For ferromagnetic material, the relative permeability (mu_(r)), versus magnetic intensity (H) has the following shape

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Answer

To analyze the relationship between relative permeability (μ_r) and magnetic intensity (H) for ferromagnetic materials, we can break down the problem into a series of steps.

### Step-by-Step Solution:

1. **Understanding Relative Permeability (μ_r)**:
- Relative permeability (μ_r) is defined as the ratio of the permeability of the material (μ) to the permeability of free space (μ_0). It indicates how easily a material can be magnetized.

2. **Starting Condition**:
- Initially, when the magnetic intensity (H) is low, the magnetization (I) of the ferromagnetic material is also low. In this region, the ratio I/H is very small. Therefore, we can express μ_r as:
\[ μ_r = 1 + \frac{I}{H} \]
- Since I/H is small, μ_r is approximately equal to 1, and it increases with increasing H.

3. **Magnetization and Saturation**:
- As H increases, the material becomes increasingly magnetized. Eventually, the material reaches a point of saturation where the magnetization (I) becomes constant. At this stage, I does not increase with further increases in H.

4. **Behavior at Saturation**:
- When the material is fully magnetized, the value of I is constant. Thus, the expression for μ_r becomes:
\[ μ_r = 1 + \frac{I_{max}}{H} \]
- Here, I_{max} is the maximum magnetization. As H continues to increase, the term I/H becomes smaller, leading to:
\[ μ_r \propto \frac{1}{H} \]
- This indicates that μ_r decreases as H increases beyond the saturation point.

5. **Graphical Representation**:
- The graph of μ_r versus H will show an initial increase as H increases, reaching a peak at saturation, and then a decrease as H continues to increase. The shape of the graph resembles a curve that rises and then falls.

### Final Graph Shape:
- The graph will start at a low value, rise steeply as H increases, reach a maximum (saturation point), and then decline as H continues to increase.