एक समान्तर पट्टिका संधारित्र की पट्टिकाओं के बीच का स्थान एक परावैद्युत से भरा जाता है जिसका परावैद्युत स्थिरांक दूरी के साथ निम्न सम्बन्ध अनुसार परिवर्तित होता है : \(K (x)= K _{ o }+\lambda x(\lambda=\) एक स्थिरांक \()\) संधारित्र की धारिता \(C\), इसकी निर्वात धारिता, \(C _{ O }\) के साथ निम्न सम्बन्ध अनुसार सम्बन्धित होगी

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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