Question

A carnot engine having efficiency \(\frac{1}{6}\), operates between the source temperature \(T_H\) and the sink temperature \(T_C\). Its efficiency increases to \(\frac{1}{3}\), when \(T_C\) is decreased by 64 K. The temperatures \(T_H\) and \(T_C\) are repectively

• A. \(374\) K , \(310\) K
• B. \(384\) K , \(320\) K
• C. \(384\) K , \(340\) K
• D. \(320\) K , \(256\) K

Hint:

Wien law gives temperature from peak wavelength; power goes as T to the fourth.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Wien's law: \(\lambda_mT = \text{constant}\). Stefan's law: power \(\propto T^4\).

Step 2: New temperature:
The peak wavelength changes from \(\lambda\) to \(\frac{2\lambda}{3}\), so \(T' = T\times\frac{\lambda}{2\lambda/3} = \frac32T\).

Step 3: Power ratio:
\[ \frac{P'}{P} = \left(\frac32\right)^4 = \frac{81}{16} \]

Final Answer:
The power increases by the factor \(\frac{81}{16}\), option (B). \[ \boxed{\frac{81}{16}} \]