# Solenoidal electric field

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

• Start with a simpler problem, calculate $\nabla\cdot \mathbf{E}$ and $\nabla \times \mathbf{E}$ for a point charge. Also, $\nabla \times \mathbf{E}=0$ isn't about the field not bending, it's about the field not shearing past itself. So while the field lines of a dipole are bent, the balance of field strength cancels. To see an example of what's going on, calculate the curl of $$\mathbf{V} = x \hat{j}.$$ – Sean E. Lake Nov 4 '16 at 4:07
• I didn't understand why ∇×E=0 is not about the field not bending? – user101134 Nov 4 '16 at 4:58
• Graph that field, $\mathbf{V}$, I gave you. The field lines are all parallel to each other, with density changing as you move in the $x$-direction. Calculate the curl, and you'll find it's the unit vector in the $z$-direction, $\hat{k}$, constant everywhere. Curl is about shear, not bending. – Sean E. Lake Nov 4 '16 at 5:05