The energy of a photon is a fundamental concept in quantum physics, and understanding how to calculate it is crucial for many applications, from spectroscopy to solar energy. In this comprehensive guide, we will delve into the details of calculating the energy of a photon, providing you with the necessary formulas, examples, and numerical problems to master this essential skill.
Understanding Photon Energy
A photon is a fundamental particle of light, and its energy is directly related to its frequency or wavelength. The energy of a photon is quantized, meaning it can only take on specific, discrete values. This quantization is a key feature of quantum mechanics and is described by Planck’s constant, a fundamental physical constant.
Calculating Photon Energy Using Frequency
When the frequency of a photon is known, you can use the following formula to calculate its energy:
E = h × f
Where:
– E is the energy of the photon (in Joules)
– h is Planck’s constant (6.626 × 10^-34 J·s)
– f is the frequency of the photon (in Hertz)
Example 1: Calculating Photon Energy from Frequency
Suppose the frequency of a photon is 4.5 × 10^14 Hz. To calculate its energy, we can plug the values into the formula:
E = (6.626 × 10^-34 J·s) × (4.5 × 10^14 Hz)
E = 2.98 × 10^-19 J
Numerical Problem 1
A photon has a frequency of 6.0 × 10^14 Hz. Calculate the energy of this photon.
Calculating Photon Energy Using Wavelength
If the wavelength of a photon is known, you can use the following formula to calculate its energy:
E = h × c / λ
Where:
– E is the energy of the photon (in Joules)
– h is Planck’s constant (6.626 × 10^-34 J·s)
– c is the speed of light in a vacuum (3.0 × 10^8 m/s)
– λ is the wavelength of the photon (in meters)
Example 2: Calculating Photon Energy from Wavelength
Suppose the wavelength of a photon is 4.8 × 10^-7 m. To calculate its energy, we can plug the values into the formula:
E = (6.626 × 10^-34 J·s) × (3.0 × 10^8 m/s) / (4.8 × 10^-7 m)
E = 4.14 × 10^-19 J
Numerical Problem 2
A photon has a wavelength of 5.0 × 10^-7 m. Calculate the energy of this photon.
Relationship between Photon Energy, Frequency, and Wavelength
The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength. This means that higher-frequency light (such as violet light) has more energy than lower-frequency light (such as red light), and that shorter-wavelength light (such as blue light) has more energy than longer-wavelength light (such as red light).
This relationship can be expressed mathematically as:
E = h × f = h × c / λ
Where:
– E is the energy of the photon (in Joules)
– h is Planck’s constant (6.626 × 10^-34 J·s)
– f is the frequency of the photon (in Hertz)
– c is the speed of light in a vacuum (3.0 × 10^8 m/s)
– λ is the wavelength of the photon (in meters)
Example 3: Comparing Photon Energies
Consider two photons, one with a frequency of 5.0 × 10^14 Hz and the other with a wavelength of 4.0 × 10^-7 m. Calculate the energy of each photon and compare them.
For the photon with a frequency of 5.0 × 10^14 Hz:
E = (6.626 × 10^-34 J·s) × (5.0 × 10^14 Hz)
E = 3.31 × 10^-19 J
For the photon with a wavelength of 4.0 × 10^-7 m:
E = (6.626 × 10^-34 J·s) × (3.0 × 10^8 m/s) / (4.0 × 10^-7 m)
E = 4.97 × 10^-19 J
Comparing the two, the photon with the shorter wavelength (4.0 × 10^-7 m) has a higher energy than the photon with the higher frequency (5.0 × 10^14 Hz).
Numerical Problem 3
A photon has a wavelength of 6.0 × 10^-7 m. Calculate the energy of this photon and compare it to the energy of a photon with a frequency of 3.0 × 10^14 Hz.
Measuring Photon Energy in Electronvolts (eV)
In addition to the Joule (J), the energy of a photon is often measured in electronvolts (eV), which is a more convenient unit for measuring the energy of individual photons. One electronvolt is equal to 1.602 × 10^-19 J, which is roughly the energy of an electron moving across a potential difference of 1 volt.
To convert the energy of a photon from Joules to electronvolts, you can use the following formula:
E (eV) = E (J) / (1.602 × 10^-19 J/eV)
Example 4: Converting Photon Energy to Electronvolts
Consider the photon with a frequency of 4.5 × 10^14 Hz, which we calculated to have an energy of 2.98 × 10^-19 J. To convert this to electronvolts, we can use the formula:
E (eV) = (2.98 × 10^-19 J) / (1.602 × 10^-19 J/eV)
E (eV) = 1.86 eV
Numerical Problem 4
A photon has an energy of 3.5 × 10^-19 J. Convert this energy to electronvolts (eV).
Conclusion
In this comprehensive guide, we have explored the various methods and formulas for calculating the energy of a photon, including using the photon’s frequency or wavelength. We have also discussed the relationship between photon energy, frequency, and wavelength, as well as the conversion between Joules and electronvolts (eV).
By mastering these concepts and techniques, you will be well-equipped to tackle a wide range of problems and applications in quantum physics, spectroscopy, and other related fields. Remember to practice the numerical problems provided to solidify your understanding and develop your problem-solving skills.
References
- How to Calculate the Quantum Energy of Light | Physics – Study.com
- How To Calculate The Energy of a Photon Given … – YouTube
- How to Calculate the Energy of a Photon | Physics – Study.com
- Photon Energy (video) – Khan Academy
- How to calculate the energy of a photon? – BYJU’S
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