Two very large, nonconducting plastic sheets, each 10.0 cm thick, carry uniform charge densities

Two very large, nonconducting plastic sheets, each 10.0 cm...

Key Concepts

Electric Field of an Infinite Plane Sheet

For an infinitely large plane with a uniform surface charge density, the electric field is constant in magnitude and directed perpendicular to the surface. This field is independent of the distance from the plane because of the symmetry in the charge distribution, and its magnitude is given by the relation E = ?/(2??) for one isolated sheet, making it a fundamental concept when analyzing systems of planar charged objects.

Superposition Principle

The superposition principle in electrostatics asserts that when multiple electric field sources are present, the net electric field at any point is the vector sum of the individual fields produced by each source. This principle is crucial for handling systems with multiple charged surfaces, as it allows the separate calculation of the field contributions from each surface, which can then be added together to obtain the overall field.

Gauss's Law

Gauss's Law states that the net electric flux through any closed surface is equal to the enclosed charge divided by the permittivity of free space (??). This principle provides a powerful method for calculating electric fields when there is a high degree of symmetry, by relating the flux of the field through a carefully chosen Gaussian surface to the total enclosed charge.

Uniform Surface Charge Density

A uniform surface charge density implies that charge is distributed evenly across the surface, leading to a constant value of charge per unit area. This uniformity is key in simplifying the calculation of electric fields, as it allows the assumption that the contributions to the field from different regions are identical in magnitude, which is essential for applying Gauss's Law effectively.

Transcript

00:01 So in this example, i just kind of re -drew the picture that the book gave us.

00:06 And we need to first begin by remembering that when we have an infinite plane, that the electric field equals sigma divided by two epsilon and not.

00:17 For each of these locations, a, b, and c, we want to find the electric field and its direction.

00:25 The easiest way to do this is to remember the principle of superposition.

00:32 That is a mathematical concept that you can understand graphically.

00:39 So here we go.

00:42 Superposition simply means add things together.

00:46 Okay, take this one at a time.

00:50 Let's look at a first.

00:53 Okay, what is happening to a? what is influencing the electric field? well, from sigma 1, there's a negative charge.

01:03 That means that i'm going to have electric field lines going to the right.

01:12 Okay, that's from sigma 1.

01:15 Going to the right at a.

01:18 What about sigma 2? sigma 2, positive 5.

01:23 Positive, this is going to radiate away.

01:31 Or, when it goes through a, it's going to the left.

01:36 What about from sigma 3? this is positive.

01:42 Positive radiates away.

01:46 When it goes to a, when it hits a's location, it's going to the left.

01:51 What about sigma 4? sigma 4 is positive, radiates away.

01:56 When it does that, when it hits a, a's region, looks like these lines go into the left.

02:04 The trick here is that we need to identify a coordinate system.

02:09 We need to identify, well, what's positive and what's negative.

02:13 I'm going to go ahead and say that two the right is an increase in x.

02:18 I'm going to say that that's positive.

02:20 Everything to the left then would be a negative contribution.

02:25 So, sigma 1 is going to have a positive contribution.

02:30 Sigma 2 would have a negative contribution.

02:39 Sigma 3, a negative contribution.

02:42 And sigma 4, but also have a negative contribution.

02:48 So to figure out where the electric field is, at a, we know in general it's this guy.

02:58 So let's go ahead and factor out a 1 over 2 epsilon 0.

03:04 Epsilon 0 is just a constant.

03:07 And then what about these sigmas? how do we add them? well, although that this is a negative 6 microculem charge, that's just telling you the direction of the field lines.

03:19 We've already taken that into account by drawing these arrows.

03:24 So we need to do a positive six contribution, microculum contribution from that sigma 1.

03:41 Sigma 2 would have a negative, so minus 5.

03:46 Sigma 3 would have a negative minus 2.

03:51 Sigma 4 would have a negative minus 4.

03:58 So overall, we would have a negative 5 microculum contribution overall for our total charge, for our total sigma, and we simply divide that by 2 epsilon not.

04:20 Plug that into your calculator, and you get a 2 .8 times 10 to the 5th, 2 .81 times 10 to the 5th.

04:32 Newton per kuom and it's negative this is the magnitude the negative sign means it's going to the left okay let's try the field of b okay eb here we go let's get rid of these lines okay so when we look at b what do we see well what's happening from sigma 1 right here this is a negative micro 6 um charge if it's negative, it's drawing in.

05:23 So from sigma 1, those lines are going to radiate to the left.

05:30 That's a negative contribution now...

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