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A spherical black body with a radius 12 cm radiates 450W power at 500 K. If the radius were halved and the temperature doubled, the power radiated in watt would be
Power is the rate of doing work or transferring energy, measured in watts (W). It is calculated as work done or energy transferred per unit time. Mathematically, power equals force times velocity or energy divided by time. It is crucial in determining efficiency in mechanical, electrical, and thermal systems.
Class 11 Physics Chapter 10 covers thermal properties of matter including temperature heat and thermal expansion. It explains concepts of specific heat capacity latent heat and the laws of thermodynamics. The chapter also discusses heat transfer methods such as conduction convection and radiation emphasizing the importance of these concepts in understanding energy transfer in various systems.
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We can apply the Stefan-Boltzmann Law, which is defined as the power P radiated by a black body. That power is proportional to the fourth power of its absolute temperature T and the surface area A. It can be represented as:
P = σ A T⁴
Where:
P is in watts and the Stefan-Boltzmann constant is approximately 5.67 × 10^(-8) W/m²K⁴ A = Surface area of the sphere
– T = absolute temperature (in Kelvin)
Given:
– Initial radius r₁ = 12 cm = 0.12 m
– Initial power P₁ = 450 W
– Initial temperature T₁ = 500 K
Surface Area Calculation:
The surface area A of a sphere is given by:
A = 4πr²
1. Initial Surface Area:
A₁ = 4π(0.12)² = 4π(0.0144) ≈ 0.18096 m²
2. New Conditions:
– The new radius r₂ = r₁/2 = 12/2 = 6 cm = 0.06 m
– The new temperature T₂ = 2 × T₁ = 2 × 500 = 1000 K
3. New Surface Area:
A₂ = 4π(0.06)² = 4π(0.0036) ≈ 0.04524 m²
New Power Calculation:
We can now determine the new power P₂ with the following expression:
P₂ = σ A₂ T₂⁴
Since the Stefan-Boltzmann constant σ is universal, we can write the new power in terms of the old power:
P₂/P₁ = (A₂ T₂⁴) / (A₁ T₁⁴)
We now substitute the values:
1. **Compute T₂⁴ and T₁⁴:
T₂⁴ = (1000)⁴ = 10¹²
T₁⁴ = (500)⁴ = 6.25 × 10¹¹
2. Compute the ratio:
P₂/P₁ = (0.04524 × 10¹²) / (0.18096 × 6.25 × 10¹¹)
Computing the above:
P₂/P₁ ≈ (0.04524 × 10¹²) / (1.12625 × 10¹¹) ≈ 0.401
3. Compute P₂:
Now, we put P₁ = 450 W:
P₂ = P₁ × (0.401) × (10¹² / 10¹¹) = 450 × 401 = 900 W
Conclusion:
Thus, if the radius were halved and the temperature doubled, the power radiated would be 900 W. The correct answer is 900.
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