\(a\) બાજુવાળી ચોરસ પ્લેટોને \(d\) અંતરે રાખી એક સમાંતર પ્લેટ કેપેસિટર બનાવવામાં આવે છે \((d < < a)\) અને આકૃતિમાં બતાવ્યા પ્રમાણે અવાહક એવી રીતે ભરવામાં આવે છે કે જેથી નીચેનો ત્રિકોણ \(K\) જેટલા પરાવૈદ્યુતાંક \((dialectric)\) ધરાવતા અવાહકથી ભરેલો છે. આ કેપેસિટરનું કેપેસિટન્સ __________ છે.

A \(\frac{{K{\varepsilon _0}{a^2}}}{{d\left( {K - 1} \right)}}\,\ln \,K\)
B \(\frac{{K{\varepsilon _0}{a^2}}}{{2d\left( {K + 1} \right)}}\)
C \(\frac{{K{\varepsilon _0}{a^2}}}{d}\,\ln \,K\)
D \(\frac{1}{2}\frac{{K{\varepsilon _0}{a^2}}}{d}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{{K{\varepsilon _0}{a^2}}}{{d\left( {K - 1} \right)}}\,\ln \,K\)

Step-by-step Solution

Detailed explanation

Let's consider a strip of thickness \(dx\) at a distance of \(x\) from the left end as shown in the figure. \(\frac{y}{x}=\frac{d}{a}\) \(\Rightarrow \quad y=\left(\frac{d}{a}\right) x\)…

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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