Some books define $H$ (the magnetic field intensity) by the relation given above. But I am having trouble conceptually understanding what this means. Is it the same thing as $B$ or something different? Can someone please elaborate and give some reasoning for the above relation?


They are different entities, but both are called the Magnetic Field depending on context.

In SI units $\vec{H}=\frac{\vec{B}}{\mu_0}-\vec{M}$.

Here $\vec{B}$ is the magnetic field as in the Lorentz force law : $\vec{F}=q\vec{E}+q\vec{v}\times \vec{B}$.

$\vec{M}$ is the Magnetization, the magnetic dipole density in a material. $\vec{H}$ is often also referred to as the Magnetic Field or just 'H'. In the former case, $\vec{B}$ is often referred to as the Magnetic Flux Density, in the latter, $\vec{B}$ is the Magnetic Field. Another difference, $\vec{B}$ is the result of current. Historically, $\vec{H}$ was thought to be the result of "Magnetic Charge" represented as poles analogous to the positive and negative point charges that give rise to the electric field.

They are frequently the same up to a proportionality constant.

In empty space $\vec{M}=0$, so $\vec{H}=\frac{\vec{B}}{\mu_0}-0.$ In some units, $\mu_0=1$ so we have $\vec{H}=\vec{B}$.

In materials with non-zero, linear Magnetic Susceptibility, $\chi_m$, $\vec{M}=\chi_m\vec{H}$, so

$\vec{B} =\mu_0(\vec{H}+\vec{M})=\mu_0\vec{H}(1+\chi_m)$

Which is simplified to $\vec{B}=\mu\vec{H}$, and so even in materials, we find a proportionality relationship between $\vec{B}$ and $\vec{H}$.

Physicists often treat two entities that are the same up to a proportionality constant as the same.

So $\vec{H}$ and $\vec{B}$ are quantitatively the same thing in a vacuum, depending on units. In such cases they differ qualitatively because of an old understanding of the source of the magnetic field.

In practice, the most important difference only occurs in materials when bound currents are induced by an applied field giving rise to a response from the material. Specifically $\vec{J}_b=\nabla \times \vec{M}$.


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