The mutual inductance, when air is replaced by a medium of relative is given by:

Options

A. `mu_(r) mu_(r) n_(1) n_(2)pir_(1)^(2)L`
B. `mu_(0) n_(1) n_(2) pir_(1)^(2)L`
C. `mu_(r) n_(1) n_(2) pir_(1)^(2) L`
D. `mu_(r) mu_(0) n_(1) n_(2) pir_(1)^(2)L`

Explanation

To solve the problem regarding the mutual inductance when air is replaced by a medium of relative permeability (μ_r), we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Mutual Inductance**:
The mutual inductance (M) between two coils is defined as the ratio of the magnetic flux (Φ) through one coil due to the current (I) in the other coil. The formula for mutual inductance in air is given by:
\[ M = \frac{\mu_0 N_1 N_2 A}{L} \]
where:
- \( \mu_0 \) = permeability of free space
- \( N_1 \) = number of turns in coil 1
- \( N_2 \) = number of turns in coil 2
- \( A \) = cross-sectional area of the coils
- \( L \) = length between the coils

2. **Replacing Air with a Medium**:
When air is replaced by a medium with relative permeability \( \mu_r \), the permeability of the medium becomes:
\[ \mu = \mu_0 \mu_r \]

3. **Revising the Formula for Mutual Inductance**:
Substituting \( \mu \) into the mutual inductance formula, we get:
\[ M = \frac{\mu_0 \mu_r N_1 N_2 A}{L} \]

4. **Final Expression for Mutual Inductance**:
Thus, the final expression for mutual inductance when air is replaced by a medium of relative permeability is:
\[ M = \mu_0 \mu_r \frac{N_1 N_2 A}{L} \]

5. **Identifying the Answer**:
The answer corresponds to the expression derived above, which is the mutual inductance in the presence of a medium with relative permeability.