The dimensional formula for magnetic permeability `mu` is :

Options

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

Answer

To find the dimensional formula for magnetic permeability \( \mu \), we can follow these steps:

### Step 1: Understand the relationship involving magnetic permeability
We start with the Biot-Savart law, which relates the magnetic field \( B \) to the current \( I \) and the distance \( r \):
\[ B = \frac{\mu_0 I \, dl}{r^2} \]
where \( \mu_0 \) is the permeability of free space, \( I \) is the current, \( dl \) is an elemental length, and \( r \) is the distance.

### Step 2: Rearranging the equation to find \( \mu_0 \)
From the equation, we can express \( \mu_0 \):
\[ \mu_0 = \frac{B \cdot r^2}{I \cdot dl} \]

### Step 3: Identify the dimensions of each term
- The dimension of magnetic field \( B \) can be derived from the Lorentz force equation:
\[ F = Q \cdot v \cdot B \]
where \( F \) is force, \( Q \) is charge, and \( v \) is velocity. The dimensions are:
- Force \( F \): \( [F] = M L T^{-2} \)
- Charge \( Q \): \( [Q] = A \cdot T \)
- Velocity \( v \): \( [v] = L T^{-1} \)

### Step 4: Deriving the dimension of magnetic field \( B \)
Rearranging the Lorentz force equation gives:
\[ B = \frac{F}{Q \cdot v} \]
Substituting the dimensions:
\[ [B] = \frac{M L T^{-2}}{(A \cdot T)(L T^{-1})} = \frac{M L T^{-2}}{A L T^{-1}} = \frac{M}{A T} \]

### Step 5: Substitute back to find \( \mu_0 \)
Now substituting the dimension of \( B \) into the expression for \( \mu_0 \):
\[ \mu_0 = \frac{B \cdot r^2}{I \cdot dl} \]
Here, \( r \) has dimensions of length \( [L] \) and \( dl \) also has dimensions of length \( [L] \):
\[ [\mu_0] = \frac{\left(\frac{M}{A T}\right) \cdot L^2}{A \cdot L} = \frac{M L^2}{A^2 T} \]

### Step 6: Final expression for magnetic permeability \( \mu \)
Since \( \mu \) is often expressed in terms of \( \mu_0 \) and relative permeability \( \mu_r \), and for free space \( \mu = \mu_0 \):
\[ [\mu] = [\mu_0] = \frac{M L^2}{A^2 T} \]

### Conclusion
Thus, the dimensional formula for magnetic permeability \( \mu \) is:
\[ \mu = M L^2 A^{-2} T^{-1} \]