The quantity of radiation received by earth
from the sun is 1.4 kW / m2. Assuming that
sun is an ideal radiator, calculate the surface
temperature of the sun. The ratio of the
radius of earth's orbit to the radius of the
sun is 216.
3 answers
5825
Please send whole solution of this question
The first thing we need to find is the total power emitted by the sun, which we can do using the equation for the surface area of a sphere:
Surface area of sun = 4πr^2
where r is the radius of the sun. We can rearrange this to find r in terms of the surface area:
r = √(Surface area of sun / 4π)
The radius of the sun is about 109 times larger than the radius of the earth, so we can use the ratio of the areas of two spheres to relate the surface area of the sun to the power received by the earth:
Ratio of areas = (Radius of earth / Radius of sun)^2
Ratio of areas = (1 / 109)^2
Ratio of areas = 1 / 11881
Power received by earth = Power emitted by sun / Ratio of areas
1.4 kW / m^2 = Power emitted by sun / 1 / 11881
Power emitted by sun = 1.4 kW / m^2 × 1 / 11881
Power emitted by sun = 1.176 × 10^10 watts
Now we can use the Stefan-Boltzmann law, which relates the power emitted by an object to its temperature:
Power emitted by sun = σ × Surface area of sun × Temperature^4
where σ is the Stefan-Boltzmann constant, equal to 5.67 × 10^-8 watt / m^2 / K^4. We can rearrange this equation to solve for the surface temperature of the sun:
Temperature = √(Power emitted by sun / (σ × Surface area of sun))
Temperature = √(1.176 × 10^10 watts / (5.67 × 10^-8 watt / m^2 / K^4 × 4π × (r_sun)^2))
Temperature = 5780 K
Therefore, the surface temperature of the sun is approximately 5780 K.
Surface area of sun = 4πr^2
where r is the radius of the sun. We can rearrange this to find r in terms of the surface area:
r = √(Surface area of sun / 4π)
The radius of the sun is about 109 times larger than the radius of the earth, so we can use the ratio of the areas of two spheres to relate the surface area of the sun to the power received by the earth:
Ratio of areas = (Radius of earth / Radius of sun)^2
Ratio of areas = (1 / 109)^2
Ratio of areas = 1 / 11881
Power received by earth = Power emitted by sun / Ratio of areas
1.4 kW / m^2 = Power emitted by sun / 1 / 11881
Power emitted by sun = 1.4 kW / m^2 × 1 / 11881
Power emitted by sun = 1.176 × 10^10 watts
Now we can use the Stefan-Boltzmann law, which relates the power emitted by an object to its temperature:
Power emitted by sun = σ × Surface area of sun × Temperature^4
where σ is the Stefan-Boltzmann constant, equal to 5.67 × 10^-8 watt / m^2 / K^4. We can rearrange this equation to solve for the surface temperature of the sun:
Temperature = √(Power emitted by sun / (σ × Surface area of sun))
Temperature = √(1.176 × 10^10 watts / (5.67 × 10^-8 watt / m^2 / K^4 × 4π × (r_sun)^2))
Temperature = 5780 K
Therefore, the surface temperature of the sun is approximately 5780 K.