A spherical black body with a radius 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

We can solve for this using Stefan-Boltzmann law, which relates the power radiated by a black body with the following relationship:
P = σ A T⁴
where,
P = power radiated
σ = Stefan-Boltzmann constant
A = surface area of the sphere
T = temperature of the sphere
Now, the surface area of the sphere is given as
A = 4 π r²
Initially,
– The radius of the sphere is r = 12 cm = 0.12 m,
– The temperature is T = 500 K,
– The power radiated is P = 450 W.

Let us first calculate how much power is given out initially using the Stefan-Boltzmann law:

P₁ = σ A₁ T₁⁴

Now, when the radius is halved or become r₂ = 0.06 m and the temperature is doubled or become T₂ = 1000 K, then the new power will be as follows:

P₂ = σ A₂ T₂⁴

Since the area A is proportional to r², we can write the ratio of the new power to the initial power as:

P₂ / P₁ = (A₂ / A₁) × (T₂⁴ / T₁⁴)

Substitute the expressions for the areas and temperatures:

P₂ / P₁ = (r₂² / r₁²) × (T₂⁴ / T₁⁴)

Substitute the values:

P₂ / P₁ = (0.06² / 0.12²) × (1000⁴ / 500⁴)

Simplify:

P₂ / P₁ = (1/4) × (16) = 4

Therefore:

P₂ = 4 × P₁ = 4 × 450 W = 1800 W

Answer: 1800 W

Click for more info:
https://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-10/