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Two spheres A and B of same material have radii 1m and 4m and temperature 1000 K and 2000 K respectively. Which one of the following statement is correct.
The energy radiated by sphere A is
The energy radiated by sphere A is
- a)greater than that of spheres B
- b)less than that of sphere B
- c)equal to that of sphere B
- d)equal to double that of sphere B
Correct answer is option 'C'. Can you explain this answer?
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Energy radiated by a sphere
The energy radiated by a sphere can be calculated using the Stefan-Boltzmann law, which states that the power radiated per unit area by a black body is proportional to the fourth power of its absolute temperature. Mathematically, it can be expressed as:
E = εσAT^4
Where:
E is the energy radiated
ε is the emissivity of the material (assumed to be 1 for simplicity in this case)
σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4)
A is the surface area of the sphere
T is the absolute temperature of the sphere
Therefore, the energy radiated by a sphere is directly proportional to the surface area and the fourth power of the temperature.
Comparison between sphere A and B
1. Surface area:
The surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere.
- Sphere A: Surface area = 4π(1)^2 = 4π
- Sphere B: Surface area = 4π(4)^2 = 64π
2. Temperature:
- Sphere A: T = 1000 K
- Sphere B: T = 2000 K
Calculating the energy radiated
Using the formula for energy radiated, we can calculate the energy radiated by each sphere.
- Sphere A: E_A = εσA_T^4 = 1 * 5.67 x 10^-8 * 4π * (1000)^4
- Sphere B: E_B = εσA_T^4 = 1 * 5.67 x 10^-8 * 64π * (2000)^4
Simplifying the equations:
- Sphere A: E_A = 4.52 x 10^11π
- Sphere B: E_B = 1.15 x 10^16π
Comparison of energy radiated
As we can see, the energy radiated by sphere A is approximately 4.52 x 10^11π, while the energy radiated by sphere B is approximately 1.15 x 10^16π.
Since the value of π is constant, we can compare the two energies directly.
- Sphere A: E_A ≈ 4.52 x 10^11
- Sphere B: E_B ≈ 1.15 x 10^16
Therefore, it is clear that the energy radiated by sphere A is much less than that of sphere B. Hence, option 'C' is correct, and the energy radiated by sphere A is equal to that of sphere B.
The energy radiated by a sphere can be calculated using the Stefan-Boltzmann law, which states that the power radiated per unit area by a black body is proportional to the fourth power of its absolute temperature. Mathematically, it can be expressed as:
E = εσAT^4
Where:
E is the energy radiated
ε is the emissivity of the material (assumed to be 1 for simplicity in this case)
σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4)
A is the surface area of the sphere
T is the absolute temperature of the sphere
Therefore, the energy radiated by a sphere is directly proportional to the surface area and the fourth power of the temperature.
Comparison between sphere A and B
1. Surface area:
The surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere.
- Sphere A: Surface area = 4π(1)^2 = 4π
- Sphere B: Surface area = 4π(4)^2 = 64π
2. Temperature:
- Sphere A: T = 1000 K
- Sphere B: T = 2000 K
Calculating the energy radiated
Using the formula for energy radiated, we can calculate the energy radiated by each sphere.
- Sphere A: E_A = εσA_T^4 = 1 * 5.67 x 10^-8 * 4π * (1000)^4
- Sphere B: E_B = εσA_T^4 = 1 * 5.67 x 10^-8 * 64π * (2000)^4
Simplifying the equations:
- Sphere A: E_A = 4.52 x 10^11π
- Sphere B: E_B = 1.15 x 10^16π
Comparison of energy radiated
As we can see, the energy radiated by sphere A is approximately 4.52 x 10^11π, while the energy radiated by sphere B is approximately 1.15 x 10^16π.
Since the value of π is constant, we can compare the two energies directly.
- Sphere A: E_A ≈ 4.52 x 10^11
- Sphere B: E_B ≈ 1.15 x 10^16
Therefore, it is clear that the energy radiated by sphere A is much less than that of sphere B. Hence, option 'C' is correct, and the energy radiated by sphere A is equal to that of sphere B.
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Two spheres A and B of same material have radii 1m and 4m and temperature 1000 K and 2000 K respectively. Which one of the following statement is correct.The energy radiated by sphere A isa)greater than that of spheres Bb)less than that of sphere Bc)equal to that of sphere Bd)equal to double that of sphere BCorrect answer is option 'C'. Can you explain this answer?
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