Constant magnetic field, magnetics

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  • Magnetic field of a point charge q moving with non-relativistic velocity v:$$B=\frac{\mu}{4\pi}\frac{q[vr]}{r^3}.$$
  • Biot-Savart law:$$dB=\frac{\mu_0}{4\pi} \frac{[jr]}{r^3}\,dV, dB=\frac{\mu_0}{4\pi}\frac{I[dl,r]}{r^3}.$$
  • Circulation of a vector B and Gauss's theorem for it:$$\oint B \,dx=\mu_0 I, \oint B\,dS=0.$$
  • Lorentz force:$$F=qE+q[vB].$$
  • Ampere force:$$dF=[jB]\, dV, dF=I [dI, B].$$
  • Force and moment of forces acting on a magnetic dipole $p_m =ISn$: $$F=p_m \frac{\partial B}{\partial n}, N=[p_m B],$$ where OBIOn is the derivative of a vector B with respect to the dipole direction.
  • Circulation of magnetization J:$$\oint j\, dr = I'$$where I' is the total molecular current.
  • Vector H and its circulation:$$H=\frac{B}{\mu_0}- j, \oint H \, dr=I,$$ where I is the algebraic sum of macroscopic currents.
  • Relations at the boundary between two magnetics:$$B_{1n}=B_{2n}, H_{\tau_1}=H_{\tau_2}$$
  • For the case of magnetics in which J = xH:$$B=\mu \mu_0 H, \mu = 1+x .$$

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