આકૃતિમાં દર્શાવ્યા પ્રમાણે \(\varepsilon \; emf \;\) ધરાવતી બેટરી સાથે \(L\) ઇન્ડક્ટર અને \(R\) અવરોધ શ્રેણીમાં જોડેલા છે.\(t=0\) સમયે કળ બંધ છે.\(\mathrm{t}=0\) અને \(\mathrm{t}=\mathrm{t}_{\mathrm{c}}\;( \mathrm{t}_{\mathrm{c}}=\)પરિપથનો સમય અચળાંક) વચ્ચે બેટરીમાથી કેટલો વિજભાર બહાર આવ્યો હશે?

A \(\frac{\varepsilon \mathrm{L}}{\mathrm{R}^{2}}\left(1-\frac{1}{\mathrm{e}}\right)\)
B \(\frac{\varepsilon \mathrm{R}}{\mathrm{e} \mathrm{L}^{2}}\)
C \(\frac{\varepsilon \mathrm{I}}{\mathrm{R}^{2}}\)
D \(\frac{\varepsilon \mathrm{I}}{\mathrm{e} \mathrm{R}^{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{\varepsilon \mathrm{I}}{\mathrm{e} \mathrm{R}^{2}}\)

Step-by-step Solution

Detailed explanation

\(\mathrm{i}=\mathrm{i}_{0}\left(1-\mathrm{e}^{-\mathrm{R}t / \mathrm{L}}\right)=\mathrm{i}_{0}\left(1-\mathrm{e}^{-t / T_{\mathrm{C}}}\right)\) \(\mathrm{q}=\int_{0}^{\mathrm{T}_{\mathrm{C}}} \mathrm{i} \mathrm{dt}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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