Electrostatics – Why the Electric Field $\vec{E}$ is Constant for an Infinite 2D Sheet of Constant Charge?

electric-fieldselectrostaticsgauss-law

So I'm reading a text on electricity and it talks about using the integral to compute the total charge of a collection of points, which I mostly understand. But then we get to finding the electric field due to a charged collection of points and I find things that don't make sense to me. For instance, for an infinite sheet of constant charge, the text says that the electric field is constant on any one side of the sheet. But that seems intuitively wrong to me, since I would think the field should be stronger the closer a point is to the sheet. I mean, if I'm standing 10 meters from a sheet, holding a charged particle, and walk closer to the sheet, I'd think the particle would react more strongly. I follow the mathematical derivation fairly well, which leads me to think I must not be thinking about the physics correctly. Can anyone help make this make sense?

Best Answer

for an infinite sheet of constant charge, the text says that the electric field is constant on any one side of the sheet. But that seems intuitively wrong to me, since I would think the field should be stronger the closer a point is to the sheet.

There's a geometric scaling argument at hand, and you probably need to appreciate Gauss' Law to get a real sense of it. It follows the same thinking that the $1/r^2$ law does.

inverse square law

In this argument for $1/r^2$ field strength from a point particle, it is seen that the solid angle that "A" occupies decreases at larger radii. If you consider a charged ball, then think about squishing it from a 3D object into a 2D piece of paper. That represents the area-based charge density.

An unsaid assumption for this line of thinking is that the field strength is determined by:

  • The solid angle occupied by charged material

times

  • The 2D charge density presented by that material

So extend this to an infinite sheet. No matter how close or far away it is, if you're looking parallel to the sheet, the sheet occupies exactly half of your field of vision. Furthermore, the surface charge density and angles of the sheet are also irrelevant of your normal distance.

Illustratively, you could apply this via the image above. Just remove the "A", and consider that this is an infinite sheet. As you expand the distance, you expand the area you sweep, but the angle times the charge density is invariant. Thus, the field is constant with distance.

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