The electron is moving with a velocity `c/3`. The de-Broglie wavelength of electron is observed to be equal to a moving photon. The ratio of kinetic energy of electron to that energy of photon is found to be a:6. Find the value of a.

Explanation

To solve the problem, we need to find the value of \( a \) given that the ratio of the kinetic energy of the electron to the energy of the photon is \( \frac{a}{6} \).

### Step-by-step Solution:

1. **Identify the velocity of the electron**:
The electron is moving with a velocity \( v = \frac{c}{3} \).

2. **Calculate the de-Broglie wavelength of the electron**:
The de-Broglie wavelength \( \lambda_e \) of the electron is given by the formula:
\[ \lambda_e = \frac{h}{mv} \]
where \( m \) is the mass of the electron and \( v \) is its velocity. Substituting \( v = \frac{c}{3} \):
\[ \lambda_e = \frac{h}{m \cdot \frac{c}{3}} = \frac{3h}{mc} \]

3. **Determine the wavelength of the photon**:
According to the problem, the de-Broglie wavelength of the electron is equal to the wavelength of the moving photon, \( \lambda_p \). Therefore:
\[ \lambda_p = \lambda_e = \frac{3h}{mc} \]

4. **Calculate the energy of the photon**:
The energy \( E \) of a photon is given by:
\[ E = \frac{hc}{\lambda_p} \]
Substituting \( \lambda_p = \frac{3h}{mc} \):
\[ E = \frac{hc}{\frac{3h}{mc}} = \frac{mc^2}{3} \]

5. **Calculate the kinetic energy of the electron**:
The kinetic energy \( KE \) of the electron is given by:
\[ KE = \frac{1}{2} mv^2 \]
Substituting \( v = \frac{c}{3} \):
\[ KE = \frac{1}{2} m \left(\frac{c}{3}\right)^2 = \frac{1}{2} m \cdot \frac{c^2}{9} = \frac{mc^2}{18} \]

6. **Find the ratio of kinetic energy of the electron to the energy of the photon**:
We need to find the ratio \( \frac{KE}{E} \):
\[ \frac{KE}{E} = \frac{\frac{mc^2}{18}}{\frac{mc^2}{3}} = \frac{1/18}{1/3} = \frac{3}{18} = \frac{1}{6} \]

7. **Relate the ratio to \( a \)**:
According to the problem, this ratio is given as \( \frac{a}{6} \):
\[ \frac{1}{6} = \frac{a}{6} \]
Therefore, we can conclude that:
\[ a = 1 \]

### Final Answer:
The value of \( a \) is \( 1 \).