Cylindrical Rod A very long conducting cylindrical rod of...

Cylindrical Rod A very long conducting cylindrical rod of length $L$ with a total charge $+q$ is surrounded by a conducting cylindrical shell (also of length $L$ ) with total charge $-2 q$, as shown in Fig. $24-32$. Use Gauss' law to find (a) the electric field at points outside the conducting shell, (b) the distribution of charge on the shell, and (c) the electric field in the region between the shell and rod. Neglect end effects.

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the total charge enclosed by that surface. It is particularly useful for calculating the electric field when the charge distribution has symmetry, such as in cylindrical systems.

Cylindrical Symmetry

Cylindrical symmetry implies that the physical setup is invariant under rotations about and translations along the axis of a cylinder. This symmetry simplifies the analysis of electric fields and charge distributions, as it reduces complex integrals to algebraic equations when applying Gauss’s Law.

Properties of Conductors in Electrostatics

In electrostatic equilibrium, the electric field inside a conductor is zero, and any excess charge resides on its surface. This principle is critical when analyzing multi-conductor systems, as the distribution of charge adjusts in response to the presence of nearby charges, ensuring that no field exists within the conducting material.

Charge Distribution on Conductors

The distribution of charge on conductors is governed by the need to maintain electrostatic equilibrium. Charges will redistribute themselves on the inner and outer surfaces of a conductor to cancel any internal fields. This redistribution is key to solving problems where conductors are influenced by nearby charges, leading to induced charges that depend on geometry and symmetry.

Electrostatic Shielding

Electrostatic shielding refers to the phenomenon where a conductor blocks external static electric fields, ensuring that the field within a cavity in the conductor remains zero. This concept is important when dealing with systems that involve nested conductors, as it helps in determining the field distribution in regions separated by conducting materials.

Transcript

00:01 Okay, so we're doing chapter 22, problem 33 here.

00:04 So it says a long cylindrical shell of radius big r0 and length little l, with r0 much smaller than l, so it's very long rod.

00:16 And it possesses a uniform surface charge density, sigma.

00:20 So the charge is uniform across the surface of the cylindrical shell.

00:25 So for part a, we want to determine the electric field at points r greater than r not, so outside of the cylinder.

00:38 So if we follow similarly how example 226 goes, we can draw a gaussian cylinder around our wire here, around the cylindrical shell of radius big r, and we should see that the electric field is going to be distributed perpendicular, to this surface at all points.

01:08 And that's what we need for gauss's law to apply.

01:12 So let's write out gauss's law here.

01:13 So we need the electric field if it's perpendicular to the surface at all points.

01:18 We then can pull it out and evaluate just the surface area.

01:27 And this then becomes e, the surface area of our cylindrical shell.

01:33 Then at this point is given by 2 pi r times the length l.

01:40 And this equals the charge enclosed over epsilon not.

01:46 From that we can rearrange and solve for what e is, and this is given as the surface density times the area enclosed, the surface area enclosed here, over 2 pi, epsilon, big r .l...

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