a) Total power output: [tex]3.845\cdot 10^{26} W[/tex]
b) The relative percentage change of power output is 1.67%
c) The intensity of the radiation on Mars is [tex]540 W/m^2[/tex]
Explanation:
a)
The intensity of electromagnetic radiation is given by
[tex]I=\frac{P}{A}[/tex]
where
P is the power output
A is the surface area considered
In this problem, we have
[tex]I=1360 W/m^2[/tex] is the intensity of the solar radiation at the Earth
The area to be considered is area of a sphere of radius
[tex]r=1.5\cdot 10^{11} m[/tex] (distance Earth-Sun)
Therefore
[tex]A=4\pi r^2 = 4 \pi (1.5\cdot 10^{11})^2=2.8\cdot 10^{23}m^2[/tex]
And now, using the first equation, we can find the total power output of the Sun:
[tex]P=IA=(1360)(2.8\cdot 10^{23})=3.845\cdot 10^{26} W[/tex]
b)
The energy of the solar radiation is directly proportional to its frequency, given the relationship
[tex]E=hf[/tex]
where E is the energy, h is the Planck's constant, f is the frequency.
Also, the power output of the Sun is directly proportional to the energy,
[tex]P=\frac{E}{t}[/tex]
where t is the time.
This means that the power output is proportional to the frequency:
[tex]P\propto f[/tex]
Here the frequency increases by 1 MHz: the original frequency was
[tex]f_0 = 60 MHz[/tex]
so the relative percentage change in frequency is
[tex]\frac{\Delta f}{f_0}\cdot 100 = \frac{1}{60}\cdot 100 =1.67\%[/tex]
And therefore, the power also increases by 1.67 %.
c)
In this second case, we have to calculate the new power output of the Sun:
[tex]P' = P + \frac{1.67}{100}P =1.167P=1.0167(3.845\cdot 10^{26})=3.910\cdot 10^{26} W[/tex]
Now we want to calculate the intensity of the radiation measured on Mars. Mars is 60% farther from the Sun than the Earth, so its distance from the Sun is
[tex]r'=(1+0.60)r=1.60r=1.60(1.5\cdot 10^{11})=2.4\cdot 10^{11}m[/tex]
Now we can find the radiation intensity with the equation
[tex]I=\frac{P}{A}[/tex]
Where the area is
[tex]A=4\pi r'^2 = 4\pi(2.4\cdot 10^{11})^2=7.24\cdot 10^{23} m^2[/tex]
And substituting,
[tex]I=\frac{3.910\cdot 10^{26}}{7.24\cdot 10^{23}}=540 W/m^2[/tex]
Learn more about electromagnetic radiation:
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