Dimension of `mu_0 in_0//c^2` will be

Options

A. `M^0LT^(-2)`
B. `M_0L^4T^(-4)`
C. `M_0L^(-4)T^4`
D. none of the above

Answer

To find the dimension of \(\frac{\mu_0 \epsilon_0}{c^2}\), we can follow these steps:

### Step 1: Understand the relationship between \(\mu_0\), \(\epsilon_0\), and \(c\)
We know that the speed of light \(c\) is given by the formula:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]
Squaring both sides, we get:
\[ c^2 = \frac{1}{\mu_0 \epsilon_0} \]
This implies:
\[ \mu_0 \epsilon_0 = \frac{1}{c^2} \]

### Step 2: Substitute \(\mu_0 \epsilon_0\) in the expression
We need to find the dimension of:
\[ \frac{\mu_0 \epsilon_0}{c^2} \]
Using the relationship derived, we can substitute:
\[ \frac{\mu_0 \epsilon_0}{c^2} = \frac{1/c^2}{c^2} = \frac{1}{c^4} \]

### Step 3: Find the dimension of \(c\)
The dimension of the speed of light \(c\) is given by:
\[ [c] = [L][T]^{-1} = LT^{-1} \]

### Step 4: Calculate the dimension of \(c^4\)
Now, we calculate the dimension of \(c^4\):
\[ [c^4] = (LT^{-1})^4 = L^4 T^{-4} \]

### Step 5: Find the dimension of \(\frac{1}{c^4}\)
Thus, the dimension of \(\frac{1}{c^4}\) is:
\[ \left[\frac{1}{c^4}\right] = \frac{1}{L^4 T^{-4}} = L^{-4} T^{4} \]

### Final Answer
Therefore, the dimension of \(\frac{\mu_0 \epsilon_0}{c^2}\) is:
\[ [L^{-4} T^{4}] \]