Magnetostatics: Expressing magnetic flux density B in terms of field strength H, magnetization M, and the vacuum permeability μ0 For a magnetic core characterized by relative permeability μr, magnetization (magnetic dipole moment per unit volume) M, and applied magnetic field strength H, which general SI relationship gives the flux density B?

Introduction / Context:
This question checks your understanding of the fundamental SI relationship connecting the magnetic flux density B, the magnetic field strength H, and the material response expressed through the magnetization M. Distinguishing the general law from special linear-material forms is crucial in electromagnetics and materials engineering.

Given Data / Assumptions:

• SI unit system is used.
• H is the applied magnetic field intensity (A/m).
• M is magnetization (A/m), i.e., dipole moment per unit volume.
• μ0 is the permeability of free space.
• Materials may or may not be linear; μr is only meaningful for linear, isotropic behavior.

Concept / Approach:
The general constitutive law in SI is B = μ0 (H + M). This identity always holds, independent of linearity. When, and only when, the material is linear and isotropic, M is proportional to H via magnetic susceptibility χm, giving M = χm H and μr = 1 + χm. In that special case, one may write B = μ0 μr H, but that expression comes from substituting M = (μr − 1) H into the general law.

Step-by-Step Solution:
Start with the general SI definition: B = μ0 (H + M).Recognize that this is valid for all materials, linear or nonlinear.If the medium is linear: M = χm H ⇒ B = μ0 (1 + χm) H = μ0 μr H.But the correct universal expression that includes M explicitly is B = μ0 (H + M).

Verification / Alternative check:
From definitions, magnetization current densities give rise to the M term; grouping free and bound contributions results in B = μ0 (H + M). Any linear simplification must reduce to this identity when M is reintroduced.

Why Other Options Are Wrong:
Option (a) lacks μ0 and is dimensionally inconsistent. Option (c) duplicates μr and M in a way that double counts polarization. Option (d) uses a minus sign before M, which contradicts the standard SI definition.

Common Pitfalls:
Confusing the always-true identity B = μ0 (H + M) with the linear-only shortcut B = μ0 μr H; forgetting units and dimensions when μ0 is omitted.

Final Answer:
B = μ0 (H + M)