The dimensional formula of ` sqrt((mu_(0))/(epsilon_(0)))` is

Options

A. `[ ML^(2) T^(-3) A^(2)]`
B. `[M^(0) LT^(-1) A^(0)]`
C. `[ ML^(2) T^(-3) A^(-2)]`
D. `[ M^(-1) L^(-2) T^(3) A^(2) ]`

Explanation

To find the dimensional formula of \( \sqrt{\frac{\mu_0}{\epsilon_0}} \), we will follow these steps:

### Step 1: Understand the relationship between \( c \), \( \mu_0 \), and \( \epsilon_0 \)
We know that the speed of light \( c \) is given by the equation:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]
Squaring both sides gives:
\[ c^2 = \frac{1}{\mu_0 \epsilon_0} \]
This can be rearranged to express \( \mu_0 \) and \( \epsilon_0 \):
\[ \mu_0 \epsilon_0 = \frac{1}{c^2} \]

### Step 2: Substitute \( \mu_0 \) and \( \epsilon_0 \)
We can express \( \mu_0 \) and \( \epsilon_0 \) in terms of their dimensional formulas. The dimensional formula for \( \mu_0 \) (the permeability of free space) is:
\[ [\mu_0] = \frac{M L}{T^2 A^2} \]
And the dimensional formula for \( \epsilon_0 \) (the permittivity of free space) is:
\[ [\epsilon_0] = \frac{A^2 T^4}{M L} \]

### Step 3: Find the dimensional formula of \( \frac{\mu_0}{\epsilon_0} \)
Now, we can find the dimensional formula of \( \frac{\mu_0}{\epsilon_0} \):
\[ \frac{\mu_0}{\epsilon_0} = \frac{\frac{M L}{T^2 A^2}}{\frac{A^2 T^4}{M L}} = \frac{M^2 L^2}{T^6} \]

### Step 4: Take the square root
Now, we need to take the square root of \( \frac{\mu_0}{\epsilon_0} \):
\[ \sqrt{\frac{\mu_0}{\epsilon_0}} = \sqrt{\frac{M^2 L^2}{T^6}} = \frac{M L}{T^3} \]

### Final Result
Thus, the dimensional formula of \( \sqrt{\frac{\mu_0}{\epsilon_0}} \) is:
\[ [M L T^{-3}] \]