Curl of an Electrostatic Field

Question

According to Maxwell's equations:
$$ vec{nabla}timesvec{E} = -frac{partial B}{partial t}$$
But if we have an electric field that is dependent upon both the radial distance and angles, we can get a non-zero curl. Does that mean that the rotor of the electric field due to any static configuration of charges, even though sometimes in the form $vec{E}(r,theta,varphi)$ and not just $vec{E}(r)$, be zero?

John Dumancic · Accepted Answer

There are two questions here: can the electric field of a static collection of charges have non-zero curl, and do any electric fields of the form $mathbf{E}(r,theta,phi)$ describe the electric field of a static collection of charges?
As for the first question, for a static collection of charges, the magnetic field does not change with time. Therefore, one has $$nablatimesmathbf{E}=0$$
Thus, the electric field for a static collection of charges must have a curl of zero; if you write down an electric field with a non-zero curl, that electric field is guaranteed to not be the electric field of any static collection of charges.
As for the second question, yes, there are electric fields dependent on angles that describe physical electrostatic situations. Consider the electric field of a static, charged tri-axial ellipsoid; the field here most certainly changes with both $theta$ and $phi$, but since it describes an electrostatic collection of charge, the field will have a curl of zero.

Curl of an Electrostatic Field

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