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•10²⁶/4•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

(a) To calculate the surface temperature of the sun, we can use the Stefan-Boltzmann law, which states that the power radiated by a perfect blackbody is given by the equation:

P = σ * A * T^4

where P is the power radiated, A is the surface area of the object, T is the temperature of the object in Kelvin, and σ is the Stefan-Boltzmann constant.

Given that the sun radiates a power of 3.80x10^26 W and has a radius of 7.00x10^8 m, we can calculate the surface area A of the sun as follows:

A = 4 * π * r^2
A = 4 * π * (7.00x10^8)^2

Using the value of the Stefan-Boltzmann constant σ = 5.67x10^-8 W/(m^2*K^4), we can rearrange the equation to solve for the surface temperature T:

T = (P / (σ * A))^0.25

Substituting the given values:

T = (3.80x10^26 W / (5.67x10^-8 W/(m^2*K^4) * 4 * π * (7.00x10^8)^2))^0.25

Calculating this equation will give us the surface temperature of the sun in Kelvin.

(b) The power radiated per square meter of the sun's surface can be calculated by dividing the total power radiated by the surface area of the sun:

Power per square meter = P / A

Given that P = 3.80x10^26 W and A is already calculated in part (a), we can substitute these values into the equation to find the power radiated per square meter.

(c) To find out how much power is received on Earth at a distance of 1.50x10^11 m away, we can use the inverse square law, which states that the power per unit area decreases with the square of the distance.

Power at Earth = Power per square meter / (4 * π * r^2)

Where r is the distance from the sun to Earth. Substituting the values, we can calculate the power received on Earth in watts per square meter.

To calculate the surface temperature of the sun (part a), we can use the Stefan-Boltzmann law, which relates the power radiated by a blackbody to its surface temperature. The Stefan-Boltzmann law is given by:

P = σ * A * ε * T^4

Where:
P is the power radiated (3.80 * 10^26 W),
σ is the Stefan-Boltzmann constant (5.67 * 10^-8 W/m^2K^4),
A is the surface area of the sun (4πr^2, where r is the radius of the sun),
ε is the emissivity (1) since the sun is a perfect blackbody, and
T is the surface temperature we want to find.

(a) Calculating the surface temperature (T) of the sun:
First, we need to calculate the surface area of the sun:

A = 4π * (7.00 * 10^8 m)^2

Then, rearranging the Stefan-Boltzmann law formula, we can solve for T:

T = (P / (σ * A * ε))^0.25

Substituting the given values:

T = (3.80 * 10^26 W / (5.67 * 10^-8 W/m^2K^4 * 4π * (7.00 * 10^8 m)^2 * 1))^0.25

Calculating this expression will give us the surface temperature (T) of the sun in Kelvin.

(b) To calculate the power radiated per square meter of the sun's surface, we divide the total power radiated by the sun (3.80 * 10^26 W) by its surface area (4πr^2).

Power per square meter (P') = P / A

Substituting the given values:

P' = (3.80 * 10^26 W) / (4π * (7.00 * 10^8 m)^2)

Calculating this expression will give us the power radiated per square meter of the sun's surface.

(c) The solar constant represents the power received per square meter at the distance of the earth (1.50 * 10^11 m) from the sun. To calculate it, we use the inverse square law, which states that the intensity (power per unit area) decreases with the square of the distance:

Pearth = Psun / (4πd^2)

Where:
Pearth is the power received per square meter at the distance of the earth,
Psun is the total power radiated by the sun (3.80 * 10^26 W),
d is the distance from the sun to the earth (1.50 * 10^11 m).

Substituting the given values:

Pearth = (3.80 * 10^26 W) / (4π * (1.50 * 10^11 m)^2)

Calculating this expression will give us the power per square meter received at the distance of the earth from the sun, which is known as the solar constant.