Which of the following does not have the dimensions of velocity ? ( Given `epsilon_(0)` is the permittivity of free space , `mu_(0)` is the permeability of free space , `v` is frequency , `lambda` is wavelength , `P` is the pressure , and `rho` is density , `k` is wave number , `omega` is the the angular frequency) (1)`omega` `k` (2)`v``lambda` (3)1/`sqrt(epsilon_(0) mu_(0))` (4)`sqrt(P/rho)`

Options

A. `omega k`
B. `v lambda`
C. `(1)/( sqrt(epsilon _(0) mu_(0)))`
D. ` sqrt((P)/( rho))`

Explanation

To determine which of the given options does not have the dimensions of velocity, we need to analyze each option one by one. The dimensions of velocity are given by \( [V] = LT^{-1} \).

### Step 1: Analyze Option (1) \( \omega k \)

- **Given**: \( \omega \) is angular frequency and \( k \) is wave number.
- **Dimensional Formula of \( \omega \)**:
\[ [\omega] = T^{-1} \]
- **Dimensional Formula of \( k \)**:
\[ [k] = L^{-1} \]
- **Combining**:
\[ [\omega k] = [\omega][k] = T^{-1} \cdot L^{-1} = L^{-1}T^{-1} \]
- **Conclusion**: This does not have the dimensions of velocity.

### Step 2: Analyze Option (2) \( v \lambda \)

- **Given**: \( v \) is frequency and \( \lambda \) is wavelength.
- **Dimensional Formula of \( v \)**:
\[ [v] = T^{-1} \]
- **Dimensional Formula of \( \lambda \)**:
\[ [\lambda] = L \]
- **Combining**:
\[ [v \lambda] = [v][\lambda] = T^{-1} \cdot L = LT^{-1} \]
- **Conclusion**: This has the dimensions of velocity.

### Step 3: Analyze Option (3) \( \frac{1}{\sqrt{\epsilon_0 \mu_0}} \)

- **Given**: \( \epsilon_0 \) is the permittivity of free space and \( \mu_0 \) is the permeability of free space.
- **Dimensional Formula of \( \epsilon_0 \)**:
\[ [\epsilon_0] = M^{-1}L^{-3}T^{4}A^{2} \]
- **Dimensional Formula of \( \mu_0 \)**:
\[ [\mu_0] = ML T^{-2} A^{-2} \]
- **Combining**:
\[ [\epsilon_0 \mu_0] = [\epsilon_0][\mu_0] = (M^{-1}L^{-3}T^{4}A^{2})(ML T^{-2} A^{-2}) = L^{-2}T^{2} \]
- **Taking the square root**:
\[ \left[\frac{1}{\sqrt{\epsilon_0 \mu_0}}\right] = \frac{1}{\sqrt{L^{-2}T^{2}}} = LT^{-1} \]
- **Conclusion**: This has the dimensions of velocity.

### Step 4: Analyze Option (4) \( \sqrt{\frac{P}{\rho}} \)

- **Given**: \( P \) is pressure and \( \rho \) is density.
- **Dimensional Formula of \( P \)**:
\[ [P] = ML^{-1}T^{-2} \]
- **Dimensional Formula of \( \rho \)**:
\[ [\rho] = ML^{-3} \]
- **Combining**:
\[ \left[\frac{P}{\rho}\right] = \frac{ML^{-1}T^{-2}}{ML^{-3}} = L^{2}T^{-2} \]
- **Taking the square root**:
\[ \left[\sqrt{\frac{P}{\rho}}\right] = \sqrt{L^{2}T^{-2}} = LT^{-1} \]
- **Conclusion**: This has the dimensions of velocity.

### Final Conclusion

The option that does not have the dimensions of velocity is **Option (1) \( \omega k \)**.

---