SSM WWW In Fig. 23-56, a nonconducting spherical shell of...

SSM WWW In Fig. $23-56,$ a nonconducting spherical shell of in- ner radius $a=2.00 \mathrm{cm}$ and outer ra- dius $b=2.40 \mathrm{cm}$ has (within its thick- ness) a positive volume charge density $\rho=A / r,$ where $A$ is a constant and $r$ is the distance from the center of the shell. In addition, a small ball of charge $q=45.0 \mathrm{fC}$ is located at that center. What value should $A$ have if the electric field in the shell $(a \leq r \leq$ b) is to be uniform?

SSM WWW In Fig. $23-56,$ a
nonconducting spherical shell of in-
ner radius $a=2.00 \mathrm{cm}$ and outer ra-
dius $b=2.40 \mathrm{cm}$ has (within its thick-
ness) a positive volume charge
density $\rho=A / r,$ where $A$ is a constant
and $r$ is the distance from the center
of the shell. In addition, a small ball of
charge $q=45.0 \mathrm{fC}$ is located at that
center. What value should $A$ have if
the electric field in the shell $(a \leq r \leq$
b) is to be uniform?

Key Concepts

Gauss's Law

An essential principle in electromagnetism, Gauss's Law relates the net electric flux through a closed surface to the enclosed charge. It is particularly useful in problems with symmetric charge distributions, as it simplifies the calculation of the electric field by reducing complex integrals to simple algebraic equations.

Superposition Principle

This principle states that the total electric field produced by multiple charges is the vector sum of the individual fields generated by each charge. In systems with both discrete charges and continuous charge distributions, the superposition principle allows one to calculate the resultant electric field by adding their contributions.

Volume Charge Distribution

Describes how electric charge is distributed throughout a volume, typically represented by a charge density that may vary with position. Understanding this concept is crucial when the charge density is not constant, as it requires integration over the volume to determine the total enclosed charge.

Spherical Symmetry

A condition where a system's properties are invariant under any rotation about a central point. Spherical symmetry greatly simplifies the analysis of electric fields because the field depends only on the radial distance from the center, allowing the use of spherical coordinates in calculations.

Uniform Electric Field

A uniform electric field is one that has the same magnitude and direction at every point within a given region. Achieving and analyzing a uniform electric field is important in various applications, and it often requires specific arrangements or distributions of charge to ensure that the resulting field remains constant over the region of interest.

Transcript

00:02 Okay, so i like this problem.

00:05 Basically, here you have, again, the same players in the story, a shell, a charge in the middle.

00:15 However, the shell has a nice property that its density changes as you go outward.

00:22 So it gets smaller and smaller.

00:25 So row equals a over r.

00:31 And you're given that this charge in the middle is 45 femtoculomes and you're given a so plot spoiler alert i think that's the only one that you actually need just two centimeters and never too early to convert a number to si so 0 .02 meters and the goal is to get figure out what a is such that this um the field is spatially uniform even though the charge density isn't.

01:18 So basically first we need to solve for the field within here.

01:22 So we're going to use galses law to do that.

01:25 I'm going to do that on the next page.

01:29 Oh, why are there three? okay, well now i'm on two.

01:33 Okay, so gouse's law, i'll redraw the sphere.

01:41 You want to put your gaussian surface within here.

01:47 And so this is, now we're going to be able to see.

01:50 Evaluating the electric field at some radius are.

01:55 So start with our usual gauces law.

01:59 So e times the integral of e .da equals q and closed over epsilon not.

02:10 And a is, we chose our galsian surface nicely such that it was, excuse me, such that the integral is a constant.

02:26 So it doesn't change river space.

02:31 And the integral of da is just 4 pi r squared.

02:35 The dot product of e and da are the same everywhere along our surface.

02:41 And so, yeah, we get this left side, which you've probably seen before.

02:46 But q enclosed is a little more complicated because q is changing in space.

02:51 So we do need to integrate to figure out that.

02:53 So i'm going to kind of do this side integral over here.

02:56 I'll do it in blue because i kind of anticipate a lot of stuff on this page.

03:03 So q and closed, it's just in general, it's the integral of the density times a dv -d -volume element.

03:12 And our density is a over our, oh, look at that integral sign.

03:17 And the element of, like the differential element for spherical coordinates, especially, especially one that's symmetric, asymmutally, and with respect to all angles, basically, that's going to be r -squared 4 -pi dr.

03:41 So hopefully you've seen that in some kind of calculus class.

03:46 So that cancels that, and then we can pull our 4 -pi out.

03:51 And the bounds of our integral are from r to a, because that's where the charge is.

03:57 So it's just an integral of r...

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