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

Options

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

Explanation

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

### Step 1: Understand the Constants
The constants \(\mu_0\) (the permeability of free space) and \(\epsilon_0\) (the permittivity of free space) have specific dimensions in terms of mass (M), length (L), and time (T).

### Step 2: Write the Dimensions of \(\mu_0\) and \(\epsilon_0\)
The dimensions of \(\mu_0\) are given by:
\[ [\mu_0] = \frac{M}{L \cdot T^2} \]
The dimensions of \(\epsilon_0\) are given by:
\[ [\epsilon_0] = \frac{L^3}{M \cdot T^4} \]

### Step 3: Multiply the Dimensions
Now we need to find the dimensions of the product \(\mu_0 \epsilon_0\):
\[ [\mu_0 \epsilon_0] = [\mu_0] \cdot [\epsilon_0] = \left(\frac{M}{L \cdot T^2}\right) \cdot \left(\frac{L^3}{M \cdot T^4}\right) \]

### Step 4: Simplify the Expression
When we multiply these dimensions, we get:
\[ [\mu_0 \epsilon_0] = \frac{M \cdot L^3}{M \cdot L \cdot T^6} = \frac{L^2}{T^6} \]

### Step 5: Find the Dimensions of \((\mu_0 \epsilon_0)^{-1/2}\)
Now we need to find the dimensions of \((\mu_0 \epsilon_0)^{-1/2}\):
\[ [(\mu_0 \epsilon_0)^{-1/2}] = \left(\frac{L^2}{T^6}\right)^{-1/2} = \frac{T^3}{L} \]

### Final Answer
Thus, the dimensions of \((\mu_0 \epsilon_0)^{-1/2}\) are:
\[ \frac{T^3}{L} \]

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