Solenoidal electric field

Question

In electrostatic electric field in a system is always irrotational  ∇×E=0.   And divergence of electric field is non zero ∇.E=ρ/ε   but in some cases divergence of electric field is also zero  ∇.E=0  such as in case of dipole  I had calculated and got that ∇.E=0 for a dipole
So in case of this dipole divergence and curl both are zero 
So what does it mean when a vector fieLd do not diverge and not rotational at all
So what kind of nature it has??
∇×E=0 , ∇.E=0.
So it means the electric field is both solenoidal  and irrotational  ,but how can these two conditions satisfy simultaneously? If a vector field is solenoidal then it has to rotate ,must have some curliness

But in pic of a dipole I can see that  the electric field is bending or rotating

Then what does it mean about zero curl (∇×E=0)?
I can see the electric field is rotational

akhmeteli · Answer

div E does not vanish everywhere for the dipole, you should get delta-functions in the points where there are charges.

saketbyte · Answer

For better understanding of an irrotational and rotational field I am attaching two video links about Vorticity(Vorticity is the curl of velocity of fluid flow) which cleared my concept to a good extent. The term here to emphasise is CURL and not rotation, using the word rotation gives the vague sense of it but curl actually is the more technically correct term.

Vorticity part 1
Vorticity part 2

Two points to understand crystal clear from link 1 are-
a. A clear straight stream of water in a laminar flow is having rotationality(precisely Curl of Velocity is not zero), even though it appears to be just flowing in a straight line.
b. A spinning tight vortex with a hole at the centre of the basin has zero rotationality(precisely Curl of Velocity is zero.) even though it appears to spin quite nicely.
Hence, curling or say circulation is a better term to understand this whole phenomenon.
You can use this online vector field visualiser and plot functions like xi-yj, xj or xi+yj to understand rotational and solenoidal vector fields.

Solenoidal electric field

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