For an isotropic medium `B, mu, H` and `M` are related as (where `B`, `mu_(0)`, H and M have their usual meaning in the context of magnetic material

Options

A. `B-M=muH_0`
B. M=mu_0(2H+B)
C. `H=mu_0(2M+B) `
D. `B=mu_0(M+H) "`

Answer

To solve the problem regarding the relationship between magnetic field \( B \), magnetic permeability \( \mu \), magnetic field intensity \( H \), and magnetization \( M \) in an isotropic medium, we can follow these steps:

### Step 1: Understand the Definitions
- **Magnetic Field \( B \)**: The total magnetic field in the material.
- **Magnetic Permeability \( \mu_0 \)**: The permeability of free space (vacuum).
- **Magnetic Field Intensity \( H \)**: The applied magnetic field intensity.
- **Magnetization \( M \)**: The magnetic moment per unit volume of the material.

### Step 2: Write the Relationship
In an isotropic medium, the relationship between these quantities can be expressed as:
\[ B = \mu_0 (H + M) \]

### Step 3: Substitute Definitions
From the definitions, we know:
- The magnetic field \( B \) can be expressed as the sum of the magnetic field due to the external field \( B_0 \) and the magnetic field due to magnetization \( B_m \):
\[ B = B_0 + B_m \]
Where:
- \( B_0 = \mu_0 H \)
- \( B_m = \mu_0 M \)

### Step 4: Combine the Equations
Substituting \( B_0 \) and \( B_m \) into the equation for \( B \):
\[ B = \mu_0 H + \mu_0 M \]

### Step 5: Factor Out \( \mu_0 \)
We can factor out \( \mu_0 \) from the right-hand side:
\[ B = \mu_0 (H + M) \]

### Step 6: Conclusion
Thus, we have derived the relationship for an isotropic medium:
\[ B = \mu_0 (H + M) \]