Question

Two spherical black bodies radiate the same amount of heat per second. If their temperatures are \( T_1 \) and \( T_2 \), and their radii are \( R_1 \) and \( R_2 \), respectively, find the relation between their temperatures and radii.

• A. \( T_1 = \sqrt{2} T_2 \)
• B. \( T_1 = 2 T_2 \)
• C. \( T_1 = \frac{T_2}{\sqrt{2}} \)
• D. \( T_1 = \sqrt{3} T_2 \)

Hint:

When comparing the radiation of heat by two bodies, use the Stefan-Boltzmann law and equate the radiated power to solve for the unknown variable. Make sure to apply the correct formulas for the surface area of spherical bodies.

Answer 1

The Correct Option is A

The problem involves two spherical black bodies radiating the same amount of heat per second. According to Stefan-Boltzmann Law, the power radiated \( P \) by a spherical black body is given by:

\[ P = \sigma A T^4 \]

where \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the temperature of the black body.

For a sphere, the surface area \( A = 4\pi R^2 \). Thus, the power radiated by each sphere can be given as:

\[ P_1 = \sigma (4\pi R_1^2) T_1^4 \]

\[ P_2 = \sigma (4\pi R_2^2) T_2^4 \]

Since both bodies radiate the same amount of heat per second, we equate \( P_1 \) and \( P_2 \):

\[ \sigma (4\pi R_1^2) T_1^4 = \sigma (4\pi R_2^2) T_2^4 \]

We can cancel \(\sigma\) and \(4\pi\) from both sides:

\[ R_1^2 T_1^4 = R_2^2 T_2^4 \]

Rearranging the equation, we get:

\[ \left(\frac{T_1}{T_2}\right)^4 = \left(\frac{R_2}{R_1}\right)^2 \]

Taking the fourth root on both sides:

\[ \frac{T_1}{T_2} = \left(\frac{R_2}{R_1}\right)^{1/2} \]

Now, if we assume the relation \( T_1 = \sqrt{2} T_2 \), it implies:

\[ \frac{T_1}{T_2} = \sqrt{2} \]

Thus, \(\left(\frac{R_2}{R_1}\right)^{1/2} = \sqrt{2}\).

Since both sides match, the correct relation is indeed \( T_1 = \sqrt{2} T_2 \).