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) and permeability (μ), we will follow these steps:

### Step 1: Write the formula for magnetic flux (φ)
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:
\[ \phi = B \cdot A \]

### Step 2: Determine the dimensions of magnetic flux (φ)
The magnetic field (B) has dimensions of:
\[ [B] = M^1 L^{-1} T^{-2} A^1 \]
The area (A) has dimensions of:
\[ [A] = L^2 \]
Thus, the dimensions of magnetic flux (φ) can be calculated as:
\[ [\phi] = [B] \cdot [A] = (M^1 L^{-1} T^{-2} A^1) \cdot (L^2) = M^1 L^{1} T^{-2} A^1 \]

### Step 3: Write the expression for permeability (μ)
Permeability (μ) is defined in terms of the magnetic field (B) and the current (I) as follows:
\[ B = \mu \cdot \frac{I}{R^2} \]
From this, we can express permeability (μ) as:
\[ \mu = \frac{B \cdot R^2}{I} \]

### Step 4: Determine the dimensions of permeability (μ)
Using the dimensions of B and I, we can find the dimensions of permeability (μ):
- The dimensions of current (I) are:
\[ [I] = A^1 \]
- The dimensions of distance (R) are:
\[ [R] = L^1 \]
Thus, the dimensions of permeability (μ) can be calculated as:
\[ [\mu] = \frac{[B] \cdot [R^2]}{[I]} = \frac{(M^1 L^{-1} T^{-2} A^1) \cdot (L^2)}{A^1} = M^1 L^{1} T^{-2} \]

### Step 5: Calculate the ratio of magnetic flux (φ) to permeability (μ)
Now we can find the dimensions of the ratio of magnetic flux to permeability:
\[ \frac{\phi}{\mu} = \frac{M^1 L^{1} T^{-2} A^1}{M^1 L^{1} T^{-2}} = A^1 \]

### Step 6: Write the final answer
The dimensions of the ratio of magnetic flux (φ) and permeability (μ) are:
\[ [A] = A^1 \]