2.2.4 The Curl of E

We (briefly) review the curl of the Electric Field. (Hint: it's 0.)

[swanh]

My EM professor is really horrible and I am honestly using your videos as my "lectures". Thank you for the work you do and your wonderful explanations!

[MrAyangan]

Wow! I had never thought that E field was such a special field. All its components are interconnected. This is truly amazing.

[hershyfishman]

2:49 This is unclear in the video. You could go in any direction on the line integral, including theta or phi. Griffith's point is that the single charge in this example is centered at the origin, so the direction of E is only in the r direction.

[jg]

(2) Stoke's theorem. If E was curly SOMEWHERE, then there would be SOME closed path integral E dot dl that would be Non-zero. Since the closed path integral of E dot dl is zero for ALL paths, there can not be ANY curliness to E. Or, since the point charge produces a non-curly field, and E can only be a sum (integral or discrete) of point charges, and the curl distributes across sums, then the resulting field MUST have 0 curliness to it, and so the path integral is also zero everywhere.

[benhaenraets]

I'm not convinced by your "obvious" argument at 4:52. If the surface integral of (del x E) dotted with da is 0, doesn't this only imply that for each da, the curl of E can be any vector perpendicular to da? For example, if you had a flat surface, which every area element having da = (1, 1, 1), the if the curl of E = (-1, 1, 0), then this still satisfies the right hand side.

[redteamdarkspear]

Why is it mathematically necessary for the curl to be 0 if E.dl=0? I see why the reverse is true (if Del X E = 0, (Del X E) . da must equal 0 so E.dl must equal 0). However, what mathematically necessitates that the curl equal 0 just because E.dl = 0? If we look at the right side of Stokes's theorem, (Del X E).da=0, there are a few ways that it could equal 0 without the curl equaling 0. The area could be 0, or there could just be no component of Del X E that is perpendicular to the area.

[jg]

Good question. Unfortunately, I'm having a hard time following your argument in a comment. Let me see if I can help you understand.

I'm going to try and post a video response tonight.

[deepigabharathi]

Since electric field is the negative gradient of potential, and curl(grad( fn)) is zero can we prove by curl (-grad(v)) is zero

[AncientAncestor]

But how can we draw the conclusion that curl E is zero everywhere when Er is discontinuous at r =0 ? We had the same problem when it came to the divergence of E and that turned out to be equal to rho divided by epsilon nought. Why is it okay to disregard r=0 when it comes to the curl, and not when it comes to the divergence?

[카이-bq]

i love you!!!! thank you!!! so much!!!!

[parthasarathym]

What if there are two point charges?...
I found some pictures in internet that showed flux around a dipole which seemed to be curly...

[jg]

OK, I think the gap in your understanding is two-fold.

(1) I don't think you have an intuitive grasp of calculus. da can NEVER be zero, because da is the bit that gets smaller and smaller and approaches zero, but cannot reach it.

[ProfessorSteez]

Do you know of any good resources on YT on learning integrating variables directly related to E&M?

I can understand the concepts ok, but have problems when it comes to setting up the integrating variables to calculate E or V (such as polar spherical or polar cylindrical in 2D vs. 3D).

[swarooppalai]

can anyone explain how to write the small element da in spherical coordinates? Thank you.