The dimensions of `(1)/(sqrt(mu_(0)epsi_(0)))` are the same as that of

Options

A. velocity
B. time
C. capacitance
D. distance

Answer

To solve the question, we need to determine the dimensions of the expression \( \frac{1}{\sqrt{\mu_0 \epsilon_0}} \) and compare it to the dimensions of the given options.

### Step-by-Step Solution:

1. **Understand the Constants**:
- \( \mu_0 \) is the permeability of free space, and its dimensions are given by:
\[ [\mu_0] = \frac{[M][L]^3}{[T]^2[A]^2} \]
- \( \epsilon_0 \) is the permittivity of free space, and its dimensions are given by:
\[ [\epsilon_0] = \frac{[T]^4[A]^2}{[M][L]^3} \]

2. **Calculate the Product \( \mu_0 \epsilon_0 \)**:
- Now, we multiply the dimensions of \( \mu_0 \) and \( \epsilon_0 \):
\[ [\mu_0 \epsilon_0] = \left(\frac{[M][L]^3}{[T]^2[A]^2}\right) \left(\frac{[T]^4[A]^2}{[M][L]^3}\right) \]
- Simplifying this gives:
\[ [\mu_0 \epsilon_0] = \frac{[M][L]^3 \cdot [T]^4 \cdot [A]^2}{[T]^2[A]^2 \cdot [M][L]^3} = \frac{[T]^4}{[T]^2} = [T]^2 \]

3. **Take the Square Root**:
- Now, we take the square root of the product:
\[ \sqrt{\mu_0 \epsilon_0} = \sqrt{[T]^2} = [T] \]

4. **Find the Dimensions of \( \frac{1}{\sqrt{\mu_0 \epsilon_0}} \)**:
- Therefore, the dimensions of \( \frac{1}{\sqrt{\mu_0 \epsilon_0}} \) are:
\[ \left[\frac{1}{\sqrt{\mu_0 \epsilon_0}}\right] = \frac{1}{[T]} = [T]^{-1} \]

5. **Comparison with Options**:
- Now we compare \( [T]^{-1} \) with the dimensions of the options provided:
- **Velocity**: \( [L][T]^{-1} \)
- **Time**: \( [T] \)
- **Capacitance**: \( [M]^{-1}[L]^{-2}[T]^{4}[A]^{2} \)
- **Distance**: \( [L] \)

- The dimensions of \( \frac{1}{\sqrt{\mu_0 \epsilon_0}} \) match with the dimensions of velocity, as velocity has the dimension \( [L][T]^{-1} \).

### Conclusion:
The dimensions of \( \frac{1}{\sqrt{\mu_0 \epsilon_0}} \) are the same as that of **velocity**.