The dimensional formula for permeability of free space, `mu_(0)` is

Options

A. `[MLT^(-2)A^(-2)]`
B. `[ML^(-1)T^(2)A^(-2)]`
C. `[ML^(-1)T^(-2)A^(2)]`
D. `[MLT^(-2)A^(-1)]`

Explanation

To find the dimensional formula for the permeability of free space, denoted as \( \mu_0 \), we can start from the relationship given by Ampère's law in magnetostatics. The formula can be expressed in terms of magnetic field \( B \), current \( I \), and distance \( r \).

1. **Understanding the relationship**:
The permeability of free space \( \mu_0 \) is related to the magnetic field \( B \) produced by a current \( I \) at a distance \( r \). The formula can be expressed as:
\[ B = \frac{\mu_0 I}{2\pi r} \]
Rearranging this gives:
\[ \mu_0 = \frac{B \cdot 2\pi r}{I} \]

2. **Identifying dimensions**:
- The dimension of magnetic field \( B \) is given as:
\[ [B] = [M][T^{-2}][A^{-1}] \quad \text{(where M is mass, T is time, A is current)} \]
- The dimension of current \( I \) is:
\[ [I] = [A] \]
- The dimension of distance \( r \) is:
\[ [r] = [L] \]

3. **Substituting dimensions into the formula**:
Now substituting the dimensions into the equation for \( \mu_0 \):
\[ [\mu_0] = \frac{[B] \cdot [r]}{[I]} = \frac{[M][T^{-2}][A^{-1}] \cdot [L]}{[A]} \]

4. **Simplifying the expression**:
This simplifies to:
\[ [\mu_0] = [M][L][T^{-2}][A^{-2}] \]

5. **Final result**:
Therefore, the dimensional formula for the permeability of free space \( \mu_0 \) is:
\[ [\mu_0] = [M^1 L^1 T^{-2} A^{-2}] \]