Question

A solid sphere at a temperature \( T \) K is cut into two hemispheres. The ratio of energies radiated by one hemisphere to the whole sphere per second is:

• A. \( 1:1 \)
• B. \( 1:2 \)
• C. \( 3:4 \)
• D. \( 1:4 \)

Hint:

Use Stefan-Boltzmann law \( E = \sigma A T^4 \) for radiation problems.
- When cutting a sphere into hemispheres, include both the curved and flat surfaces in area calculations.

Answer 1

The Correct Option is C

Using Stefan-Boltzmann law, the power radiated by a body is given by: \[ E = \sigma A T^4 \] where \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the temperature. 1. Surface Area of the Sphere: The total surface area of the sphere is: \[ A_{\text{sphere}} = 4\pi R^2 \] The energy radiated per second by the sphere: \[ E_{\text{sphere}} = \sigma (4\pi R^2) T^4 \] 2. Surface Area of One Hemisphere: Each hemisphere consists of: - A curved surface of area \( 2\pi R^2 \) - A flat circular base of area \( \pi R^2 \) So, the total surface area of one hemisphere is: \[ A_{\text{hemisphere}} = 2\pi R^2 + \pi R^2 = 3\pi R^2 \] The energy radiated per second by one hemisphere: \[ E_{\text{hemisphere}} = \sigma (3\pi R^2) T^4 \] 3. Ratio of Energies: \[ \frac{E_{\text{hemisphere}}}{E_{\text{sphere}}} = \frac{3\pi R^2}{4\pi R^2} = \frac{3}{4} \] Hence, the required ratio is \( 3:4 \). Thus, the correct answer is \(\boxed{3:4}\).