In the $RC$ circuit shown, switch is closed at $t = 0$ . Graphs showing variation of potential $(V_R)$ across resistor and potential $(V_C)$ across capacitor are given. Time constant of circuit is approximately equal to.....$ms$
Diffcult
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$\mathrm{V}_{\mathrm{R}}=\varepsilon \cdot \mathrm{e}^{-t / \mathrm{RC}}$
${{\rm{V}}_C} = \varepsilon \left[ {1 - {{\rm{e}}^{ - t/{\rm{RC}}}}} \right]$
At $t=100 \mathrm{\,ms}$
${{\rm{V}}_{\rm{R}}} = {{\rm{V}}_{\rm{C}}} \Rightarrow {{\rm{e}}^{ - t/{\rm{RC}}}} = 1 - {{\rm{e}}^{ - t/{\rm{RC}}}} \Rightarrow 2{{\rm{e}}^{ - t/{\rm{RC}}}} = 1$
$ \Rightarrow {{\rm{e}}^{ - t/{\rm{RC}}}} = 1/2 \Rightarrow {{\rm{e}}^{ - t/{\rm{RC}}}} = 2 \Rightarrow \frac{{\rm{t}}}{{{\rm{RC}}}} = \ell {\rm{n}}2$
$ \Rightarrow \frac{{100}}{{{\rm{RC}}}} = \ell {\rm{n}}2 \Rightarrow {\rm{RC}} = \frac{{100}}{{\ell {\rm{n}}(2)}} = 145.45\,{\rm{ms}}$
(as $\ln 2 = 0.693 \simeq 0.7$)
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