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Assuming the sun to have a spherical out...

Assuming the sun to have a spherical outer surface of radius `r` radiating like a black body at temperature `t^(@)C`. The power received by a unit surface (normal to the incident rays) at a distance `R` from the centre of the sun is
where `sigma` is the Stefan's constant.

A

`(4 pi r^2sigmat^4)/(R^2)`

B

`(r^2sigma(t+273)^(4))/(4 piR^2)`

C

`(16 pi^(2)r^(2)sigmat^4)/(R^2)`

D

`(r^2sigma(t+273)^(4))/(R^2)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem of finding the power received by a unit surface area at a distance \( R \) from the center of the sun, we can follow these steps: ### Step 1: Understand the Concept of Intensity Intensity (\( I \)) is defined as the power (\( P \)) received per unit area. According to Stefan-Boltzmann law, the total power radiated by a black body is given by: \[ P = \sigma A T^4 \] where \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area of the black body, and \( T \) is the absolute temperature in Kelvin. ### Step 2: Calculate the Surface Area of the Sun The surface area \( A \) of the sun, which we assume to be a sphere, is given by: \[ A = 4 \pi r^2 \] where \( r \) is the radius of the sun. ### Step 3: Convert the Temperature to Kelvin The temperature given is \( t \) degrees Celsius. To convert this to Kelvin, we use the formula: \[ T = t + 273 \] For example, if \( t = 80 \) degrees Celsius, then: \[ T = 80 + 273 = 353 \text{ K} \] ### Step 4: Calculate the Total Power Radiated by the Sun Using the Stefan-Boltzmann law, the total power radiated by the sun can be expressed as: \[ P = \sigma (4 \pi r^2) (T)^4 \] ### Step 5: Determine the Intensity at a Distance \( R \) The intensity \( I \) at a distance \( R \) from the center of the sun is given by: \[ I = \frac{P}{4 \pi R^2} \] Substituting the expression for \( P \): \[ I = \frac{\sigma (4 \pi r^2) (T)^4}{4 \pi R^2} \] ### Step 6: Simplify the Expression The \( 4 \pi \) terms cancel out: \[ I = \frac{\sigma r^2 T^4}{R^2} \] ### Final Expression Thus, the power received by a unit surface area at a distance \( R \) from the center of the sun is: \[ I = \frac{\sigma r^2 (T + 273)^4}{R^2} \]

To solve the problem of finding the power received by a unit surface area at a distance \( R \) from the center of the sun, we can follow these steps: ### Step 1: Understand the Concept of Intensity Intensity (\( I \)) is defined as the power (\( P \)) received per unit area. According to Stefan-Boltzmann law, the total power radiated by a black body is given by: \[ P = \sigma A T^4 \] where \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area of the black body, and \( T \) is the absolute temperature in Kelvin. ...
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Knowledge Check

  • Assuming the Sun to be a spherical body of radius R at a temperature of TK, evalute the total radiant powerd incident of Earth at a distance r from the sun where r_0 is the radius of the Earth and sigma is Stefan's constant.

    A
    `4pir_0^2R^2sigma(T^4)/(r^2)`
    B
    `pir_0^2R^2sigma(T^4)/(r^2)`
    C
    `r_0^2R^2sigma(T^4)/(4pir^2)`
    D
    `R^2sigma(T^4)/(r^2)`
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    A
    inversely proportional to `(S+R)^(2)`
    B
    inversely proportional to `R^(2)`.
    C
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  • Inner and outer surfaces of a cylinder are maintained at temperature 4T and T respectively. Heat flows radially from inner surface of radius R to outer surface of radius 4R. Radial distance from the center where temperature is 2T.

    A
    2 R
    B
    `2^(5//3)R`
    C
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    D
    3 R
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