Question

The relation between efficiency ((\eta)) of Carnot engine and coefficient of performance ((\beta)) of refrigerator is

• A. (\eta = \frac{1}{1 + \beta})
• B. (\eta = \frac{1}{1 - \beta})
• C. (\eta = \frac{\beta}{1 - \beta})
• D. (\eta = \frac{1 + \beta}{\beta})

Hint:

Carnot cycle COP ($\beta$) is always greater than efficiency ($\eta$).

Answer 1

The Correct Option is A

Step 1: Definitions
Efficiency $\eta = \frac{W}{Q_H}$.
COP $\beta = \frac{Q_C}{W}$.

Step 2: Derivation
$\beta = \frac{Q_C}{Q_H - Q_C}$.
$1 + \beta = 1 + \frac{Q_C}{Q_H - Q_C} = \frac{Q_H}{Q_H - Q_C} = \frac{Q_H}{W} = \frac{1}{\eta}$.

Step 3: Conclusion
$\eta = \frac{1}{1 + \beta}$.
Final Answer: (A)