સમાંતર પ્લેટ કેપેસીટરની પ્લેટ વચ્ચે ડાઈઇલેક્ટ્રિક મૂકવામાં આવે છે. જેનો ડાઈઇલેક્ટ્રિક અચળાંક નીચે મુજબ બદલાય છે. \(K(x) = K_0 + \lambda x\) (\(\lambda =\) અચળાંક) શૂન્યાવકાશમાં કેપેસીટરનું મૂલ્ય \(C_0\) હોય તો \(C_0\)ના સ્વરૂપમાં કેપેસીટન્સ \(C\) કેટલું મળે?

A \(C\, = \,\frac{{\lambda d}}{{\ln \,(1 + {K_0}\lambda d)}}{C_0}\)
B \(C\, = \,\frac{{\lambda }}{{d.\ln \,(1 + {K_0}\lambda d)}}{C_0}\)
C \(C\, = \,\frac{{\lambda d}}{{\ln \,(1 + \lambda d/{K_0})}}{C_0}\)
D \(C\, = \,\frac{\lambda }{{d.\ln \,(1 + {K_0}/\lambda d)}}{C_0}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(C\, = \,\frac{{\lambda d}}{{\ln \,(1 + \lambda d/{K_0})}}{C_0}\)

Step-by-step Solution

Detailed explanation

The value of diclectric constant is given as \(\mathrm{K}=\mathrm{K}_{0}+\lambda \mathrm{x}\) And, \(\mathrm{V}=\int_{0}^{\mathrm{d}} \mathrm{Edr}\) \(\mathrm{v}=\int_{0}^{\mathrm{d}} \frac{\sigma}{\mathrm{K}} \mathrm{dx}\)…

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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