Electromagnetic induction, Maxwell's equations

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• Faraday's law of electromagnetic induction:$$\mathscr{E}_i=-\frac{d\Phi}{dt}$$
• In the case of a solenoid and doughnut coil:$$\Phi=N \Phi_{1},$$ where $N$ is the number of turns, $\Phi_1$ is the magnetic flux through each turn.
• Inductance of a solenoid:$$L=\mu \mu_0 n^2 V$$
• Intrinsic energy of a current and interaction energy of two currents:$$W=\frac{LI^2}{2}, W_{12}=L_{12}I_{1}I_{2}.\tag{4}$$
• Volume density of magnetic field energy:$$w=\frac{B^2}{2 \mu \mu_0} =\frac{BH}{2}.$$
• Displacement current density:$$\vec{j}_{dis}=\frac{\partial \vec{B}}{\partial t}.$$
• Maxwell's equations in differential form: $$\vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}, \vec{\nabla}.\vec{B}=0, \\ \vec{\nabla} \times \vec{H} = \vec{j} + \frac{\partial \vec{D}}{\partial t}, \vec{\nabla}.\vec{D}=\rho,$$ where $\vec{\nabla} \times \equiv$ rot (the rotor) and $\vec{\nabla}. \equiv$ div (the divergence).
• Field transformation formulas for transition from a reference frame $K$ to a reference frame $K^\prime$ moving with the velocity $v_0$ relative to it. In the case  $v_0 \ll c$ $$\vec{E^\prime}=\vec{E}+\vec{v_0}\times \vec{B}, \vec{B^\prime}=\vec{B}- \frac{\vec{v_0}\times \vec{E}}{c^2}.$$In the general case $$\vec{E^\prime_\parallel} = \vec{E_\parallel},    \vec{B^\prime_\parallel}=\vec{B_\parallel},$$ $$\vec{E^\prime_\bot}=\frac{\vec{E_\bot}+\vec{v_0}\times\vec{B}}{\sqrt{ 1-( \frac{v_0 }{c})^2}}, \vec{B^\prime_\bot}=\frac{\vec{B_\bot}-(\vec{v_0}\times\vec{E})/c^2}{\sqrt{ 1-( \frac{v_0 }{c})^2}}$$where the symbols $\parallel$ and $\bot$ denote the field components, respectively parallel and perpendicular to the vector $v_0$.