The dimensions of `(mu_(0)epsilon_(0))^(-1//2)` are

Options

A. `[L^(1//2) T^(-1//2)]`
B. `[L^(-1)T]`
C. `[LT^(-1)]`
D. `[L^(1//2)T^(1//2)]`

Explanation

To find the dimensions of \((\mu_0 \epsilon_0)^{-1/2}\), we will follow these steps:

### Step 1: Understand the Constants
- \(\mu_0\) is the permeability of free space.
- \(\epsilon_0\) is the permittivity of free space.

### Step 2: Write the Dimensions of \(\mu_0\) and \(\epsilon_0\)
- The dimension of permeability \(\mu_0\) is given by:
\[ [\mu_0] = \frac{M}{L T^2} \]
where \(M\) is mass, \(L\) is length, and \(T\) is time.

- The dimension of permittivity \(\epsilon_0\) is given by:
\[ [\epsilon_0] = \frac{L^3 T^4}{M} \]

### Step 3: Multiply the Dimensions of \(\mu_0\) and \(\epsilon_0\)
Now, we will multiply the dimensions of \(\mu_0\) and \(\epsilon_0\):
\[ [\mu_0 \epsilon_0] = \left(\frac{M}{L T^2}\right) \left(\frac{L^3 T^4}{M}\right) \]

### Step 4: Simplify the Expression
When we multiply these dimensions, we get:
\[ [\mu_0 \epsilon_0] = \frac{M \cdot L^3 \cdot T^4}{L \cdot T^2 \cdot M} = \frac{L^3 T^4}{L T^2} = L^{3-1} T^{4-2} = L^2 T^2 \]

### Step 5: Find the Dimensions of \((\mu_0 \epsilon_0)^{-1/2}\)
Now, we need to find the dimensions of \((\mu_0 \epsilon_0)^{-1/2}\):
\[ [\mu_0 \epsilon_0]^{-1/2} = (L^2 T^2)^{-1/2} = L^{-1} T^{-1} \]

### Final Answer
Thus, the dimensions of \((\mu_0 \epsilon_0)^{-1/2}\) are:
\[ [L^{-1} T^{-1}] \]