A material particle with a rest mass `m_o` is moving with a velocity of light c. Then, the wavelength of the de Broglie wave associated with it is

Options

A. `((h)/(m_oc))`
B. zero
C. `infty`
D. `((m_oc)/(h))`

Explanation

To solve the problem, we need to find the de Broglie wavelength associated with a material particle that has a rest mass \( m_0 \) and is moving at the speed of light \( c \).

### Step-by-Step Solution:

1. **Understanding the Particle's Mass**:
- The particle is moving at the speed of light, which means it is a relativistic particle. In relativity, the mass of a particle increases with its velocity.
- The relativistic mass \( m \) is given by the formula:
\[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]
- Here, \( m_0 \) is the rest mass, \( v \) is the velocity of the particle, and \( c \) is the speed of light.

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

3. **Substituting the Velocity**:
- Since the particle is moving at the speed of light, we set \( v = c \).
- Substituting \( v \) into the relativistic mass formula:
\[ m = \frac{m_0}{\sqrt{1 - \frac{c^2}{c^2}}} = \frac{m_0}{\sqrt{1 - 1}} = \frac{m_0}{\sqrt{0}} \]
- This results in \( m \) approaching infinity, as division by zero is undefined.

4. **Calculating the de Broglie Wavelength**:
- Now substituting \( m \) and \( v \) into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{m \cdot c} \]
- Since \( m \) is approaching infinity, the wavelength \( \lambda \) becomes:
\[ \lambda = \frac{h}{\infty} = 0 \]

5. **Conclusion**:
- Therefore, the de Broglie wavelength associated with a material particle moving at the speed of light is:
\[ \lambda = 0 \]

### Final Answer:
The wavelength of the de Broglie wave associated with the particle is \( \lambda = 0 \).