In Q.72, if the velocity of electron is 25% of the velocity of photon, then `(E_e)/(E_(ph))` equals

Options

A. 0.04356
B. 0.044
C. 0.0472
D. 0.125

Explanation

To solve the problem of finding the ratio of the energy of an electron to the energy of a photon when the velocity of the electron is 25% of the velocity of the photon, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the velocities**:
- The velocity of photons, \( v_{ph} \), is the speed of light, \( c \).
- The velocity of the electron, \( v_e \), is given as 25% of the velocity of the photon:
\[ v_e = 0.25c \]

2. **Write the energy of a photon**:
- The energy of a photon can be expressed as:
\[ E_{ph} = h \nu \]
- Using the relationship between frequency (\( \nu \)) and wavelength (\( \lambda \)), we can also write:
\[ E_{ph} = \frac{hc}{\lambda} \]
- However, we can also express it in terms of momentum. The momentum of a photon is given by:
\[ p_{ph} = \frac{E_{ph}}{c} \]
- Thus, the energy of a photon can also be expressed as:
\[ E_{ph} = p_{ph} \cdot c \]

3. **Write the energy of an electron**:
- The kinetic energy of the electron is given by:
\[ E_e = \frac{1}{2}mv_e^2 \]
- Substituting \( v_e = 0.25c \):
\[ E_e = \frac{1}{2}m_e(0.25c)^2 = \frac{1}{2}m_e \cdot \frac{c^2}{16} = \frac{m_e c^2}{32} \]

4. **Relate the momentum of the photon and the electron**:
- The momentum of the photon is:
\[ p_{ph} = \frac{E_{ph}}{c} \]
- The momentum of the electron can be expressed as:
\[ p_e = m_e v_e = m_e \cdot 0.25c \]
- Setting the two momenta equal gives:
\[ \frac{E_{ph}}{c} = m_e \cdot 0.25c \]
- Rearranging gives:
\[ E_{ph} = 0.25 m_e c^2 \]

5. **Find the ratio of the energies**:
- Now we can find the ratio \( \frac{E_e}{E_{ph}} \):
\[ \frac{E_e}{E_{ph}} = \frac{\frac{m_e c^2}{32}}{0.25 m_e c^2} \]
- Simplifying this expression:
\[ \frac{E_e}{E_{ph}} = \frac{1}{32} \div \frac{1}{4} = \frac{1}{32} \cdot \frac{4}{1} = \frac{4}{32} = \frac{1}{8} \]

6. **Convert to decimal**:
- The ratio \( \frac{1}{8} \) can be expressed as a decimal:
\[ \frac{1}{8} = 0.125 \]

### Final Answer:
Thus, the ratio \( \frac{E_e}{E_{ph}} \) equals \( 0.125 \).

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