A thin, flat sheet of charge has a uniform surface charge density σ(σ/ 2 on each side). (a) Sket

A thin, flat sheet of charge has a uniform surface charge...

A thin, flat sheet of charge has a uniform surface charge density $\sigma(\sigma / 2$ on each side). (a) Sketch the field lines due to the sheet. (b) Sketch the field lines for an infinitely large sheet with the same charge density. (c) For the infinite sheet, how does the field strength depend on the distance from the sheet? [Hint: Refer to your field line sketch.] (d) For points close to the finite sheet and far from its edges, can the sheet be approximated by an infinitely large sheet? [Hint: Again, refer to the field line sketches.] (e) Use Gauss's law to show that the magnitude of the electric field near a sheet of uniform charge density $\sigma$ is $E=\sigma /\left(2 \epsilon_{0}\right)$.

A thin, flat sheet of charge has a uniform surface charge density $\sigma(\sigma / 2$ on each side). (a) Sketch the field lines due to the sheet. (b) Sketch the field lines for an infinitely large sheet with the same charge density.
(c) For the infinite sheet, how does the field strength depend on the distance from the sheet? [Hint: Refer to your field line sketch.] (d) For points close to the finite sheet and far from its edges, can the sheet be approximated by an infinitely large sheet? [Hint: Again, refer to the field line sketches.] (e) Use Gauss's law to show that the magnitude of the electric field near a sheet of uniform charge density $\sigma$ is $E=\sigma /\left(2 \epsilon_{0}\right)$.

Key Concepts

Electrostatic Symmetry

Symmetry in a charge distribution simplifies the calculation and understanding of electric fields. In the case of a flat, uniformly charged sheet, the symmetry means that the electric field must be perpendicular to the sheet and uniformly distributed, which is a key reason why the infinite sheet has a constant field magnitude at all distances.

Field Lines

Field lines provide a visual representation of the electric field, indicating both direction and relative magnitude by their orientation and density. For an infinite sheet, the field lines are parallel and uniformly spaced, reflecting a uniform field, while for a finite sheet, the lines spread out near the edges, indicating variations in field strength due to edge effects.

Gauss's Law

Gauss’s law states that the net electric flux through any closed surface is proportional to the total enclosed electric charge. This principle is especially useful when dealing with symmetric charge distributions, as it allows for straightforward calculation of the electric field by choosing an appropriate Gaussian surface.

Electric Field of a Charged Sheet

A uniformly charged sheet produces an electric field that is perpendicular to its surface. When the sheet is approximated as infinite, the field exhibits uniform magnitude and direction on both sides, independent of the distance from the sheet, aside from a sign change. This concept is fundamental in understanding fields created by planar charge distributions.

Transcript

00:01 So if we have a thin sheet of uniform surface charge density, we'll call it sigma.

00:09 And they say sigma over two split in half, right? one on the top, one on the bottom to make a total of sigma.

00:16 First thing we want to do is we want to draw the field lines.

00:21 And you can imagine that the start off coming straight off the sheet.

00:27 And on the other side, you know, down.

00:35 As they get further from the sheet, they're going to deviate.

00:38 In fact, as they get closer to the edge, they're already going to be deviating and looping around.

00:50 And why is that? so when you have two charges next to each other that are both the same, say they're both positive, separately they're going to have radial fields coming out of them.

01:05 The one from the right, it's got an up and left component.

01:08 The one on the left has an up and right component.

01:09 When they're very close to each other, these are going to cancel out.

01:12 And symmetrically around, they're going to cancel out.

01:15 But we know that like charges repel each other.

01:20 Basically what the field lines look like when they're close enough to each other is.

01:24 Over on the side, they'll still look the same, somewhat radial, but they'll be sort of pushing the other field lines away from each other because we want zero electric field right down the line there.

01:41 Basically, you have an infinite number of charges doing this with each other, all in the middle there.

01:47 They're all going to be canceling out so that they point straight up until they get further and further away from each other.

01:51 So we can start getting this deviation of electric field lines off to the side.

01:57 And off on the very edge, now this component over here might be canceled by the one to its left, but there's nothing on its right to cancel out these components for electric fields.

02:11 There is something directly below it, though.

02:15 So we'll have the electric field lines going outward, but not crossing, obviously.

02:21 So they start off going straight out from the sheet, and then as you get further away, they sort of pull away from each other so that there's a line here that doesn't get crossed.

02:34 There's a line here that doesn't get crossed.

02:39 Now the question is, what about an infinitely large sheet? let's pretend this is infinitely large...

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