The sun is 1.5 1011 m from the earth.
(a) Over what surface area is solar radiation spread at this distance?
(b) What is the total power radiated by the sun? (The sun delivers energy to the surface of the earth at a rate of approximately 1400 W/m2.)
(c) Treating the sun as a body of emissivity 1, estimate its temperature. The sun's radius is 6.96 108 m.
K
a. A = 4•π•r^2 = 4•π•(1.5•10^11)^2 = =9•π•10^22 m^2.
b. Power density is p =1400 W/m^2.
The total power is
P = p•A =1400 • 9•π•10^22 =
=3.96•10^26 W.
c. Stefan-Boltzmann Law
R = σ•T^4,
R =P/A1 = P/(4• π•r1^2), is the black body irradiance,
σ =5.67• 10^-8 W•m^-2•K^-4 is the Stefan-Boltzmann constant,
T =[P/(A1• σ)]^1/4 ={3.96•10^26/[4• π•(6.96• 10^8)^2]}^1/4 = 5800 K.
To answer these questions, we can use some formulas and known values. Let's go step by step:
(a) Over what surface area is solar radiation spread at this distance?
To calculate the surface area over which solar radiation is spread, we can use the formula for the surface area of a sphere:
Surface Area = 4πr^2
Given that the distance between the sun and the earth is 1.5 x 10^11 m, we can substitute this value into the formula:
Surface Area = 4π(1.5 x 10^11)^2
Calculating that gives us the surface area over which the solar radiation is spread.
(b) What is the total power radiated by the sun?
To calculate the total power radiated by the sun, we need to know the power per unit area (which is given as 1400 W/m^2) and the surface area calculated in the previous step.
Total Power = Power per unit area x Surface Area
Substituting the values gives us the total power radiated by the sun.
(c) Treating the sun as a body of emissivity 1, estimate its temperature.
To estimate the temperature of the sun, we can use the Stefan-Boltzmann law, which relates the power radiated by a blackbody (like the sun) to its temperature:
Power = σ * A * T^4
Here, σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/(m^2*K^4)), A is the surface area of the sun, and T is the temperature of the sun.
Using the total power calculated in the previous step, we can rearrange the formula to solve for T:
T = (Power / (σ * A))^ (1/4)
Substituting the values will give us an estimate of the sun's temperature.
(a) To find the surface area over which solar radiation is spread at a distance of 1.5 * 10^11 m, we can use the formula for the surface area of a sphere.
The surface area of a sphere is given by the formula:
A = 4πr^2
where A is the surface area and r is the radius of the sphere.
Given that the sun is 1.5 * 10^11 m from the earth, the radius of the sphere (distance from the center of the sun to its surface) is equal to the distance from the earth to the sun:
r = 1.5 * 10^11 m
Substituting this value into the formula, we get:
A = 4π(1.5 * 10^11)^2
Calculating this, we find:
A ≈ 2.82743339 * 10^23 m^2
Therefore, the surface area over which solar radiation is spread at this distance is approximately 2.82743339 * 10^23 square meters.
(b) To find the total power radiated by the sun, we can multiply the power delivered to the surface of the earth by the surface area.
Given that the sun delivers energy to the surface of the earth at a rate of approximately 1400 W/m^2, we can multiply this by the surface area calculated in part (a):
Total power radiated = Power per unit area * Surface area
= 1400 W/m^2 * 2.82743339 * 10^23 m^2
Calculating this, we get:
Total power radiated ≈ 3.95640875 * 10^26 W
Therefore, the total power radiated by the sun is approximately 3.95640875 * 10^26 watts.
(c) To estimate the temperature of the sun, we can use the Stefan-Boltzmann law, which relates the total power radiated by a black body to its temperature.
The Stefan-Boltzmann law is given by the equation:
P = σAεT^4
where P is the total power radiated, σ is the Stefan-Boltzmann constant (approximately 5.67 * 10^-8 W/m^2K^4), A is the surface area of the black body, ε is the emissivity (assumed to be 1 for the sun), and T is the temperature in Kelvin.
We can rearrange this equation to solve for temperature:
T^4 = P / (σAε)
Substituting the values we found in parts (a) and (b) into the equation, we get:
T^4 ≈ (3.95640875 * 10^26 W) / (5.67 * 10^-8 W/m^2K^4 * 2.82743339 * 10^23 m^2 * 1)
Calculating this, we find:
T^4 ≈ 5.96803 * 10^15 K^4
Taking the fourth root of both sides, we get:
T ≈ 1.666529475 * 10^12 K
Therefore, the estimated temperature of the sun is approximately 1.666529475 * 10^12 Kelvin.