# I Static magnetic field from time-varying electric field

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1. Feb 29, 2016

### EmilyRuck

Hello!
In this thread, in this answer, my statement "A time-varying electric field creates a magnetic field which is time-varying itself" was refuted.

Because I never observed this before, I would like to discuss about it. As far as I know, Maxwell's equations are valid always together, that is contemporary. Suppose that impressed currents are 0 and that we are in a linear, homogeneous medium. So, yes, Ampère's law does not cause a variation for the magnetic field

$\nabla \times \mathbf{H} = \epsilon \displaystyle \frac{\partial \mathbf{E}}{\partial t}$

but the co-existence of this equation with

$\nabla \times \mathbf{E} = - \mu \displaystyle \frac{\partial \mathbf{H}}{\partial t}$

implies that, when one field (the electric one or the magnetic one) varies with time, it will create the other, varying with time too.
If it is incorrect, could you give me a more clear explanation? And could you give an example of a time-varying electric field which creates a static magnetic field?
Thank you anyway,

Emily

2. Feb 29, 2016

### blue_leaf77

$\nabla \times \mathbf{E} = 0$ does not always imply that $\mathbf{E}=0$.

3. Feb 29, 2016

### Staff: Mentor

Consider the field induced from $E=(kt,0,0)$

4. Feb 29, 2016

### blue_leaf77

Consider $\mathbf{E} = A_0t\hat{x}$ and $\mathbf{H} = \epsilon A_0 (z\hat{y} + 2y\hat{z})$, it should satisfy the two Maxwell equations with the curl.

5. Mar 2, 2016

### EmilyRuck

Thank you both. Your examples provide a static magnetic field.
The requirements seem to be (observing $\nabla \times \mathbf{H} = \epsilon \partial \mathbf{E} / \partial t$):
- a constant $\partial \mathbf{E} / \partial t$ and so a linear time-dependency for the $\mathbf{E}$ components.
But this will imply (observing $\nabla \times \mathbf{E} = - \mu \partial \mathbf{H} / \partial t = 0$):
- a field $\mathbf{E}$ which is still conservative.
Are these necessary and sufficient conditions for having a static magnetic field with a time-varying electric field?

Another important case is the sinusoidal one. But I think the above requirements could never be satisfied with that time-variation. At least in this case, a (sinusoidal) time-varying electric-field always produces a (sinusoidal) time-varying magnetic field and vice-versa.
Is it right or are there some other exceptions?