Magnetic field in an air cavity within a magnetic core (uniform field assumption) A solenoid core of relative permeability μr contains a small cylindrical cavity of area dA and length dl. If the desired flux density inside the cavity equals the core flux density B0, what must be true about the magnetic field strength inside the cavity relative to the field in the surrounding core?

Introduction / Context:
Air cavities in magnetic materials are common in sensors and magnetic circuit analysis. Because flux density B relates to field strength H through material permeability, ensuring a specified B inside a cavity requires understanding how H differs across materials.

Given Data / Assumptions:

• Core region: B_core = μ0 * μr * H_core.
• Cavity region (air): B_cav = μ0 * H_cav.
• We desire B_cav = B0, equal to the core’s B0 (assumed uniform for the argument).
• Linear, isotropic materials and small cavity so fringing is neglected.

Concept / Approach:
Since permeability in air is μ0 and in core is μ0 * μr (μr > 1), the same B requires larger H in the lower-permeability region. Thus, to keep B_cav equal to B0, the magnetic field strength in the cavity must be larger than that in the core by a factor approximately μr.

Step-by-Step Solution:
Set equality of desired flux densities: B_cav = B_core = B0.Use constitutive relations: μ0 * H_cav = μ0 * μr * H_core.Cancel μ0 and solve: H_cav = μr * H_core.Conclusion: The cavity requires a stronger H than the surrounding core to achieve the same B.

Verification / Alternative check:
Finite-element simulations of small air gaps in magnetic circuits show elevated H inside the gap relative to the core for continuous B across the interface (normal component of B is continuous neglecting surface currents).

Why Other Options Are Wrong:
Equal fields would yield B_cav < B_core. Weaker field inside the cavity would reduce B further. “Arbitrary” and “zero” ignore constitutive physics.

Common Pitfalls:

• Confusing continuity: normal B is continuous, while tangential H may change with permeability jump.
• Forgetting that air has far lower permeability, demanding higher H to reach the same B.

Final Answer:
Field inside the cavity must be stronger than the field in the core