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Two identical solid spheres have the same temperature. One of the sphere is cut into two identical pieces. The intact sphere radiates an energy Q during a given small time interval. During the same interval, the two hemispheres radiate a total energy Q'. The ratio Q'/Q is equal to :
- a)2.0
- b)4.0
- c)2/3
- d)1.5
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Two identical solid spheres have the same temperature. One of the sphe...
Heat radiated (at temp same temp) ∝ A
⇒ Q ∝ 4πR2 and Q' ∝ (4πR2 + 2 x πR2)
⇒
Here πR2 is extra surface area of plane surface of the hemisphere
⇒ Q ∝ 4πR2 and Q' ∝ (4πR2 + 2 x πR2)
⇒
Here πR2 is extra surface area of plane surface of the hemisphere
Most Upvoted Answer
Two identical solid spheres have the same temperature. One of the sphe...
Explanation:
When a solid sphere is cut into two identical pieces, each piece will have half the volume of the original sphere but the same surface area. Let's assume the original sphere has a radius of R.
Step 1: Calculate the surface area of the original sphere:
The surface area of a sphere is given by the formula:
A = 4πR²
Step 2: Calculate the surface area of each hemisphere:
When the sphere is cut into two identical hemispheres, each hemisphere will have half the surface area of the original sphere. Therefore, each hemisphere will have a surface area of:
A_hemisphere = (1/2) * 4πR² = 2πR²
Step 3: Calculate the energy radiated by the intact sphere:
The energy radiated by a sphere is given by the Stefan-Boltzmann law:
E = σεAT⁴
where E is the energy radiated, σ is the Stefan-Boltzmann constant, ε is the emissivity of the material, A is the surface area, and T is the temperature in Kelvin.
Since both spheres have the same temperature, emissivity, and time interval, we can say that the energy radiated by the intact sphere is Q.
Step 4: Calculate the energy radiated by each hemisphere:
Since each hemisphere has half the surface area of the intact sphere, the energy radiated by each hemisphere can be calculated as:
E_hemisphere = σε(2πR²)T⁴
Step 5: Calculate the ratio of the energy radiated:
Now, let's calculate the ratio Q/Q_hemisphere:
Q/Q_hemisphere = Q / (E_hemisphere + E_hemisphere)
= Q / (2 * E_hemisphere)
= Q / (2 * σε(2πR²)T⁴)
= 1 / (4 * σεπR²T⁴)
Step 6: Simplify the expression:
Using the value of the Stefan-Boltzmann constant (σ = 5.67 x 10⁻⁸ W/m²K⁴) and assuming the emissivity (ε) and temperature (T) to be constant, we can simplify the expression further:
Q/Q_hemisphere = 1 / (4 * σεπR²T⁴)
= 1 / (4 * 5.67 x 10⁻⁸ * ε * πR² * T⁴)
= 1.75 x 10⁷ / (εR²T⁴)
Step 7: Compare the simplified expression with the given options:
The ratio Q/Q_hemisphere is equal to 1.75 x 10⁷ / (εR²T⁴). This expression does not match any of the given options (a), b), c), or d)). Therefore, the correct answer cannot be determined based on the given options.
When a solid sphere is cut into two identical pieces, each piece will have half the volume of the original sphere but the same surface area. Let's assume the original sphere has a radius of R.
Step 1: Calculate the surface area of the original sphere:
The surface area of a sphere is given by the formula:
A = 4πR²
Step 2: Calculate the surface area of each hemisphere:
When the sphere is cut into two identical hemispheres, each hemisphere will have half the surface area of the original sphere. Therefore, each hemisphere will have a surface area of:
A_hemisphere = (1/2) * 4πR² = 2πR²
Step 3: Calculate the energy radiated by the intact sphere:
The energy radiated by a sphere is given by the Stefan-Boltzmann law:
E = σεAT⁴
where E is the energy radiated, σ is the Stefan-Boltzmann constant, ε is the emissivity of the material, A is the surface area, and T is the temperature in Kelvin.
Since both spheres have the same temperature, emissivity, and time interval, we can say that the energy radiated by the intact sphere is Q.
Step 4: Calculate the energy radiated by each hemisphere:
Since each hemisphere has half the surface area of the intact sphere, the energy radiated by each hemisphere can be calculated as:
E_hemisphere = σε(2πR²)T⁴
Step 5: Calculate the ratio of the energy radiated:
Now, let's calculate the ratio Q/Q_hemisphere:
Q/Q_hemisphere = Q / (E_hemisphere + E_hemisphere)
= Q / (2 * E_hemisphere)
= Q / (2 * σε(2πR²)T⁴)
= 1 / (4 * σεπR²T⁴)
Step 6: Simplify the expression:
Using the value of the Stefan-Boltzmann constant (σ = 5.67 x 10⁻⁸ W/m²K⁴) and assuming the emissivity (ε) and temperature (T) to be constant, we can simplify the expression further:
Q/Q_hemisphere = 1 / (4 * σεπR²T⁴)
= 1 / (4 * 5.67 x 10⁻⁸ * ε * πR² * T⁴)
= 1.75 x 10⁷ / (εR²T⁴)
Step 7: Compare the simplified expression with the given options:
The ratio Q/Q_hemisphere is equal to 1.75 x 10⁷ / (εR²T⁴). This expression does not match any of the given options (a), b), c), or d)). Therefore, the correct answer cannot be determined based on the given options.
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Two identical solid spheres have the same temperature. One of the sphe...
C
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Two identical solid spheres have the same temperature. One of the sphere is cut into two identical pieces. The intact sphere radiates an energy Q during a given small time interval. During the same interval, the two hemispheres radiate a total energy Q. The ratio Q/Q is equal to :a)2.0b)4.0c)2/3d)1.5Correct answer is option 'D'. Can you explain this answer?
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Two identical solid spheres have the same temperature. One of the sphere is cut into two identical pieces. The intact sphere radiates an energy Q during a given small time interval. During the same interval, the two hemispheres radiate a total energy Q. The ratio Q/Q is equal to :a)2.0b)4.0c)2/3d)1.5Correct answer is option 'D'. Can you explain this answer? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about Two identical solid spheres have the same temperature. One of the sphere is cut into two identical pieces. The intact sphere radiates an energy Q during a given small time interval. During the same interval, the two hemispheres radiate a total energy Q. The ratio Q/Q is equal to :a)2.0b)4.0c)2/3d)1.5Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two identical solid spheres have the same temperature. One of the sphere is cut into two identical pieces. The intact sphere radiates an energy Q during a given small time interval. During the same interval, the two hemispheres radiate a total energy Q. The ratio Q/Q is equal to :a)2.0b)4.0c)2/3d)1.5Correct answer is option 'D'. Can you explain this answer?.
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