The wavelengths of a photon, an electron and a uranium nucleus are same. Maximum energy will be of

Options

A. photon
B. electron
C. it will depend on their properties and wavelength
D. uranium nucleus

Answer

To solve the problem of determining which particle (photon, electron, or uranium nucleus) has the maximum energy when they all have the same wavelength, we can use the de Broglie wavelength formula and the relationship between energy and mass.

### Step-by-Step Solution:

1. **Understand the de Broglie Wavelength Formula**:
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle. For a non-relativistic particle, momentum \( p \) can be expressed as:
\[ p = mv \]
where \( m \) is the mass and \( v \) is the velocity of the particle.

2. **Relate Wavelength to Energy**:
The energy \( E \) of a particle can be related to its momentum using the formula:
\[ E = \frac{p^2}{2m} \]
Substituting \( p = mv \) into this equation gives:
\[ E = \frac{(mv)^2}{2m} = \frac{mv^2}{2} \]
However, for our purposes, we will use the relationship derived from the de Broglie wavelength:
\[ E = \frac{h^2}{2m\lambda^2} \]

3. **Analyze the Energy Expression**:
From the equation \( E = \frac{h^2}{2m\lambda^2} \), we can see that energy \( E \) is inversely proportional to the mass \( m \) of the particle when the wavelength \( \lambda \) is constant. This means that as the mass decreases, the energy increases.

4. **Compare the Masses**:
- A photon has zero rest mass.
- An electron has a small mass (approximately \( 9.11 \times 10^{-31} \) kg).
- A uranium nucleus has a significantly larger mass (approximately \( 3.95 \times 10^{-25} \) kg).

5. **Determine Maximum Energy**:
Since the energy is inversely proportional to mass, the particle with the smallest mass will have the maximum energy. Among the three particles:
- Photon (mass = 0)
- Electron (mass = \( 9.11 \times 10^{-31} \) kg)
- Uranium nucleus (mass = \( 3.95 \times 10^{-25} \) kg)

The photon, having the smallest mass, will have the maximum energy.

### Conclusion:
The maximum energy will be of the **photon**.