A spherical black body with a radius of 12 cm radiates 450 W power at 500 K. If the radius were halved and the temperature doubled, the power radiated in watt would be (a)225 (b)450 (c) 900 (d)1800
Answers
The answer is (d)1800 W.
A spherical black body with a radius of 12 cm radiates 450 W power at 500 K.
The radius is halved and the temperature is doubled.
- We know that , heat radiated per unit time by a black body is given by,
E = σAT⁴ where σ is the Stefan's constant of radiation, A is the surface area and T is the temperature of the body
- For spherical body, E=σ.T⁴.4πR²
- From this equation, we can see that the heat radiated is directly proportional to the temperature as well as the radius.
- Let E₁, T₁ and R₁ be the initial parameters and E₂, T₂ and R₂ be the final parameters.
- R₁ = 12 cm = 0.12 m
- Therefore, E₂/E₁ = (T₂/T₁)⁴.(R₂/R₁)²
E₂/450 = (1000/500)⁴ × (0.06/0.12)²
E₂ = 450 × 2⁴ × (1/2)²
E₂ = 1800 W
We know that the Power emitted by a black body , P = σAT⁴ where A is area of the body and T is the temperature.
r - initial radius of the spherical body = 12 cm
r' - final radius of the spherical body = 6 cm
T - initial temperature = 500 K
T' - final temperature = 1000 K
Initial power = P = σAT⁴ = σ(4πr²)T⁴ = 450 W
Final power = P' = σA'T'⁴ = σ(4πr'²)T'⁴
P/P' = (r/r')² × (T/T')⁴
=> P/P' = (12/6)² (500/1000)⁴
=> P' = 4P
=> P' = 1800 W
The power radiated wil be equal ro 1800 W and the option d) is correct.