The dimensions of `mu_(0)` are

Options

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

Answer

To find the dimensions of the magnetic permeability \( \mu_0 \), we will follow these steps:

### Step 1: Understand the definition of magnetic permeability
Magnetic permeability \( \mu_0 \) is defined as the proportionality constant between the magnetic induction \( B \) and the magnetic field intensity \( H \). Mathematically, this can be expressed as:
\[ B = \mu_0 H \]

### Step 2: Identify the SI units of magnetic induction and magnetic field intensity
The SI unit of magnetic induction \( B \) is the Tesla (T), and the SI unit of magnetic field intensity \( H \) is the Ampere per meter (A/m).

### Step 3: Write the relationship in terms of units
From the equation \( B = \mu_0 H \), we can express \( \mu_0 \) in terms of the units of \( B \) and \( H \):
\[ \mu_0 = \frac{B}{H} \]

### Step 4: Substitute the SI units into the equation
Substituting the units into the equation gives:
\[ \mu_0 = \frac{\text{Tesla}}{\text{A/m}} = \frac{\text{T}}{\text{A/m}} \]

### Step 5: Convert Tesla to base SI units
The Tesla can be expressed in terms of base SI units as follows:
\[ 1 \text{ T} = 1 \frac{\text{kg}}{\text{s}^2 \cdot \text{A}} = \text{kg} \cdot \text{s}^{-2} \cdot \text{A}^{-1} \]

### Step 6: Substitute the units of Tesla into the equation for \( \mu_0 \)
Now substituting the unit of Tesla into the equation for \( \mu_0 \):
\[ \mu_0 = \frac{\text{kg} \cdot \text{s}^{-2} \cdot \text{A}^{-1}}{\text{A/m}} = \frac{\text{kg} \cdot \text{s}^{-2} \cdot \text{A}^{-1}}{\text{A} \cdot \text{m}^{-1}} = \text{kg} \cdot \text{s}^{-2} \cdot \text{A}^{-2} \cdot \text{m} \]

### Step 7: Write the final dimensions of \( \mu_0 \)
Now, we can express the dimensions of \( \mu_0 \):
\[ \mu_0 = \text{M} \cdot \text{L} \cdot \text{T}^{-2} \cdot \text{A}^{-2} \]

Thus, the dimensions of \( \mu_0 \) are:
\[ \text{M} \cdot \text{L} \cdot \text{T}^{-2} \cdot \text{A}^{-2} \]

### Conclusion
The correct answer is:
\[ \text{Dimensions of } \mu_0 = [M L T^{-2} A^{-2}] \]