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).

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