A photon , an electron and a uranium nucleus all have the same wavelength . The one with the most energy

Options

A. is the photon
B. is the electron
C. is the uranium nucleus
D. depends upon the wavelength and the properties of the particle

Explanation

To determine which particle (photon, electron, or uranium nucleus) has the most energy when they all have the same wavelength, we can analyze the energy of each particle using their respective formulas.

### Step-by-Step Solution:

1. **Understanding Energy and Wavelength Relationship**:
- The energy of a photon is given by the formula:
\[ E = \frac{hc}{\lambda} \]
where \(E\) is the energy, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength.

2. **Photon Energy Calculation**:
- Since the photon is massless, its energy is solely dependent on its wavelength. For a photon with wavelength \(\lambda\), its energy is:
\[ E_{\text{photon}} = \frac{hc}{\lambda} \]

3. **Energy of an Electron**:
- The energy of a moving electron can be expressed in terms of its kinetic energy:
\[ E_{\text{electron}} = \frac{p^2}{2m} \]
where \(p\) is the momentum and \(m\) is the mass of the electron. The momentum \(p\) can also be related to wavelength by:
\[ p = \frac{h}{\lambda} \]
- Substituting \(p\) into the kinetic energy formula gives:
\[ E_{\text{electron}} = \frac{(h/\lambda)^2}{2m} = \frac{h^2}{2m\lambda^2} \]

4. **Energy of a Uranium Nucleus**:
- Similarly, for the uranium nucleus, we can use the same kinetic energy formula:
\[ E_{\text{uranium}} = \frac{p^2}{2M} \]
where \(M\) is the mass of the uranium nucleus. Using the same momentum relation:
\[ E_{\text{uranium}} = \frac{(h/\lambda)^2}{2M} = \frac{h^2}{2M\lambda^2} \]

5. **Comparing Energies**:
- Now we have:
- \(E_{\text{photon}} = \frac{hc}{\lambda}\)
- \(E_{\text{electron}} = \frac{h^2}{2m\lambda^2}\)
- \(E_{\text{uranium}} = \frac{h^2}{2M\lambda^2}\)

- Since \(\lambda\) is constant for all three, we can compare the energies based on their mass:
- The photon has no mass (effectively zero).
- The electron has a small mass \(m\).
- The uranium nucleus has a much larger mass \(M\).

6. **Conclusion**:
- The energy is inversely proportional to the mass when comparing the electron and uranium nucleus. Since the photon has no mass, it will have the highest energy among the three.
- Therefore, the particle with the most energy is the **photon**.