Question

Two bodies A and B having temperatures \( 327^\circ C \) and \( 427^\circ C \) are radiating heat to the surrounding. The surrounding temperature is \( 27^\circ C \). The ratio of rates of heat radiation of A to that of B is

• A. 0.52
• B. 0.31
• C. 0.81
• D. 0.42

Hint:

Two bodies A and B having temperatures $327C$ are radiating heat to the surrounding. The surrounding temperature is $27C$. The ratio of rates of heat radiation of A to that of B is

Answer 1

The Correct Option is A

To solve this problem, we need to compare the rates of heat radiation of two bodies, A and B, using the Stefan-Boltzmann law, which states:

\(P = \sigma \cdot A \cdot (T^4 - T_s^4)\)

where:

• \(P\) is the power radiated by the body.
• \(\sigma\) is the Stefan-Boltzmann constant.
• \(A\) is the surface area of the body.
• \(T\) is the absolute temperature of the body in Kelvin.
• \(T_s\) is the absolute temperature of the surroundings in Kelvin.

Given:

• Temperature of A, \(T_A = 327^\circ C = 327 + 273 = 600 \, \text{K}\)
• Temperature of B, \(T_B = 427^\circ C = 427 + 273 = 700 \, \text{K}\)
• Surrounding temperature, \(T_s = 27^\circ C = 27 + 273 = 300 \, \text{K}\)

The rate of heat radiation for body A is given by:

\(P_A = \sigma \cdot A \cdot (600^4 - 300^4)\)

For body B:

\(P_B = \sigma \cdot A \cdot (700^4 - 300^4)\)

We need the ratio of the rates of heat radiation, \(\frac{P_A}{P_B}\):

\(\frac{P_A}{P_B} = \frac{600^4 - 300^4}{700^4 - 300^4}\)

Calculating the terms, we get:

\(600^4 = 1.296 \times 10^{11}\)

\(700^4 = 2.401 \times 10^{11}\)

\(300^4 = 8.1 \times 10^9\)

Substitute back into the ratio:

\(\frac{P_A}{P_B} = \frac{(1.296 \times 10^{11}) - (8.1 \times 10^9)}{(2.401 \times 10^{11}) - (8.1 \times 10^9)}\)

\(\frac{P_A}{P_B} \approx \frac{1.2159 \times 10^{11}}{2.3209 \times 10^{11}} \approx 0.524\)

The ratio of the rates of heat radiation of A to B is approximately 0.52, which matches the correct answer given in the options.