Question

The solar constant measured by Earth satellites is roughly \(1400 . W / m^{2}\) Though the Sun emits light of different wavelengths, the peak of the wavelength spectrum is at \(500, \mathrm{nm}\) a) Find the corresponding photon frequency. b) Find the corresponding photon energy. c) Find the number flux of photons (number of photons per unit area per unit time) arriving at Earth, assuming that all light emitted by the Sun has the same peak wavelength.

Short answer

b) What is the corresponding photon energy for the Sun's emitted light with a peak wavelength of 500 nm? c) What is the number flux of photons arriving at Earth for the Sun's emitted light with a peak wavelength of 500 nm and a solar constant of 1400 W/m^2?

Step-by-step solution

Calculate the photon frequency

Using the speed of light formula, we can find the photon frequency. Given the speed of light c = 3.00 × 10^8 m/s and the wavelength λ = 500 nm (converted to meters by multiplying it by 10^-9), we can find the frequency using the formula: \(c = \lambda \times \nu\). \(3.00 \times 10^8 m/s = (500 \times 10^{-9} m) \times \nu\) Now, solve for \(\nu\): \(\nu = \frac{3.00 \times 10^8 m/s}{500 \times 10^{-9} m} = 6.00 \times 10^{14} Hz\)

Calculate the photon energy

Now that we have the photon frequency, we can find the photon energy using the energy formula: \(E = h \times \nu\), where \(h\) is the Planck constant (\(6.63 × 10^{-34} Js\)). \(E = (6.63 \times 10^{-34} Js)(6.00 \times 10^{14} Hz) = 3.98 \times 10^{-19} J\)

Calculate the number flux of photons

Finally, we can find the number flux of photons using the photon flux formula: \(I = N \times E\). We are given the solar constant \(I = 1400 W/m^2\) and the photon energy \(E = 3.98 \times 10^{-19} J\). Now solve for \(N\): \(1400 W/m^2 = N \times 3.98 \times 10^{-19} J\) \(N = \frac{1400 W/m^2}{3.98 \times 10^{-19} J} = 3.52 \times 10^{21} photons/m^2s\) So the answers are: a) The corresponding photon frequency is \(6.00 \times 10^{14} Hz\). b) The corresponding photon energy is \(3.98 \times 10^{-19} J\). c) The number flux of photons arriving at Earth is \(3.52 \times 10^{21} photons/m^2s\).

Key concepts

Photon Frequency

When we discuss light in terms of physics, it's essential to realize that it behaves both as a wave and as a particle, introducing the concept of photons. Each photon, the fundamental particle of light, carries a certain frequency, denoted as \( u \), which is directly related to its wavelength \( \lambda \). The frequency of a photon can be calculated using the relationship \( c = \lambda \times u \), where \( c \) is the speed of light (approximately \( 3.00 \times 10^8 m/s \)).

The step-by-step solution shows that for a photon with a wavelength of 500 nm, which corresponds to the peak emission from the sun, its frequency can be found by rearranging the formula to \( u = \frac{c}{\lambda} \). This gives us a photon frequency of \( 6.00 \times 10^{14} Hz \). Understanding photon frequency is crucial because it determines the energy of the photon and, subsequently, the type of electromagnetic radiation, from radio waves to gamma rays.

Photon Energy

Delving into the quantum realm, energy is a central topic, especially when discussing photons, the quantum units of light. The energy \( E \) of a photon is tied to its frequency \( u \) through Planck's equation, which is \( E = h \times u \), where \( h \) is the Planck constant \( (6.63 \times 10^{-34} Js) \).

According to the exercise, once the frequency of the photon is determined, applying Planck's equation reveals the energy carried by a single photon at the peak wavelength of sunlight to be \( 3.98 \times 10^{-19} J \). This photon energy is significant in many applications, such as calculating the number flux of photons or understanding the potential chemical reactions photons might induce when interacting with materials, as in solar panels or photosynthesis.

Number Flux of Photons

In terms of quantifying the flow of photons, the concept of number flux comes into play. The number flux of photons \( N \) is the count of photons passing through a unit area per unit time. This measure is crucial for applications ranging from solar energy to radiative heat transfer.

The number flux is computed using the solar constant, which is the power per unit area received from the Sun, in association with the photon energy. The formula \( I = N \times E \) is used for this purpose, where \( I \) denotes the solar constant and \( E \) the energy per photon. For the sun's peak emission wavelength, the calculation gives a number flux of \( 3.52 \times 10^{21} photons/m^2s \), as shown in the solution. This understanding helps in imagining how many tiny packets of energy hit a given area in a second, which is essential for evaluating the efficiency of solar panels or the intensity of sunlight exposure.