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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?
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
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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.
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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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