An infinitely long cylindrical conductor has radius R and...

An infinitely long cylindrical conductor has radius $R$ and uniform surface charge density $\sigma$. (a) In terms of $\sigma$ and $R,$ what is the charge per unit length $\lambda$ for the cylinder? (b) In terms of $\sigma,$ what is the magnitude of the electric field produced by the charged cylinder at a distance $r>R$ from its axis? (c) Express the result of part (b) in terms of $\lambda$ and show that the electric field outside the cylinder is the same as if all the charge were on the axis. Compare your result to the result for a line of charge in Example 22.6 (Section 22.4 ).

Key Concepts

Cylindrical Symmetry

Cylindrical symmetry occurs when a system is invariant under rotations about a central axis and translations along that axis. In problems involving long, uniformly charged cylinders, this symmetry allows one to choose a cylindrical Gaussian surface that simplifies the electric field calculation, as the field has the same magnitude at every point equidistant from the axis and points radially outward from the cylinder.

Surface Charge Density and Linear Charge Density

Surface charge density describes the amount of charge per unit area on a surface, while linear charge density refers to the charge per unit length along a line. In the context of a charged cylinder, the surface charge density can be converted to a linear charge density by multiplying it by the circumference of the cylinder. This conversion is crucial for relating properties of two-dimensional charge distributions to those of one-dimensional representations, which simplifies the analysis of the external electric field.

Gauss's Law

This is a fundamental law of electromagnetism stating that the net electric flux through a closed surface is proportional to the enclosed electric charge. It is particularly useful for calculating electric fields when high degrees of symmetry are present. By applying Gauss's law, one can relate the electric field to the amount of charge enclosed within a Gaussian surface that is chosen to exploit the symmetry of the problem.

Equivalence of Charge Distribution in External Field

For conductors with a uniform charge distribution, the electric field outside the conductor can often be treated as if all the charge were concentrated along the central axis. This principle, derived using Gauss's law, shows that the external field depends only on the total charge per unit length rather than on the detailed geometry of the charge distribution. It is analogous to results for line charges and helps to simplify comparisons between different configurations.

Transcript

00:02 So when answering kaos is law problems, you want to begin with writing it explicitly.

00:09 It says that the closed loop integral of e.

00:13 D.

00:14 A.

00:16 Equals the total charge.

00:17 Enclosed, divided by epsilon, not i was long is simply constant.

00:22 This guy's otherwise called fi the electric flux.

00:29 If you have a very, very long line of charge, then usually we say it has some linear charge density random.

00:39 And this is simply the charge on that line, divided by the length how long it iss or the region that you're looking at typically the surface charge area density sigma.

00:58 So for part a, we're kind of looking for sigma sigma is charge her area, but what is the area when you have a very, very small wire? well, in order to answer this question, you have to think of a cylinder.

01:23 When you have a cylinder, it has some base.

01:30 Maybe this is a distance d and some height.

01:35 Maybe the height is l.

01:38 The circumference of this is to hi, dean.

01:46 And so the surface area of this object, which is indeed what we would plug in right here.

01:55 It's simply the base times the hot two pi d times l.

02:06 Okay, so now that we kind of understand that it's a little bit easier to rearrange things.

02:14 So we have this l the length that we go on a cylinder.

02:21 We also have this l when we have this wire, let's explicitly plug it in...