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Question:

can h field only reside in material region?

A steady h-field can be defined through the physical properties of the material region. Furthermore, at the interface with air, the field is continuous. Is the field present in air, and is it still defined using the properties of the earlier magnetic region?

Answer:

H is the Magnetic field strength, in which any material is placed flux density (B) of the material is B/H ?. For Free Space (vacuum), B H ? 1. Free space is taken as reference. Air also can be taken as free space, almost. At abrupt interfaces the continuity has to be there in conformity with Maxwell's Equations.
Can I get a ride?!? ;)

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