\frac{\mathit{1}}{RC}\right){e}^{\mathit{-}t\mathit{/}RC}``
`` \,\mathrm{\,Displacement\,}\,\mathrm{\,resistance\,},{R}_{d}=\frac{{E}_{\mathit{0}}}{{i}_{\,\mathrm{\,d\,}}}-R``
`` \Rightarrow {R}_{d}=\frac{{E}_{\mathit{0}}}{\frac{{E}_{\mathit{0}}}{R}{\,\mathrm{\,e\,}}^{\mathit{-}t\mathit{/}RC}}-R``
`` \Rightarrow {R}_{d}=R{\,\mathrm{\,e\,}}^{t\mathit{/}RC}-R``
`` \mathit{\Rightarrow }{R}_{\mathit{d}}\mathit{=}R({\,\mathrm{\,e\,}}^{\mathit{t}\mathit{/}\mathit{R}\mathit{C}}-1)``
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