# Electromagnetic Induction

• Faraday’s law.  Induced emf in a closed loop $\mathcal{E}=-\frac{d\Phi_B}{dt}$, $\Phi_B$ – magnetic flux through the loop.
• Lenz’s law.  An induced current or emf always tends to oppose or cancel out the change that caused it.
• Motional emf.  $\mathcal{E}=\oint(\vec{v}\times\vec{B})\cdot d\vec{l}$.
• Induced electric fields.  When an emf is induced by a changing magnetic flux through a stationary conductor, there is an induced nonconservative electric field $\vec{E}$: $\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi_B}{dt}$.
• Displacement current.  A time-varying electric electric field generates displacement current $i_D$, which acts as a source of magnetic field in exactly the same way as conduction current: $i_D=\epsilon\frac{d\Phi_E}{dt}$.
• Maxwell’s equations.  The relationships between electric and magnetic fields and their sources:

$\oint\vec{E}\cdot d\vec{A}=\frac{Q_{encl}}{\epsilon_0}$ (Gauss’s law for $\vec{E}$ fields)
$\oint\vec{B}\cdot d\vec{A}=0$ (Gauss’s law for $\vec{B}$ fields)
$\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi_B}{dt}$ (Faraday’s law)
$\oint\vec{B}\cdot d\vec{l}=\mu_0(i_C+\epsilon_0\frac{d\Phi_E}{dt})_{encl}$ (Ampere’s law including displacement current).