What is said time varying magnetic field? I have heard a lot about it and the Internet is not willing to give me any answers. I assume time varying is a qualification, I was reading around here and heard about an induced magnetic field so I take it that is another qualification? What does that mean? Thank you.
[Physics] Time Varying Magnetic Field
electromagnetismmagnetic fields
Related Solutions
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.
What matters is the rate of change of the magnetic flux through a closed loop (or an open surface): $$ {\cal E}=-\frac{d}{dt}\int_C \vec B\cdot d\vec S $$ It doesn't matter how this change in flux is produced.
You can start, for instance, with a static magnetic field produced by a wire carrying a constant current, and drag a square frame at constant speed away from this wire, and the result will be a change in flux, and thus an EMF, and thus an associated electric field. In this case, the change in flux is produced by the change in the strength of $\vec B$ over the frame when the frame is moved, even if $\vec B$ is not time-varying.
Alternatively, you can have a fixed frame but a time-dependent $\vec B$ so that $\vec B\cdot d\vec S$ changes in time. You can also deform the frame in a constant magnetic field so the change in flux is not contained in the changing $d\vec S$. You can also change the angle between $\vec B$ and $d\vec S$ at some rate so that $\vec B\cdot d\vec S$ changes in time .
All of the above result in induction, and thus in an EMF, and thus in an electric field.
Best Answer
Time-varying means that as time, $t$, increases, the magnetic field changes. One of the more common representations is a sinusoidal wave:
(image from the linked Wikipedia page). Though the image above says $x$, the relation between $x$ and $\sin(x)$ is what is important.
With magnetic fields, Maxwell's equations, $$ \nabla\cdot\mathbf E=\frac\rho{\epsilon_0} \quad \nabla\cdot\mathbf B=0 \\ \nabla\times\mathbf E=0 \quad \nabla\times\mathbf B=\mu_0\mathbf J $$ (where $\mathbf E$ is the electric field, $\mathbf B$ the magnetic field, $\mathbf J$ the current density, $\epsilon_0$ the vacuum permittivity, $\rho$ the charge density, and $\mu_0$ the vacuum permeability) then become $$ \nabla\cdot\mathbf E=\frac\rho{\epsilon_0} \quad \nabla\cdot\mathbf B=0 \\ \nabla\times\mathbf E=\color{blue}{-\frac{\partial\mathbf B}{\partial t}} \quad \nabla\times\mathbf B=\mu_0\mathbf J+\color{blue}{\frac{1}{c^2}\frac{\partial\mathbf E}{\partial t}} $$ where the blue-colored terms show that the two fields induce each other when changing in space and/or time.