Question

A spherical black body of radius 12 cm at a temperature of \(T\) K radiates a power of 400 W. If the radius of the sphere is doubled and absolute temperature is halved, then the power radiated by the body is

• A. 100 W
• B. 200 W
• C. 400 W
• D. 1600 W

Hint:

Power radiated depends on surface area and the fourth power of temperature: \(P \propto r^2 T^4\). Scale both to compute new power quickly.

Answer 1

The Correct Option is A

Power radiated by a black body is given by the Stefan–Boltzmann law: \[ P = \sigma A T^4 = \sigma (4\pi r^2) T^4 \] Initial power: \[ P_1 = \sigma (4\pi r^2) T^4 \] New radius \(= 2r\), new temperature \(= \frac{T}{2}\) New power: \[ P_2 = \sigma (4\pi (2r)^2) \left(\frac{T}{2}\right)^4 = \sigma (4\pi \cdot 4r^2) \cdot \frac{T^4}{16} = \sigma (4\pi r^2) T^4 \cdot \frac{4}{16} = \frac{1}{4} P_1 \] \[ P_2 = \frac{1}{4} \times 400 = 100 \, \text{W} \]