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.

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.

To calculate the surface temperature of the sun, we can use the Stefan-Boltzmann Law, which relates the power emitted by a black body to its temperature.

The power emitted per unit area by a black body is given by the formula P = σT^4, where P is the power in watts per square meter (W/m^2), σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m^2K^4), and T is the temperature in Kelvin (K).

We are given that the quantity of radiation received by the Earth from the sun is 1.4 kW/m^2 (or 1.4 x 10^3 W/m^2). This represents the power emitted by the sun per unit area at the distance of the Earth.

Let's denote the surface temperature of the sun by Ts.

Using the ratio of the radius of Earth's orbit to the radius of the sun (216), we can calculate the power emitted by the sun (Ps):

Ps = P * (4π * R^2) = (1.4 x 10^3) * (4π * R^2)

Where R is the radius of the sun and can be calculated as R = 216 * RE, where RE is the radius of Earth's orbit.

Now we can equate the power emitted by the sun (Ps) to the power emitted per unit area by a black body (P = σT^4):

(1.4 x 10^3) * (4π * R^2) = σ * Ts^4

Simplifying the equation:

Ts^4 = [(1.4 x 10^3) * (4π * R^2)] / σ

Now we solve for Ts by taking the fourth root on both sides of the equation:

Ts = [(1.4 x 10^3) * (4π * R^2) / σ]^(1/4)

Substituting the values of R and σ, we can calculate Ts.