# Varying electric field?

Maybe a silly question, but anyway:

We always hear about "electromagnetic induction", that is when a time varying magnetic flux induces an electric field.

Does the same thing happen when there is a varying electric field? Does a time varying electric field result in an induced magnetic field?

Yes varying electric field induce a magnetic field as stated by the Maxwell's fourth equation (an extension of the Ampère's law). In free space that is: $$\nabla\times\vec{B}=\mu_0\vec{J}+\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}$$ An example can be any type of electromagnetic wave, where $\vec{E}$ (and also $\vec{B}$) is varying with time, a simple one would be a plane-wave where $\vec{E}=\vec{E}_0\cos(\omega t-kz)$ with $\vec{E}_0$ is a costant vector (more here).

In a dielectric material the Maxwll's equation becomes: $$\nabla\times\vec{H}=\vec{J}_f+\frac{\partial\vec{D}}{\partial t}$$ where $\vec{D}=\epsilon_0\vec{E}+\vec{P}$ and so $\frac{\partial\vec{D}}{\partial t}=\epsilon_0\frac{\partial\vec{E}}{\partial t}+\frac{\partial\vec{P}}{\partial t}$. The additional term $\frac{\partial\vec{P}}{\partial t}$ keeps track of any moving charge due to the polarization of the material.

Yes. One of Maxwell's equations is:

$\nabla \times \vec{B} = \mu_o (\vec{J} + \epsilon_o \frac{\partial \vec{E}}{\partial t})$

Where $\vec{J}$ is a current (if one exists), and $\vec{E}$ is the electric field.

• Yes I knew that, but are there any physical examples of it? Commented Jan 23, 2014 at 23:17
• I mean we always say that when a current starts flowing in a wire, its varying magnetic field induces a back EMF (proportional to the system's self inductance). Isn't the electric field also changing? Doesn't it take some time to be felt at further and further distances? Commented Jan 23, 2014 at 23:19
• When the current is first turned on, there is a change in the electric field will itself cause a magnetic field as you expect. But, this is fairly brief as once the current is constant then the electric field is constant and doesn't change. As for a physical example... The best example I can think of is light. Light is the propagation of changes in the electric field which themselves cause changes in the magnetic field which then cause changes in the electric field (and so on...). Commented Jan 23, 2014 at 23:27
• "Yes I knew that, but are there any physical examples of it?" , other than, for example, the light by which you read this comment? @user37677 Commented Jan 24, 2014 at 0:45
• When electric field oscillates in capacitor in the LC-circuit, there is some oscillating magnetic field around the capacitor (and wire and the coil) too. Commented Jan 24, 2014 at 13:58