Questions: Magnetic Susceptibility and Permeability

Question 1

An iron core with relative permeability μ_r = 5000 is inserted into a solenoid with n turns per meter carrying current I. How does the magnetic flux density B inside change compared to the air-core solenoid?

• A. B increases by a factor of 5000, because B = μ_r μ₀ nI
• B. B stays the same; the iron core only affects H, not B
• C. B decreases because iron is diamagnetic and opposes applied fields
• D. B increases by a factor of √5000 because permeability enters as a square root

Answer: (A)

For a linear magnetic material, B = μH = μ_r μ₀ H. In a solenoid, H = nI regardless of the core material (H is determined by free currents). So B = μ_r μ₀ nI — a factor of μ_r larger than the air-core case B = μ₀ nI. With μ_r = 5000, the field is enhanced 5000-fold. This is why transformer and electromagnet cores are made of iron: high permeability concentrates the magnetic flux enormously. Iron is ferromagnetic, not diamagnetic, so it enhances rather than opposes the field.

Question 2

What physical quantity does the auxiliary field H represent in a magnetic material, and how does it differ from B?

• A. H represents the total magnetic field including both free currents and magnetization; B represents only the free-current contribution
• B. H is defined by H = B/μ₀ − M and represents the field due to free currents alone, excluding the contribution of the material's own magnetization
• C. H and B represent the same physical quantity measured in different unit systems
• D. H is the externally applied field before the material is inserted; B is the field after the material is inserted

Answer: (B)

H is defined by H = B/μ₀ − M, which isolates the contribution from free currents (those under experimental control, like current in a coil) from the magnetization M (the material's own magnetic response). B is the total field — the one entering the Lorentz force law — and includes both contributions: B = μ₀(H + M). This three-field framework (B, H, M) is necessary because materials magnetize in response to H, and that magnetization feeds back into B. H provides a handle on the externally imposed driving; B captures the physical result.

Question 3

Diamagnetic materials are attracted toward regions of stronger magnetic field because χₘ < 0 means they acquire a magnetization in the same direction as the applied field.

• T. True
• F. False

Answer: False

Both clauses are wrong. Diamagnetic materials are *repelled* from regions of stronger field (they are weakly expelled from magnets). And χₘ < 0 means M = χₘH is antiparallel to H — the magnetization opposes the applied field, not aligns with it. This opposing magnetization is why diamagnets are repelled: a magnet induces a magnetization that pushes back. Paramagnetic materials (χₘ > 0) acquire magnetization aligned with H and are attracted toward stronger fields. Superconductors are the extreme diamagnetic case (χₘ = −1), expelling all field from their interior (Meissner effect).

Question 4

The permeability μ and susceptibility χₘ satisfy μ = μ₀(1 + χₘ), so a material with χₘ = 0 (no magnetic response) has permeability μ = μ₀, the same as free space.

• T. True
• F. False

Answer: True

By definition, μ = μ₀(1 + χₘ). When χₘ = 0, the material acquires no magnetization in response to an applied field (M = χₘH = 0), so B = μ₀(H + M) = μ₀H — exactly the free-space relation. A material with χₘ = 0 is magnetically transparent: it neither enhances nor weakens the field inside. Relative permeability μ_r = 1 + χₘ = 1 in this case, confirming the material behaves like vacuum.

Question 5

Why are three distinct fields (B, H, and M) needed to describe magnetism in materials, and what does each represent physically?

Model answer: B is the total magnetic flux density — the physical field entering the Lorentz force law and Faraday's law. M is the magnetization — the magnetic dipole moment per unit volume of the material, representing the material's own magnetic response. H is the auxiliary field (H = B/μ₀ − M), representing the contribution from free currents alone, independent of the material's magnetization. Three fields are needed because materials magnetize: an externally applied field (described by H, driven by free currents) induces a magnetization M in the material, which adds to the total field B. Without distinguishing what we control (H) from what the material contributes (M) and what results (B), the feedback between field and material response cannot be cleanly expressed.

The three-field framework is forced by the physics of magnetized media. In free space, B and H are proportional (B = μ₀H) and one field suffices. In a material, the material's own dipoles contribute to B in addition to whatever currents we impose, creating the feedback loop M = χₘH and B = μ₀(H + M). The H field is the clean input (controlled by free currents via ∇ × H = J_free), M is the material's response, and B is the measurable output. Collapsing H and B to a single field would make the constitutive relation circular and hide the causal structure of how materials respond to applied fields.