Static magnetic field from time-varying electric field

[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.

Actually, this is not correct.

From Ampere's law, the curl of B is proportional to the time rate of change of E (and current density if there's one). But this curl of B need not have a time varying solution as well. It can easily be a magnetostatic field.

Zz.

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

 

[blue_leaf77]

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

 

[Dale]

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

 

[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.

 

[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?