The energy of a photon, whose momentum is 10 MeV/c, where c is the speed of light, is- given by
10 MeV
Calculating Photon Energy from Momentum
Understanding the relationship between energy and momentum is fundamental in physics, especially when dealing with particles like photons that travel at the speed of light. For massless particles such as photons, the energy (\(E\)) and momentum (\(p\)) are directly related by the speed of light (\(c\)).
What is a Photon?
A photon is an elementary particle, the quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves. Photons are massless and always move at the speed of light in vacuum.
Energy and Momentum of a Photon
For any particle, the relativistic energy (\(E\)) and momentum (\(p\)) are related by the equation:
\(E^2 = (pc)^2 + (m_0c^2)^2\)
where \(m_0\) is the rest mass of the particle and \(c\) is the speed of light. For a photon, the rest mass \(m_0\) is zero (\(m_0 = 0\)). Substituting \(m_0 = 0\) into the equation, we get:
\(E^2 = (pc)^2 + (0 \cdot c^2)^2\)
\(E^2 = (pc)^2\)
Taking the square root of both sides, we find the relationship for a photon:
\(E = pc\)
This simple equation tells us that the energy of a photon is directly proportional to its momentum, with the speed of light being the constant of proportionality.
Step-by-Step Calculation
We are given the momentum of the photon:
\(p = 10 \text{ MeV/c}\)
We need to find the energy (\(E\)) of the photon. Using the formula \(E = pc\):
\(E = (10 \text{ MeV/c}) \times c\)
Notice that the unit 'c' in the denominator of the momentum cancels out with the speed of light 'c' we multiply by:
\(E = 10 \text{ MeV} \cdot \frac{c}{c}\)
\(E = 10 \text{ MeV} \cdot 1\)
\(E = 10 \text{ MeV}\)
Thus, the energy of the photon with a momentum of 10 MeV/c is 10 MeV.
Summary of Calculation
| Quantity | Value | Units |
|---|---|---|
| Momentum (\(p\)) | 10 | MeV/c |
| Speed of light (\(c\)) | c | - |
| Energy (\(E = pc\)) | \(10 \times c\) | (MeV/c) \(\times\) c = MeV |
| Calculated Energy (\(E\)) | 10 | MeV |
The calculation confirms that the energy of the photon is equal to the numerical value of its momentum when the momentum is expressed in units of Energy/c (like MeV/c) and then multiplied by c.
Revision Table: Key Photon Concepts
| Concept | Description | Formula (for Photon) |
|---|---|---|
| Photon | Massless elementary particle; quantum of light/EM field | \(m_0 = 0\) |
| Energy (\(E\)) | Capacity to do work; associated with the frequency of light (\(E=h\nu\)) and momentum | \(E = pc\) |
| Momentum (\(p\)) | Measure of mass in motion; associated with wavelength (\(p=h/\lambda\)) and energy | \(p = E/c\) |
| Speed of Light (\(c\)) | Constant speed of all massless particles in vacuum | \(\approx 3 \times 10^8\) m/s |
Additional Information: Energy and Momentum in Physics
The relationship \(E=pc\) for photons is a special case derived from Einstein's relativistic energy-momentum relation. This relation is crucial in high-energy physics and describes how energy and momentum are connected for particles traveling at relativistic speeds, particularly for massless particles where rest energy (\(m_0c^2\)) is zero.
Another important formula relating photon energy to its frequency (\(\nu\)) is \(E=h\nu\), where \(h\) is Planck's constant. Similarly, the momentum of a photon is related to its wavelength (\(\lambda\)) by \(p=h/\lambda\). Combining these two relations with the wave speed equation \(c = \lambda\nu\), we can also derive \(E=pc\):
- From \(E=h\nu\) and \(p=h/\lambda\), we have \(\nu = E/h\) and \(\lambda = h/p\).
- Substitute these into \(c = \lambda\nu\): \(c = (h/p) \times (E/h)\).
- \(c = E/p\).
- Rearranging gives \(E = pc\).
This shows the consistency between the particle nature (energy, momentum) and wave nature (frequency, wavelength) of light, connected through Planck's constant and the speed of light.
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