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

• 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:

• 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?

• What do you mean in a) by ..magnetic field around it? Mar 21, 2022 at 20:01
• @Triatticus I mean 'around' the Electric field. I am assuming here that the electric field is known constant and stationary relative to my frame of reference Mar 22, 2022 at 6:33
• I mean that you are asking if a curl free and uniform E field causes a nonzero circulation B field. If that's so maxwells equations already state the contrary of that. In a current free region the only source for a circulating magnetic field would be a variable electric field. Mar 22, 2022 at 12:20

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) 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

• a)Why cant the magnetic field be constant and irrotational? For the Constant Irrotational magnetic field, the curl of the magnetic field as zero, and also the time rate of change of magnetic field will be zero. Mar 21, 2022 at 16:58
• @GRANZER you're asking two different questions here. Does a static, irrotational E field produce a B field of any kind? No. Can a B field exist alongside a static, irrotational E field? Yes. Mar 22, 2022 at 21:33