Which of the following expressions is correct with respect to relative permeability?

Question

Examoogle · Accepted Answer

Option A: $B = \mu_r \mu_0 H$

Explanation & Concepts:
The magnetic flux density ($B$) in a magnetic material is directly proportional to the magnetic field intensity ($H$). The constant of proportionality is the absolute permeability of the medium ($ \mu$), which is the product of the permeability of free space ($\mu_0$) and the relative permeability of the medium ($\mu_r$). Thus, the correct expression is $B = \mu_r \mu_0 H$.

Working Principle: When a magnetic field intensity $H$ is applied to a material, it causes magnetic flux lines to form inside the material. The ease with which these flux lines are established is determined by the material's magnetic permeability. Relative permeability $\mu_r$ measures how much more permeable the material is compared to vacuum (free space).

Key Formulas: $B = \mu H = \mu_0 \mu_r H$ — constitutive relation for magnetic fields; $\mu_r = \frac{\mu}{\mu_0}$ — definition of relative permeability; $\mu_0 = 4\pi 	imes 10^{-7} 	ext{ H/m}$ — permeability of free space

Key Points: Relative permeability $\mu_r$ is a dimensionless, unitless quantity.; For air/vacuum, $\mu_r = 1$; for ferromagnetic materials, $\mu_r \gg 1$.; Absolute permeability $\mu$ has units of Henry per meter (H/m) or Webers per Ampere-meter (Wb/A·m).

Applications: Transformer core design and material selection.; Analysis of magnetic circuits in electrical machines (motors and generators).

Additional Info: Standard value of permeability of free space: $\mu_0 = 4\pi 	imes 10^{-7} 	ext{ H/m}$.. Option B incorrectly places field intensity $H$ in the denominator.. Option C and Option D incorrectly introduce $H^2$, which violates the linear constitutive relationship $B \propto H$ for isotropic materials.

Answer: A is correct — $B = \mu_r \mu_0 H$ correctly represents the relationship between magnetic flux density ($B$), permeability ($ \mu = \mu_r \mu_0$), and magnetic field intensity ($H$).

Exam Tip: In magnetic circuits, $\mu_r \mu_0$ plays an analogous role to electrical conductivity ($\sigma$) in electric circuits when comparing $B = \mu H$ to Ohm's law density form $J = \sigma E$.

Examoogle · Answer

Option B: $B = \frac{\mu_r \mu_0}{H}$

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Option C: $B = \mu_r \mu_0 H^2$

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Option D: $B = \frac{\mu_r \mu_0}{H^2}$

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