# Curl of electric field due to dipole?

The field lines of electric field of dipole do seem to curl if seen geometrically. If water flow lines were of such shape, a ball would have rotated in it, doesn't this interpretation of non-zero curl apply here? But we know that curl of electrostatic field is zero. So, how do we conceive it geometrically? I am convinced of it from all mathematical arguments, maybe a more visual meaning of curl would help?

• The curl of a vector field isn't the same thing as the curvature of the field lines. This seems like a pure math question, which would get a good answer on Math.SE. Apr 13 '20 at 19:55
• If water flow lines were of such shape , a ball would have rotated in it. I don’t think that’s true. Apr 13 '20 at 19:56
• Can you plz elaborate? @G.Smith Apr 13 '20 at 20:20
• Are you asking why for instance in this image the curl is zero even where it seem like the electric field lines are "curving"? Apr 13 '20 at 20:42
• @Charlie Yes exactly Apr 13 '20 at 21:11

## Alternative definition

Whenever you try to find whether a field has a non zero curl, then imagine yourself moving a small charge (or whatever object upon which the field acts) in a very small loop (to be rigorous, it should be infinitesimal). If you end up doing some non zero work when you move the charge across the loop, then it means that the field has a non zero curl at that point ($$\vec{\nabla} \times \mathbf E\neq 0$$). But if you do no work when you move around a loop, then it means that the field does not have any curl at that point ($$\vec{\nabla} \times \mathbf E=0$$).

## Case of a dipole

Let's say that you place a dipole and thus the dipole establishes its field in the surroundings. Now, if you take any infinitesimal circle and calculate the work done while going around it, you would see that for almost half part of the circle, the field is along the direction of motion, and for almost the other half, it is opposite to the direction of motion. This way, the net work done by the field cancels out which implies that the curl of the field is zero (everywhere).

You are perceiving the curl as the curving of field. But whenever you estimate the curl of the field by the curvature, you should always consider the a whole loop instead of just going in one direction. In this case, you saw that the field is curving when you go from a point A to point B, but then you should also take into account the curvature when you come back from point B to point A. And this case, both the values exactly cancel exch other and the curl therefore, is zero. Also, if your arguments were true, then every field whose curl would be zero should exist as straight lines and shouldn't curl anywhere, but this isn't true.

• Thanks for the explanation @FakeMod . Does that mean we cannot tell about the curl of any field by looking at it's shape and anything? Also, Is there any example of a field which has non-zero curl everywhere? Apr 14 '20 at 6:22
• @SiddharthJain Curl is a local property, so it's hard to tell the curl of a field just by looking. $\mathbf E = x \hat j$ has a non zero curl ($= \hat k$) everywhere.
– user258881
Apr 14 '20 at 6:35

Perhaps you could think this a more visual representation of curl. Nothing like a dipole. Curl describes the flow of a field around a point.

To make sense of this one has to think of the limit as the size of the loop goes to zero at the point $$a$$ (curl is defined in terms of derivatives, meaning that this limit is implicit). Alternatively, think of the integral of curl over the area, as in Stoke's theorem. An example is the magnetic field around a current carrying wire, for which the curl is zero everywhere but at the wire.

• it would be better if you could explain the meaning applied to this specific situation of electric dipoles Apr 13 '20 at 21:13
• I thought that was obvious. And anyway I already did. The field of dipoles does not contain any element of a circular field like this. That is why the curl is zero. Apr 13 '20 at 21:33
• The magnetic field of a straight wire carrying a constant current has circular field lines that look just like this one. But the curl of the field is zero everywhere except at the wire. I believe you have reinforced the OP’s misunderstanding. Apr 13 '20 at 21:37
• @G.Smith, perhaps you are right. I have expanded to clarify. Apr 13 '20 at 22:14