विभिन्न पदार्थों की बनी दो छड़ों \(A\) और \(B\) को, आरेख में दर्शाये गये अनुसार आपस में वैल्ड कर जोड़ दिया गया है। इन छड़ों की ऊष्मा चालकता क्रमश: \(K _{1}\) तथा \(K _{2}\) हैं। तो, इनसे बनी संयुक्त छड़ की ऊष्मा चालकता होगी :

A \(\frac{{3{K_1}{K_2}}}{2}\)
B \(\frac{{{K_1} + {K_2}}}{3}\)
C \(\;\frac{{{K_1}{K_2}}}{{3({K_1} + {K_2})}}\)
D \(\;\frac{{{K_1} + {K_2}}}{2}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\;\frac{{{K_1} + {K_2}}}{2}\)

Step-by-step Solution

Detailed explanation

Eqiuivalent thermal conductivity of the composite rod in parallel combination will be, \(K = \frac{{{K_1}{A_1} + {K_2}{A_2}}}{{A{ _1} + {A_2}}} = \frac{{{K_1} + {K_2}}}{2}\)

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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