Electric field (in)dependent of distance?

[Tiago3434]

Hi guys, I was reading about electromagnetism, specifically about the application of Gauss' Law to an infinite charged sheet, and how its electric field doesn't depend on the distance from the sheet. I think I have finally managed to wrap my mind around the concept intuitively, based on one of Feynman's explanations: when you take a point P close to the sheet, the electric field due to the closest point of P on the sheet is really strong, and the electric field (vectors) of all other point are almost parallel to the sheet, so they end up canceling each other.

When you take another point farther away, the electric field due to the point closest to such point is weaker, but the electric fields due to the other points are almost perpendicular to the sheet, so they end up adding, and this mechanism (intuitively) would explain why the electric field is constant.

But wouldn't this explanation also be true for an infinite charged line, making it distance-independent? Is there any intuition as to why the electric field of an infinite sheet is distance irrelevant, but for an infinite line it isn't? Thanks in advance.

 

[Dale]

Is there any intuition as to why the electric field of an infinite sheet is distance irrelevant, but for an infinite line it isn't?

For me it is clear from a geometric symmetry argument. For an infinite plane the only possible direction is normal to the plane due to the symmetry. The family of lines normal to a plane don't diverge so the field strength is constant. For a infinite line, by symmetry the lines must be radial and normal to the wire. The family of such lines diverges, so the field strength goes down as you go out.

 

[Tiago3434]

Thanks, Dale, that was really helpful! I can visualize the entire phenomenon in my head much better now. But just to make sure I understand this thing, the whole thing about adding vectors that are almost perpendicular vs almost parallel doesn't apply to the line, but would for the plane? Why?

 

[Dale]

the whole thing about adding vectors that are almost perpendicular vs almost parallel doesn't apply to the line, but would for the plane?

Sorry, I am not sure what you are referring to here.

 

[pixel]

OP is asking a good question. Why the intuitive argument of constant E that he quotes from Feynman for the infinite sheet of charge in post #1 doesn't also apply to an infinite line of charge. What is the essential difference in the two cases that makes the E field of the line of charge drop off with distance?

I'll try a hand-waving argument. When you are a given distance from the infinite plane, there’s an area below you that is mainly contributing to the E field in the direction perpendicular to the plane. If you double your distance from the plane, the contribution from each element on the plane is now 1/4 what it was before, but the number of mainly contributing elements increases by a factor of 4. For the infinite line, if you double your distance the number of mainly contributing elements of the line just doubles.

 

[Tiago3434]

Wow, pixel, thanks so much! Amazing answer, really. Now I can see it intuitively in my head.

 

[pixel]

Wow, pixel, thanks so much! Amazing answer, really. Now I can see it intuitively in my head.

Thanks for the great question - it made me think about it for a while. Basically, Feynman's argument is a way of intuitively justifying the constant E field of an infinite plane after knowing the result from calculation. It is not enough by itself to distinguish the two cases you asked about. An additional argument based on the geometry of the two cases is required.