How bright would the sun appear to be to an observer on Earth if the Sun were four times farther from the Earth than it actually is?
Please express answer as a fraction of the Sun's brightness on Earth's surface, keeping in mind the three following objectives:
1. Identify the components of the electromagnetic spectrum
2. Calculate the frequency or wavelength of electromagnetic radiation;
Recognize that light has a finite speed.
3. Describe how the brightness of a light source is affected by distance.
The light is emitted in all directions of 3 dimensional space with the same intensity by the Sun. By considering the Sun as a single emitting point in space we can say that the wave front of the emitted light will be a sphere centered on the Sun. The product between the intensity of the brightness `I` and the surface of the sphere that represent the wave front `S =4*pi*R^2` is constant. (This is equivalent to the fact that the luminous flux is constant through any closed surface in space). Therefore
`I*S = C rArr I(R) =C/(4*pi*R^2)`
`I(4R) = C/(4*pi*16R^2)`
Answer: If the Earth would be 4 times further from the Sun the brightness intensity will be 16 times smaller than actually it is.
Let the distance between the sun and the earth be x.
Let the light energy received by the earth rom the sun be L.
If the distance of the sun from earth becomes four times, then obviously the time taken by the light wave to reach earth increases four times.
We know that frequency(v)=1/T.
and speed of light(c)=v*wavelength.
c is constant and is equal to 299792458 m/s.
So, frequency is inversely proportional to the wavelength.
So, if the time increases four times, frequency(v) decreases 4 times and the wavelength simultaneously increases four times.
Hence, the wavelength of the light would become four times the wavelength of the visible part of the electromagnetic spectrum i.e 4 times of 4000 to 8000 Angstorm. Now, the wavelength would be in the range of 16000 to 32000 Angstorm, which is not perceptible by the naked human eye.
Hence, the earth would receive the waves which are not optically perceptible by the humans.
Hope it helps.