Question

Consider a pair of insulating blocks with thermal resistances \( R_1 \) and \( R_2 \) as shown in the figure. The temperature \( \theta \) at the boundary between the two blocks is:

• A. \( \theta_2 = \frac{R_2}{R_1 + R_2} \theta_1 \)
• B. \( \theta_1 + \theta_2 = \frac{R_1 R_2}{R_1 + R_2} \)
• C. \( \theta_1 + \theta_2 = \frac{R_1 R_2}{R_1 + R_2} \)
• D. \( \theta_1 R_2 + R_1 \)

Hint:

In heat transfer through materials with different thermal resistances, the temperature at the boundary is proportional to the resistances.

Answer 1

The Correct Option is A

Step 1: Understanding thermal resistance.
In a steady-state heat transfer situation, the temperature at the interface is determined by the thermal resistances of the materials. The relation between the temperatures and resistances is: \[ \frac{\theta_2}{\theta_1} = \frac{R_2}{R_1 + R_2} \] Step 2: Conclusion.
Thus, the temperature \( \theta_2 \) at the boundary between the blocks is given by: \[ \theta_2 = \frac{R_2}{R_1 + R_2} \theta_1 \]
Final Answer: \[ \boxed{\theta_2 = \frac{R_2}{R_1 + R_2} \theta_1} \]