The relation connecting magnetic susceptibility `chi_(m)` and relative permeability `mu_(r)`, is

Options

A. `chi_(m)=mu_(r)+1`
B. `chi_(m)=mu_(r)-1`
C. `chi_(m)=(1)/(mu_(r))`
D. `chi_(m)=3(1+mu_(r))`

Explanation

To derive the relation connecting magnetic susceptibility (χ_m) and relative permeability (μ_r), we can follow these steps:

### Step 1: Understand the Definitions
- **Magnetic Susceptibility (χ_m)**: It is a dimensionless quantity that indicates how much a material will become magnetized in an applied magnetic field.
- **Relative Permeability (μ_r)**: It is the ratio of the permeability of a material (μ) to the permeability of free space (μ₀). It indicates how much more or less permeable a material is compared to a vacuum.

### Step 2: Write the Definition of Relative Permeability
The relative permeability is defined as:
\[ \mu_r = \frac{\mu}{\mu_0} \]
where:
- μ = permeability of the material
- μ₀ = permeability of free space

### Step 3: Relate Permeability to Magnetic Susceptibility
The permeability of a material can also be expressed in terms of magnetic susceptibility as:
\[ \mu = \mu_0 (1 + \chi_m) \]

### Step 4: Substitute into the Relative Permeability Equation
Now, substituting the expression for μ into the relative permeability formula:
\[ \mu_r = \frac{\mu_0 (1 + \chi_m)}{\mu_0} \]

### Step 5: Simplify the Equation
The μ₀ cancels out:
\[ \mu_r = 1 + \chi_m \]

### Step 6: Rearranging the Equation
From the above equation, we can express magnetic susceptibility in terms of relative permeability:
\[ \chi_m = \mu_r - 1 \]

### Final Result
Thus, the relation connecting magnetic susceptibility (χ_m) and relative permeability (μ_r) is:
\[ \mu_r = 1 + \chi_m \quad \text{or} \quad \chi_m = \mu_r - 1 \]