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When one studies electrostatics we have the electric field $\mathbf{E}$. This object usually is introduced as a field produced by a configuration of charges such that the force on another charge $Q$ at the point $a\in \mathbb{R}^3$ is $\mathbf{F} = Q\mathbf{E}(a)$.

In magnetostatics, one also has a field $\mathbf{B}$, but in the case it allows the force on a charged particle be written $\mathbf{F} = Q\mathbf{v}\times \mathbf{B}$. Although the velocity dependence thing, it is also one vector field that allows the magnetic force be obtained based on it, like the electrostatic field.

Although this similarity, I've seem in many places people calling $\mathbf{B}$ the magnetic induction field rather than magnetic field. Why is that? Is that just a terminology that has been established by convention or there is really a difference between being "a magnetic field" and "a magnetic induction field"?

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Besides the electric field $\vec E$ and the $\vec B$ field there are two other macroscopic fields, the displacement field $\vec D$ and the magnetic field $\vec H$.

In a vaccum, $\vec D= \vec E$ (up to a scaling constant) and $\vec H = \vec B$ (up to a scaling constant).

The magnetic field $\vec{H}$ is often what you make with a permanent magnetic, and it takes the net effect of all the little magnetic moment of all the pieces of matter into account and is determined by the actual batteries and such you hook up say to an electromagnetic that is made stronger by placing iron inside.

To avoid confusion, you should give the fields different names. If you noticed, I just call the $\vec B$ field the $\vec B$ field. It's not like all people are consistent with each other, but they do all use the symbol $\vec B$ for it.

Magnetic induction is a fine name since the $\vec B$ field is the field that comes up in Faraday's Law of induction: $$ \oint \vec E \cdot d \vec \ell = -\iint \frac{ \partial \vec{B}}{\partial t} \cdot d\vec a. $$ Whereas the magnetic field $\vec H$ comes up in Ampere's Law: $$ \oint \vec H \cdot d \vec \ell = \iint \vec J \cdot d\vec a. $$

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The named after Lorentz force $$\vec F = q_e \vec v_e \times \vec B$$ is part of the phenomenon of electromagnetic induction. According to the rules of vector products, the formula is changeable - for orthogonal vectors only - to the forms $$q_e \vec v_e = ( \vec B \times \vec F) / || \vec B|| ^2$$ (induction of a current in a generator) and $$\vec B = (\vec F \times q_e \vec v_e ) / || q_e \vec v_e|| ^2$$ (generation of a magnetic field by the movement of a conductor transversely to the current direction).

Not using permanent magnets this - the magnetic induction according to the last equation - is the only way to produce magnetic fields. Hope this answers our question.

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