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H-field and magnetic materials

In SI units, the H-field is related to the B-field by H ≡ B μ 0 − M. {\displaystyle \mathbf {H} \ \equiv \ {\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M}.}

In terms of the H-field, Ampere's law is ∮ H ⋅ d ℓ = ∮ (B μ 0 − M) ⋅ d ℓ = I t o t − I b = I f, {\displaystyle \oint \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}=\oint \left({\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M} \right)\cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {tot} }-I_{\mathrm {b} }=I_{\mathrm {f} },} where I represents the 'free current' enclosed by the loop so that the line integral of H does not depend at all on the bound currents.

For the differential equivalent of this equation see Maxwell's equations. Ampere's law leads to the boundary condition (H 1 ∥ − H 2 ∥) = K f × n ^, {\displaystyle \left(\mathbf {H_{1}^{\parallel }} -\mathbf {H_{2}^{\parallel }} \right)=\mathbf {K} _{\mathrm {f} }\times {\hat {\mathbf {n} }},} where K is the surface free current density and the unit normal n ^ {\displaystyle {\hat {\mathbf {n} }}} points in the direction from medium 2 to medium 1.

Similarly, a surface integral of H over any closed surface is independent of the free currents and picks out the "magnetic charges" within that closed surface: ∮ S μ 0 H ⋅ d A = ∮ S (B − μ 0 M) ⋅ d A = 0 − (− q M) = q M, {\displaystyle \oint _{S}\mu _{0}\mathbf {H} \cdot \mathrm {d} \mathbf {A} =\oint _{S}(\mathbf {B} -\mu _{0}\mathbf {M})\cdot \mathrm {d} \mathbf {A} =0-(-q_{\mathrm {M} })=q_{\mathrm {M} },}

which does not depend on the free currents.

The H -field, therefore, can be separated into two independent parts: H = H 0 + H d, {\displaystyle \mathbf {H} =\mathbf {H} _{0}+\mathbf {H} _{\mathrm {d} },}

where H is the applied magnetic field due only to the free currents and H is the demagnetizing field due only to the bound currents.

The magnetic H -field, therefore, re-factors the bound current in terms of "magnetic charges". The H field lines loop only around "free current" and, unlike the magnetic B field, begins and ends near magnetic poles as well.