A photon of green light has a wavelength of 520 nm. Find the photon’s frequency, magnitude of momentum, and energy. Express the energy in both joules and electron volts.
The Correct Answer and Explanation is:
To find the frequency, momentum, and energy of a photon of green light with a wavelength of 520 nm, we use the following fundamental physics equations:
Given:
- Wavelength (λ) = 520 nm = 520×10−9520 \times 10^{-9} m
- Speed of light (c) = 3.00×1083.00 \times 10^8 m/s
- Planck’s constant (h) = 6.626×10−346.626 \times 10^{-34} J·s
- 1 eV = 1.602×10−191.602 \times 10^{-19} J
Step 1: Frequency (f)
Use the wave equation: f=cλ=3.00×108520×10−9=5.77×1014 Hzf = \frac{c}{\lambda} = \frac{3.00 \times 10^8}{520 \times 10^{-9}} = 5.77 \times 10^{14} \text{ Hz}
Step 2: Momentum (p)
Photon momentum is given by: p=hλ=6.626×10−34520×10−9=1.275×10−27 kg\cdotpm/sp = \frac{h}{\lambda} = \frac{6.626 \times 10^{-34}}{520 \times 10^{-9}} = 1.275 \times 10^{-27} \text{ kg·m/s}
Step 3: Energy (E)
Energy of a photon: E=hf=(6.626×10−34)(5.77×1014)=3.82×10−19 JE = hf = (6.626 \times 10^{-34})(5.77 \times 10^{14}) = 3.82 \times 10^{-19} \text{ J}
Convert to electron volts: E=3.82×10−191.602×10−19≈2.38 eVE = \frac{3.82 \times 10^{-19}}{1.602 \times 10^{-19}} \approx 2.38 \text{ eV}
Final Answers:
- Frequency (f): 5.77×10145.77 \times 10^{14} Hz
- Momentum (p): 1.275×10−271.275 \times 10^{-27} kg·m/s
- Energy (E): 3.82×10−193.82 \times 10^{-19} J or 2.38 eV
Explanation
A photon is a quantum particle of light that carries energy and momentum but has no rest mass. The characteristics of a photon, such as its frequency, momentum, and energy, are directly related to its wavelength. For green light with a wavelength of 520 nm (nanometers), these properties can be determined using fundamental constants in physics.
First, the frequency of a photon is found using the wave equation f=c/λf = c/\lambda, where cc is the speed of light and λ\lambda is the wavelength. A shorter wavelength means a higher frequency, which corresponds to more energy. For 520 nm green light, this gives a frequency of about 5.77×10145.77 \times 10^{14} Hz.
Next, we calculate the momentum of the photon. Unlike objects with mass, photon momentum is given by p=h/λp = h/\lambda, where hh is Planck’s constant. Though tiny, the photon’s momentum is crucial in fields like quantum optics and even solar sail propulsion.
Finally, the photon’s energy is derived from the Planck-Einstein relation, E=hfE = hf, showing that energy is directly proportional to frequency. This energy is typically expressed in joules but is often converted into electron volts (eV) for atomic and quantum scales. For 520 nm light, the energy is approximately 3.82×10−193.82 \times 10^{-19} joules or 2.38 eV.
Understanding these values is essential in disciplines like quantum physics, chemistry (e.g., photoelectric effect), and optical engineering. They illustrate how light interacts with matter, enabling technologies such as lasers, LEDs, and solar panels.
