To determine the relative permeability of a magnetic material, we need to use the fundamental relationships between magnetic flux density, magnetic field intensity, and magnetization. The problem provides the magnetization (\(M\)) and the flux density (\(B\)) of the material.

Given Parameters

• Magnetization (\(M\)) of the material = 2350 A/m
• Magnetic flux density (\(B\)) = 3.142 mWb/m²

It is important to note that 3.142 is a common approximation for the mathematical constant \(\pi\). Therefore, we can consider the flux density \(B = \pi \times 10^{-3}\) Wb/m² (or Tesla).

Fundamental Magnetic Relationships

We will use the following key equations that describe the behavior of magnetic fields in materials:

1. The relationship between magnetic flux density (\(B\)), magnetic field intensity (\(H\)), and absolute permeability (\(\mu\)) of the material: \[B = \mu H\]
2. The absolute permeability (\(\mu\)) of a material is related to the permeability of free space (\(\mu_0\)) and the relative permeability (\(\mu_r\)) by: \[\mu = \mu_0 \mu_r\] Where \(\mu_0 = 4\pi \times 10^{-7}\) H/m is the permeability of free space.
3. The total magnetic flux density (\(B\)) in a material is also given by the sum of the flux density due to the applied field (\(\mu_0 H\)) and the flux density due to the magnetization (\(\mu_0 M\)) of the material: \[B = \mu_0 H + \mu_0 M\] This can be written as: \[B = \mu_0 (H + M)\]

Deriving the Relative Permeability Formula

Our goal is to find the relative permeability, \(\mu_r\). Let's combine the equations above.

From equation (1) and (2), we can write: \[B = \mu_0 \mu_r H\] From this, we can express the magnetic field intensity \(H\) as: \[H = \frac{B}{\mu_0 \mu_r}\]

Now, substitute this expression for \(H\) into equation (3): \[B = \mu_0 \left( \frac{B}{\mu_0 \mu_r} + M \right)\] Distribute \(\mu_0\) on the right side: \[B = \frac{\mu_0 B}{\mu_0 \mu_r} + \mu_0 M\] Simplify the first term: \[B = \frac{B}{\mu_r} + \mu_0 M\]

Now, rearrange the equation to solve for \(\mu_r\): \[B - \mu_0 M = \frac{B}{\mu_r}\] Multiply both sides by \(\mu_r\): \[\mu_r (B - \mu_0 M) = B\] Finally, isolate \(\mu_r\): \[\mu_r = \frac{B}{B - \mu_0 M}\]

Calculating Relative Permeability

Now, we can substitute the given values into the derived formula:

• \(B = 3.142 \times 10^{-3}\) Wb/m² (Using \(3.142 \approx \pi\), so \(B = \pi \times 10^{-3}\) T)
• \(M = 2350\) A/m
• \(\mu_0 = 4\pi \times 10^{-7}\) H/m

First, let's calculate the term \(\mu_0 M\): \[\mu_0 M = (4\pi \times 10^{-7} \text{ H/m}) \times (2350 \text{ A/m})\] \[\mu_0 M = 4 \times 2350 \times \pi \times 10^{-7} \text{ T}\] \[\mu_0 M = 9400 \pi \times 10^{-7} \text{ T}\] \[\mu_0 M = 0.00094 \pi \text{ T}\]

Now, substitute this into the formula for \(\mu_r\): \[\mu_r = \frac{\pi \times 10^{-3} \text{ T}}{\pi \times 10^{-3} \text{ T} - 0.00094 \pi \text{ T}}\] Factor out \(\pi\) from the numerator and denominator: \[\mu_r = \frac{\pi \times 10^{-3}}{\pi (10^{-3} - 0.00094)}\] \[\mu_r = \frac{10^{-3}}{10^{-3} - 0.00094}\] \[\mu_r = \frac{0.001}{0.001 - 0.00094}\] \[\mu_r = \frac{0.001}{0.00006}\] \[\mu_r = \frac{100}{6}\] \[\mu_r = 16.666...\]

Rounding to two decimal places, the relative permeability is approximately 16.67.

Conclusion

The calculated relative permeability of the magnetic material is 16.67. This demonstrates how the intrinsic properties of a material (magnetization) contribute to the overall flux density in the presence of an applied field, and how relative permeability quantifies a material's ability to support the formation of a magnetic field within itself.