A very long, solid cylinder with radius R has positive...

A very long, solid cylinder with radius $R$ has positive charge uniformly distributed throughout it, with charge per unit volume $\rho .$ (a) Derive the expression for the electric field inside the volume at a distance $r$ from the axis of the cylinder in terms of the charge density $\rho$ . What is the electric field at a point outside the volume in terms of the charge per unit length $\lambda$ in the cylinder? (c) Compare the answers to parts (a) and (b) for $r=R .$ (d) Graph the electric-field magnitude as a function of $r$ from $r=0$ to $r=3 R .$

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the net electric flux through a closed surface to the charge enclosed by that surface. It is especially useful in situations with high symmetry, such as a uniformly charged cylinder, because it allows one to choose an appropriate Gaussian surface that simplifies the calculation by exploiting the symmetry of the charge distribution.

Cylindrical Symmetry

Cylindrical symmetry occurs when a system’s properties are invariant under rotations about a central axis and translations along that axis. For a long charged cylinder, this symmetry implies that the electric field depends only on the radial distance from the axis and points radially outward, greatly simplifying the analysis by reducing a potentially complex three-dimensional problem to a one-dimensional one.

Uniform Charge Density

Uniform charge density means that charge is distributed evenly throughout the volume of the cylinder, which allows the calculation of the enclosed charge within any Gaussian surface to be straightforwardly determined by volume integration. This key concept is essential for applying Gauss's Law, as it ensures the charge enclosed is directly proportional to the volume of the chosen Gaussian surface.

Boundary Conditions

Boundary conditions refer to the requirements that the electric field must satisfy at interfaces or surfaces, particularly where two different regions meet. In the context of a charged cylinder, matching the electric-field expressions at the surface (where the inside and outside fields converge) ensures that there are no discontinuities in the field unless a surface charge is present, providing consistency in the physical description.

Graphical Analysis of Electric Field

Graphical analysis involves plotting the magnitude of the electric field as a function of distance from the axis to visually interpret how the field changes both inside and outside the charged cylinder. This analysis not only helps in understanding the behavior of the field in different regimes, such as the linear increase inside the cylinder and the inverse distance dependency outside, but also reinforces the underlying principles of symmetry and Gauss's Law.

Transcript

00:01 Here we're going to find the radial electric field both inside and outside a insulating cylinder with a uniform charge density inside.

00:13 And in order to do that, we're going to be using galses law, which says that the electric flux through a closed surface, which is a product of the electric field dotted into the normal to the surface, added up all the way around, is equal to the enclosed charge over epsilon knot.

00:55 So you almost never actually do the integral on the left -hand side.

01:05 For us, and we'll start off with inside the cylinder, but what to imagine is a little gaussian surface.

01:15 I'll draw that in green with an unspecified r and an unspecified length l.

01:24 So it's something that just exists in your imagination, but has the same symmetry as your object.

01:32 That's holding the charge.

01:36 And what we know is that the electric field will point outwards through that surface, perpendicular.

01:42 In the radial direction.

01:46 So we'll have the radial electric field times the surface area, 2 pi little rl, will equal to the enclosed charge, which will be the density times the volume of the small cylinder.

02:08 So the small cylinder has surface area equal to 2 pi rl, and it has voles.

02:18 Volume equal to pi r squared times l.

02:27 So the density times the volume gives us the enclosed charge, and then we have to divide by epsilon knot.

02:39 And we can do a little bit of cleaning up here.

02:43 Lots of things will cancel, including the unspecified length.

02:58 Now there is another way to specify this.

03:02 So we can say that the row is the q total over the full volume of the similar cylinder, but the full thing stretching all the way out to the edge of the insulator.

03:32 So the density is equal to row.

03:40 The row is q total over pi r squared.

03:53 So that's the full charge extending all the way out.

03:57 And we can further write this as 1 over pi squared, pi r squared, times a linear charge density.

04:10 So lambda is q total over l.

04:16 So an alternative way to write this that has the same functional form but involves the linear charge density.

04:27 Okay, or er is equal to lambda r over 2 pi epsilon r squared.

04:48 Okay, so let's take a look at outside.

04:51 And again, there will be two alternative ways to write the density.

04:56 But outside, what what we want to do is draw a bigger calcium surface.

05:08 And the difference between those, the geometry is going to be the same.

05:14 But the difference between these is that your charge distribution ends at the edge of the insulator...