Electromagnetic induction , maxwell's equations

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  • Faraday's law of electromagnetic induction:$$\mathscr{C}_i=-\frac{d\Phi}{di}\tag{1}$$
  • In the case of a solenoid and doughnut coil:$$\Phi=N \Phi_{1}  \tag{2}$$ where N is the number of turns, is the magnetic flux through each turn.
  • Inductance of a solenoid:$$L=\mu \mu_0 n^2 V   \tag{3}$$
  • 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:$$w=\frac{B^2}{2 \mu \mu_0} =\frac{BH}{2}  .\tag{5}$$
  • Displacement current density:$$\hat{j} dis=\frac{\partial B}{\partial t}.   \tag{6}$$
  • Maxwell's equations in differential form:$$\nabla \times E = - \frac{\partial B}{\partial t}, \nabla .B=0, $$ , $$\nabla \times H = j + \frac{\partial D}{\partial t}, \nabla.D=\rho, \tag{7}$$ where $\nabla \times $= rot (the rotor) and $\nabla$= div (the divergence).Field transformation formulas for transition from a reference frame K to a reference frame K' moving with the velocity vo relative to it. In the case  $v_0 << c$ $$E'=E+[v_0 B], B'=B- \frac{[v_0 E]}{c^3}\tag{8}$$ In the general case $$E'_{||} = E_{||},    B'_{||}=B_{||},$$ , $$E'=\frac{E+[v_0 B]}{\sqrt{ 1-( \frac{v_0 }{c})}}\tag{9}$$ where the symbols II and I denote the field components, respectively parallel and perpendicular to the vector vo.

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