Does a constant irrotational electric field have a magnetic field around it? 
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*i) I know from Maxwell-Faraday's law that if the electric field is irrotational then the time rate of change of magnetic field is zero.


*ii) Also, when assuming no current, I know from the Maxwell-Ampere law that a constant electric field means the curl of the magnetic field is zero.
But both laws don't talk about the magnetic field directly as Maxwell-Faraday's law talks about the 'rate-of-change' of magnetic field and the Maxwell-Ampere law talks about the 'curl' of the magnetic field.
So my questions are:

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*a) Does a constant irrotational electric field have a  magnetic field around it?


*b) Is a constant irrotational electric field the same as an electrostatic field?
PS: Refer Is there a magnetic field around a fully charged capacitor?
 A: As stated in the other post linked, there are background solutions to maxwells equations, which are determined independantly to charges and currents.
One of which COULD be a constant electric field. When I say constant, I mean the electric field has the same single vector attached to every point in space. ( overplayed onto of fields produced by charges and currents)
IF this homogenous solution was a constant E field, the associated background B field would be a time independant function too. As
$\nabla × \vec{E} = \frac{\partial \vec{B}}{\partial t}$
$0 = \frac{\partial \vec{B}}{\partial t}$
$\vec{C}(x,y,z) = \vec{B}$
This homogenous constant E field solution to maxwells equations would be different then the typical "electrostatic fields" we are used to, because in this solution $\nabla \cdot \vec{E} = 0$. Where as the typical E fields from charges, would have a non zero divergence at some point in space.
A: a) No, as you stated, as the electric field would not be varying, it would not be creating a magnetic field associated with it.
b) Electrostatic fields are electric fields that do not change with time so yes, as you said, constant irrotational electric fields are electrostatic fields
