Electron and photon are having equal wavelengths but difference masses. The ratio of their kinetic energy will be

Options

A. `E_e/E_p gt 1`
B. `E_e/E_p lt 1`
C. `E_e/E_p = 1`
D. `E_e/E_p ge 1`

Explanation

To solve the problem of finding the ratio of kinetic energy between an electron and a photon having equal wavelengths, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have an electron and a photon with equal wavelengths, denoted as \( \lambda_e = \lambda_{ph} \).
- The mass of the electron is \( m_e \) and the mass of the photon is effectively zero (photons are massless).

2. **Use de Broglie Wavelength Formula**:
- The de Broglie wavelength is given by the formula:
\[ \lambda = \frac{h}{p} \]
- Where \( h \) is Planck's constant and \( p \) is the momentum.

3. **Relate Momentum to Mass and Velocity**:
- For the electron, momentum \( p_e \) is given by:
\[ p_e = m_e v_e \]
- For the photon, momentum \( p_{ph} \) is given by:
\[ p_{ph} = \frac{E_{ph}}{c} \]
- Where \( E_{ph} \) is the energy of the photon and \( c \) is the speed of light.

4. **Set the Momenta Equal**:
- Since the wavelengths are equal, we can set the momenta equal:
\[ m_e v_e = \frac{E_{ph}}{c} \]

5. **Express Energy in Terms of Kinetic Energy**:
- The kinetic energy \( KE \) of the electron is given by:
\[ KE_e = \frac{1}{2} m_e v_e^2 \]
- The energy of the photon is given by:
\[ E_{ph} = h f = h \frac{c}{\lambda} \]

6. **Find the Ratio of Kinetic Energies**:
- To find the ratio of kinetic energies, we can express both in terms of their respective momenta:
\[ \frac{KE_e}{E_{ph}} = \frac{\frac{1}{2} m_e v_e^2}{E_{ph}} \]
- Substitute \( E_{ph} \) into the equation:
\[ \frac{KE_e}{E_{ph}} = \frac{\frac{1}{2} m_e v_e^2}{\frac{h}{\lambda} } \]

7. **Simplify the Expression**:
- Since \( \lambda \) is the same for both, we can simplify the ratio:
\[ \frac{KE_e}{KE_{ph}} = \frac{m_e v_e^2}{h \frac{c}{\lambda}} = \frac{m_e v_e^2 \lambda}{h c} \]

8. **Conclusion**:
- The ratio of the kinetic energies will depend on the velocities and the mass of the electron. Since the mass of the photon is zero, the kinetic energy of the photon is not defined in the same way. Thus, the ratio will yield a value less than 1.

### Final Answer:
The ratio of the kinetic energy of the electron to that of the photon is less than 1.