A long, straight wire has fixed negative charge with a...

A long, straight wire has fixed negative charge with a linear charge density of magnitude 3.6 $\mathrm{nC} / \mathrm{m}$ . The wire is to be enclosed by a coaxial, thin-walled nonconducting cylindrical shell of radius 1.5 $\mathrm{cm}$ . The shell is to have positive charge on its outside surface with a surface charge density $\sigma$ that makes the net external electric field zero. Calculate $\sigma$ .

A long, straight wire has fixed negative charge with a linear charge density of magnitude 3.6 $\mathrm{nC} / \mathrm{m}$ . The wire is to be enclosed by a coaxial, thin-walled nonconducting cylindrical shell of
radius 1.5 $\mathrm{cm}$ . The shell is to have positive charge on its outside surface with a surface charge density $\sigma$ that makes the net external
electric field zero. Calculate $\sigma$ .

Key Concepts

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is particularly useful in problems with high-symmetry, such as cylindrical symmetry, where it can be used to calculate the electric field by choosing appropriate Gaussian surfaces that simplify integration.

Cylindrical Symmetry

Cylindrical symmetry implies that the physical quantities of the system depend only on the radial distance from the axis, not on the angular or longitudinal coordinates. This symmetry allows the selection of Gaussian surfaces that are well-suited for applying Gauss’s Law, thereby simplifying the calculations of electric fields around line charges and cylindrical shells.

Electric Field of a Line Charge

The electric field due to an infinitely long, uniformly charged line can be derived using Gauss’s Law and is inversely proportional to the radial distance from the line. This concept is essential for understanding how the electric field behaves in the vicinity of the charged wire and provides the basis for determining the field that must be counteracted by the cylindrical shell.

Electric Field of a Charged Cylindrical Shell

A thin-walled cylindrical shell with charge distributed on its surface produces an electric field that, for points outside the shell, can be treated as though all the charge were concentrated along its axis. This result, derived using Gauss’s Law, is key to understanding how the shell’s charge influences the net external field.

Superposition Principle

The principle of superposition states that the total electric field created by multiple charge distributions is the vector sum of the electric fields due to each distribution alone. This principle is critical in combining the fields from the line charge and the cylindrical shell to achieve a net external electric field of zero.

Transcript

00:02 Okay, so this problem says you have a wire, and the wire is surrounded by a non -conducting coax.

00:16 And then you're given the linear charge density.

00:21 I'm going to call this lambda i for inner, and that's 3 .6 times 10 to the minus 9 kuloms per meter.

00:33 And the goal is to get the charge density on the outer surface.

00:43 So i'll use sigma for density, for surface density, and o for outer.

00:55 So yeah, basically the idea is i think that you have some charges here.

00:59 I think they were positive.

01:00 I'm just gonna double check the problem in another window.

01:03 So, yeah, so charge outside surface.

01:14 And then you want the net electric field to be zero.

01:17 So yeah, so you have this positive charge here and you want some, you're going to put some negative charges here, and you want the total electric field, i think on the outside to be zero.

01:31 And so the idea is you just want to have the same amount of charge on the inside as you do on the outside.

01:36 So i'm going to write this at, so this is how i approached it.

01:42 I'm sure there's different ways.

01:45 So don't think this is the only way to formally write it.

01:48 But it first struck me to say, okay, well, i need the charge per unit length to be the same on the outside as it is in the inside.

01:56 So you can say lambda inner equals lambda outer.

02:01 And so now we need to kind of put lambda outer in terms of sigma outer because that's really what we're after.

02:09 So what is sigma outer? so sigma outer is going to be the total amount of, so given some kind of cylindrical surface, it's the total amount of charge on that surface divided by the area of that surface.

02:25 And then the area of a cylindrical shell is 2 pi r times the length of that shell.

02:34 So i'll go ahead and draw that...

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