From here, I have got the definition of $\vec{H}$. However even in wikipedia and other sites, I cannot find a definition for $\vec{B}$ which shows its similarity with $\vec{H}$.

Similarity: I know for free currents, $\vec{B}$ and $\vec{H}$ are same. I also know outside the magnet, for bound currents, $\vec{B}$ and $\vec{H}$ are same.

What is the definition for $\vec{B}$ whic makes it so much similiar to $\vec{H}$?

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    $\begingroup$ $H=\frac{B}{\mu_0}-M$ where $M$ is the magnetization. $\endgroup$ – BioPhysicist Sep 14 '18 at 6:10
  • $\begingroup$ I see... I understand the formula.... But what would be the physical interpretation of $\vec{B}$.... In other words, when we have a useful quantity "$\vec{H}$" in the first place, what is the need of defining another quantity called $\vec{B}$ $\endgroup$ – N.G.Tyson Sep 14 '18 at 8:47

I would say the more "fundamental" field is actually the magnetic field $\vec B$. The definition of $\vec H$ from this is then

$$\vec H=\frac{\vec B}{\mu_0}-\vec M$$

Where $\vec M$ is the magnetization of the medium your magnetic field is in.

Both are useful depending on the context. Since the magnetization in free space is $0$, $\vec B$ is more useful when looking at fields in a vacuum or media where magnetization can be neglected. $\vec H$ is more useful when your magnetic field is in some medium where a net magnetization arises.

Either way, it's all based on how useful the quantity is. For example, we technically don't need a definition of moment of inertia, angular momentum, etc., but those definitions are very useful for discussing rotational motion. The same is true here. We technically would only need one definition, but they are both useful, so we use both. One is more useful over the other when we are looking at fields in different media (or in lack of a medium).

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