The sun radiates like a perfect blackbody with an emissivity of exactly 1.
(a) Calculate the surface temperature(K) of the sun, given it is a sphere with a 7.00 multiplied by 108 m radius that radiates 3.80 multiplied by 1026 W into 3 K space.
(b) How much power(W/m^2)does the sun radiate per square meter of its surface?
(c) How much power(W/m^2)in watts per square meter is this at the distance of the earth, 1.50 multiplied by 1011 m away? (This number is called the solar constant.)
(a)
Stefan-Boltzmann Law
R=σT⁴
R=P/A =P/4πR²
σT⁴=P/4πR²
T =forthroot{ P/4σπR²} =
=forthroot{3.8•10² /4•5.67•10⁻⁸•π•(7•10⁸)²} =5744 K
(b)
P₀= P/4πR²=3.8•10²⁶/4•π •(7•10⁸)² =6.17•10⁷ W/m²
(c)
P₁= P/4πR₀²=3.8•10²⁶/4•π •(1.5•10¹¹)² =1344 W/m²
To calculate the surface temperature (K) of the sun, we can use the Stefan-Boltzmann law, which states that the power radiated by a blackbody is directly proportional to the fourth power of its temperature.
(a) The formula to calculate the surface temperature is:
Power = 4πr^2σT^4
Where:
Power is the power radiated by the Sun,
r is the radius of the Sun (7.00 × 10^8 m),
σ is the Stefan-Boltzmann constant (5.67 × 10^(-8) W/(m^2·K^4)),
T is the surface temperature of the Sun in Kelvin.
Here, we know the power radiated by the Sun is 3.80 × 10^26 W. So, we can rearrange the formula to solve for T:
T^4 = Power / (4πr^2σ)
T^4 = (3.80 × 10^26 W) / (4π(7.00 × 10^8 m)^2 × (5.67 × 10^(-8) W/(m^2·K^4)))
T^4 ≈ 3.17 × 10^7 K^4
Taking the fourth root of both sides, we get:
T ≈ 5780 K
Therefore, the surface temperature of the Sun is approximately 5780 Kelvin.
(b) To calculate the power radiated per square meter of the Sun's surface, we divide the total power radiated by the Sun by its surface area:
Power per square meter = Power / Surface Area
Surface Area = 4πr^2
Surface Area = 4π(7.00 × 10^8 m)^2
Power per square meter = (3.80 × 10^26 W) / (4π(7.00 × 10^8 m)^2)
Power per square meter ≈ 6.32 × 10^7 W/m^2
Therefore, the Sun radiates approximately 6.32 × 10^7 Watts per square meter of its surface.
(c) The solar constant represents the amount of power per square meter received at the distance of the Earth from the Sun.
To calculate the power per square meter at the distance of the Earth, we need to account for the spreading of the radiation over larger areas. The power spreads out over a spherical surface area, so we divide the total power radiated by the Sun by the surface area of a sphere with a radius equal to the distance between the Earth and the Sun.
Power per square meter at Earth's distance = Power / Surface Area of a Sphere
Surface Area of a Sphere = 4π(distance)^2
Distance = 1.50 × 10^11 m (distance of the Earth from the Sun)
Power per square meter at Earth's distance = (3.80 × 10^26 W) / (4π(1.50 × 10^11 m)^2)
Power per square meter at Earth's distance ≈ 1.37 × 10^3 W/m^2
Therefore, at the distance of the Earth, the Sun radiates approximately 1.37 × 10^3 Watts per square meter (which is known as the solar constant).