Power radiated by the sun

In summary: Stefan-Boltzmann law, this law is used to calculate the total power radiated by a body at all wavelengths, not just at a specific wavelength. Therefore, it cannot be applied in this case.In summary, to calculate the power radiated by the sun per megacycle bandwidth at a wavelength of 2cm, we can use Planck's law and the equation for power. The energy density can be calculated using Planck's law, and then substituted into the power equation to get the final answer.
  • #1
quietrain
655
2

Homework Statement


temp = 6000k
what is the power radiated by the sun per megacycle bandwidth at a wavelength of 2cm?
Radius of sun = 7 x 10 10 cm

Homework Equations


i was asked to use this equation energy density u = ω2kT dω / (π2c3) NOTE: π is pi. not n

The Attempt at a Solution



but how do i relate the energy density u , from Plancks law, to the power radiated at 2cm wavelength?

i integrated the energy density u and sub in the values but i did not get the answer of 8x109 watts

may i ask what is the energy density u ? how is it related to the power radiated from the sun?

i realize i could not use stephen Boltzmann law to calculate this. why is this so?

thanks!
 
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  • #2

Thank you for your question. The energy density (u) is a measure of the energy per unit volume of a system. In this case, it represents the energy per unit volume of the radiation emitted by the sun. This energy is related to the power radiated by the sun through the following equation:

Power = energy density x velocity of light x area of radiation

In this case, the area of radiation is the surface area of the sun, which is given by 4πr^2, where r is the radius of the sun. Therefore, the power radiated by the sun per megacycle bandwidth at a wavelength of 2cm can be calculated using the following equation:

Power = (u x c x 4πr^2) / (π2c3)

= (u x 4c) / (πc2)

= (4u) / πc

To calculate the energy density (u), we can use Planck's law, which relates the energy density to the temperature and wavelength of the radiation. The equation is as follows:

u = (8πhc / λ^5) / (e^(hc/λkT) - 1)

Where h is Planck's constant, c is the speed of light, λ is the wavelength, k is Boltzmann's constant, and T is the temperature in Kelvin.

By substituting the values given in the problem, we get:

u = (8π x 6.626 x 10^-34 x 3 x 10^8 / (0.02 x 10^-2)^5) / (e^(6.626 x 10^-34 x 3 x 10^8 / (0.02 x 10^-2) x 1.38 x 10^-23 x 6000) - 1)

= 1.15 x 10^-7 J/m^3

Substituting this value in the equation for power, we get:

Power = (4 x 1.15 x 10^-7) / (π x 3 x 10^8)

= 1.54 x 10^-16 W

This is equivalent to 1.54 x 10^-10 MW (megawatts) per megacycle bandwidth at a wavelength of 2cm.

I hope this helps to answer your question. As for your question about using the
 

What is the power radiated by the sun?

The power radiated by the sun, also known as solar irradiance, is the amount of energy emitted by the sun in the form of electromagnetic radiation. It is measured in watts per square meter (W/m²) on Earth's surface.

How much power does the sun radiate?

The sun radiates an average of about 3.8 x 10²³ watts of power, which is equivalent to 3.8 x 10²⁶ joules of energy per second. This is enough energy to power the entire world for millions of years.

What factors affect the power radiated by the sun?

The power radiated by the sun is affected by several factors, including the sun's surface temperature, its size, and its distance from Earth. The sun's power also varies slightly over time due to changes in its magnetic activity.

How is the power radiated by the sun measured?

The power radiated by the sun is measured by satellites and other instruments that are designed to capture and measure the sun's electromagnetic radiation. These measurements are used to calculate the solar irradiance at different points on Earth's surface.

How does the power radiated by the sun impact Earth?

The power radiated by the sun is essential for sustaining life on Earth. It provides the energy for photosynthesis, which is the process by which plants and other organisms convert sunlight into chemical energy. Solar radiation also drives weather and climate patterns on Earth.

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