The dimensions of the ratio of magnetic flux `(phi)` and permeability `(mu)` are

Options

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

Answer

To find the dimensions of the ratio of magnetic flux (Φ) to permeability (μ), we can follow these steps:

### Step 1: Understand the definitions
- **Magnetic Flux (Φ)** is defined as the product of the magnetic field (B) and the area (A) through which the field lines pass. The formula is given by:
\[ Φ = B \cdot A \]

- **Permeability (μ)** is a measure of how well a material can support the formation of a magnetic field within itself. The formula for permeability in terms of magnetic field (B), current (I), and distance (r) is:
\[ μ = \frac{B \cdot 2\pi r}{I} \]

### Step 2: Write the ratio of magnetic flux to permeability
We need to find the ratio:
\[ \frac{Φ}{μ} = \frac{B \cdot A}{\frac{B \cdot 2\pi r}{I}} \]

### Step 3: Simplify the ratio
By simplifying the above expression, we can cancel out B:
\[ \frac{Φ}{μ} = \frac{A \cdot I}{2\pi r} \]

### Step 4: Determine the dimensions of each term
- **Area (A)** has dimensions:
\[ [A] = L^2 \]

- **Current (I)** has dimensions:
\[ [I] = A \]

- **Distance (r)** has dimensions:
\[ [r] = L \]

### Step 5: Substitute the dimensions into the ratio
Now substituting the dimensions into the simplified ratio:
\[ \frac{Φ}{μ} = \frac{L^2 \cdot A}{2\pi \cdot L} \]

### Step 6: Simplify the dimensions
This simplifies to:
\[ \frac{Φ}{μ} = \frac{L^2 \cdot A}{L} = L \cdot A \]

### Step 7: Write the final dimensions
Thus, the dimensions of the ratio of magnetic flux to permeability are:
\[ [M^0 L^1 T^0 A^1] \]

### Conclusion
The dimensions of the ratio of magnetic flux (Φ) to permeability (μ) are:
\[ M^0 L^1 T^0 A^1 \]