A Uniformly Charged Slab. A slab of insulating material has...

A Uniformly Charged Slab. A slab of insulating material has thickness 2$d$ and is oriented so that its faces are parallel to the $y z$ -plane and given by the planes $x=d$ and $x=-d$ . The $y$ -and $z$ -dimensions of the slab are very large compared to $d$ and may be treated as essentially infinite. The slab has a uniform positive charge density $\rho .$ (a) Explain why the electric field due to the slab is zero at the center of the slab $(x=0)$ . (b) Using Gauss's law,find the electric field due to the slab (magnitude and direction) at all points in space.

Key Concepts

Gaussian Surface Analysis

Gaussian surface analysis involves selecting an appropriate closed surface over which Gauss's Law can be applied to evaluate the electric field. The choice of surface is driven by the symmetry of the problem so that the electric field either has a constant magnitude on parts of the surface or is perpendicular or parallel to those surfaces, greatly simplifying the calculations. This method is particularly useful for deducing the electric field in or around objects with high symmetry, such as a uniformly charged slab or planar distributions.

Uniform Charge Distribution

A uniform charge distribution means that the charge density is constant throughout the object. This uniformity greatly simplifies the problem of finding the electric field, as it allows for the assumption that every part of the charge contributes similarly, and it makes using Gauss's Law more straightforward. It ensures that any variations in the electric field are due solely to the geometry of the charge distribution rather than fluctuations in the charge density.

Gauss's Law

Gauss's Law is a fundamental principle of electromagnetism which states that the electric flux through any closed surface is proportional to the enclosed electric charge. This law is particularly powerful when applied to systems with high symmetry, as it allows for the calculation of the electric field without directly performing complex integrations over the charge distribution.

Symmetry in Electrostatics

Symmetry plays a critical role in determining the behavior of the electric field in charge distributions. In highly symmetric systems, such as those with uniform charge distributions, the contributions to the electric field from different parts of the system can cancel out. This explains phenomena such as the field becoming zero at certain points, like the center of a symmetric slab, where the effects from charges on either side of the test point are equal and opposite.

Transcript

00:01 In this question, we want to find the electric field due to a uniformly charged slab.

00:06 So this charge slab extends along the yz plane and then it has a thickness of 2d.

00:22 Okay, here is minus d and then zero here.

00:32 Then we want to, there are two parts in this question.

00:37 First we want to explain why the electric fuel.

00:40 Is 0 at x equals to 0 line at the center of the slab and find the electric field everywhere.

00:50 So in this question for at x equals to 0 at the center of the slab, okay, it is symmetrical.

01:03 This line is symmetrical or it's a line of symmetry, yeah, it's a line of symmetry of the slab of the slab okay then you can see you can treat that as if there are to uniformly charge sheet slap yeah or slab being placed side by side and x equals to 0 okay so yeah and then the so the electric fuel and x equals to 0 by each slab will be so the electric fuel by each slab is parallel to the x -axis okay so you cannot point in y or z direction because it extends in the y z plane okay yeah then uh x because it goes zero uh those two those electric field are equal in magnitude opposite in direction at x equals to zero, the e is zero.

03:38 So what i'm saying is that, so i'm treating this as like there are two slaps that has placed side by side at x equals to zero and then slab one is generating a few pointing to the right and step two is pointing to the left and then you just add this two up and along the x equals to zero line and then it would just give you zero everywhere at x equals to zero okay another way to say that it's just zero by symmetry yeah it's kind of the same as saying that you are kind of treating treating this as two separate slabs okay yeah right so in part b we want to find an electric field everywhere and here we will be applying gauze's law okay, so first we will do inside the slab.

04:38 Okay, we will draw the slab here.

04:43 Okay, enlarge it.

04:49 Okay, so we have shown that the e is zero at x equals to zero.

04:55 So we can actually have our gauchian surface that looks like this.

05:01 Okay, so that on the x equals to zero line, that phase is going to have e equals to 0.

05:14 0 and we know that e is pointing like this.

05:18 Okay, and it's going to be uniform because constant on the on the x equals to like x...