Constant magnetic field, magnetics

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• Magnetic field of a point charge $q$ moving with non-relativistic velocity $\vec{v}$: $$\vec{B}=\frac{\mu_0}{4\pi}\frac{q\, \vec{v}\times \vec{r}}{r^3}.$$
• Biot-Savart law: $$d\vec{B}=\frac{\mu_0}{4\pi} \frac{\vec{j}\times\vec{r}}{r^3}dV, d\vec{B}=\frac{\mu_0}{4\pi}\frac{I, d\vec{l}\times\vec{r}}{r^3}.$$
• Circulation of a vector $\vec{B}$ and Gauss's theorem for it: $$\oint \vec{B}.d\vec{r}=\mu_0 I, \oint \vec{B}.d\vec{S}=0.$$
• Lorentz force:$$\vec{F}=q\,\vec{E}+q\,\vec{v}\times\vec{B}.$$
• Ampere force:$$d\vec{F}=\vec{j}\times\vec{B}\,dV, d\vec{F}=I\, d\vec{I}\times\vec{B}.$$
• Force and moment of forces acting on a magnetic dipole $\vec{p}_m =IS\vec{n}$: $$\vec{F}=p_m \frac{\partial \vec{B}}{\partial n}, \vec{N}=\vec{p}_m\times\vec{B},$$ where $\partial\vec{B}/\partial n$ is the derivative of a vector $vec{B}$ with respect to the dipole direction.
• Circulation of magnetization $\vec{J}$: $$\oint \vec{J}.d\vec{r} = I^\prime,$$where $I^\prime$ is the total molecular current.
• Vector $\vec{H}$ and its circulation:$$\vec{H}=\frac{\vec{B}}{\mu_0}-\vec{J}, \oint \vec{H}.d\vec{r}=I,$$ where $I$ is the algebraic sum of macroscopic currents.
• Relations at the boundary between two magnetics:$$B_{1n}=B_{2n}, H_{1\tau}=H_{2\tau}$$
• For the case of magnetics in which $\vec{J} = \chi \vec{H}$: $$\vec{B}=\mu \mu_0 \vec{H}, \mu = 1+\chi.$$