= Magnetic field
{title2=$\mathbf B$}
{wiki}
= Magnetic flux density
{synonym}
A <magnetic field> is a <vector field> $\mathbf B(\mathbf r,t)$ whose local value determines the magnetic part of the <Lorentz force> on a body with <electric charge> $q$ and <velocity> $\mathbf v$. In <SI> units,
$$
\mathbf F_B=q\mathbf v\times\mathbf B,\qquad
|\mathbf F_B|=|q|vB\sin\alpha,
$$
where $v$ is the <speed>, $B=|\mathbf B|$, and $\alpha$ is the <angle> between the <velocity> and the <magnetic field>. The <cross product> makes a nonzero magnetic <force> perpendicular to both the <velocity> and the <magnetic field>. The <right-hand rule> gives its direction for positive <electric charge>; negative <electric charge> reverses it. https://openstax.org/books/university-physics-volume-2/pages/11-2-magnetic-fields-and-lines[OpenStax, magnetic fields and forces]{ref}.
The <SI> unit of <magnetic field> is the <tesla (unit)>:
$$
1\,\mathrm T=\frac{1\,\mathrm N}{(1\,\mathrm C)(1\,\mathrm{m\,s^{-1}})}.
$$
For a stipulated example in <SI> units, take <electric charge> $q=2\,\mathrm C$, <velocity> $\mathbf v=(3,0,0)\,\mathrm{m\,s^{-1}}$, and <magnetic field> $\mathbf B=(0,0,4)\,\mathrm T$. The <cross product> gives magnetic <force> $\mathbf F_B=(0,-24,0)\,\mathrm N$. With the same <speed> directed along $+z$, the magnetic <force> is zero: motion parallel to a <magnetic field> produces no magnetic <acceleration>.
<Magnetic field lines> visualize the direction of a <magnetic field> by being tangent to it at each point. They are not generally the path of a body with <electric charge>: in a uniform <magnetic field> along $+z$, the <magnetic field lines> are straight while transverse <velocity> produces <circular motion>, and adding parallel <velocity> produces <helical motion in a uniform magnetic field>. The worked <Lorentz force> example below illustrates this distinction.
A <magnetic field> need not be static. <Magnetostatics> is the restricted case with no dependence on <time>; an <electromagnetic wave> has changing <electric field> and <magnetic field> even in vacuum. A changing <magnetic field> is coupled to the <electric field> through <Faraday's law of induction>. https://www.feynmanlectures.caltech.edu/II_20.html[Feynman lectures, II.20: electromagnetic waves in free space]{ref}. The useful separation is between local bending, described by the <Lorentz force>, and the amount of <magnetic field> crossing a <surface>, described by <magnetic flux>.
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