Electromagnetic induction

Electromagnetic induction is a phenomenon occurring in electrodynamics that can be summarized as follows:

  • A changing magnetic field induces an electric field.
  • A changing electric field induces a magnetic field.

Induced electric field

This electric field is determined much in the same way as the magnetostatic field, by way of its Curl:

∇⋅E=0,∇×E=−∂B∂t\nabla\cdot\mathbf{E}=0,\quad \nabla\times\mathbf{E}=- \frac{ \partial \mathbf{B} }{ \partial t }

These are mathematically equivalent restrictions as those put on the magnetic field, just with μ0J\mu_{0}\mathbf{J} changed to −∂B∂t-\frac{ \partial \mathbf{B} }{ \partial t }. We can even find an analog for the Biot-Savart law with

E=−14π∫1r2(∂B∂t×r^) dτ=−14π∂∂t∫B×r^r2 dτ\mathbf{E}=- \frac{1}{4\pi}\int \frac{1}{\mathfrak{r}^{2}}\left( \frac{ \partial \mathbf{B} }{ \partial t } \times \hat{\boldsymbol{\mathfrak{r}}} \right)\ d\tau=- \frac{1}{4\pi} \frac{ \partial }{ \partial t } \int \frac{\mathbf{B}\times \hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^{2}}\ d\tau

Notably, unlike the Biot-Savart law, this works even for time-dependent currents. This implies we can use Ampere's law in the same way we can for magnetic fields too:

∮E⋅dr=−dΦdt\oint \mathbf{E}\cdot d\mathbf{r}=- \frac{d\Phi}{dt}

where the quantity passing through the loop is now the change of magnetic flux instead of μ0Ienc\mu_{0}I_\text{enc}.

Induced magnetic field

For the discovery and derivation of the induced magnetic field, see Ampere's law > In electrodynamics.