Repeat Problem 22.38, but now let the conducting tube have charge per unit length -α. As in Prob

Repeat Problem 22.38, but now let the conducting tube have...

Key Concepts

Conductors in Electrostatics

Conductors in electrostatic equilibrium have the property that the electric field inside the material is zero. Charges in a conductor redistribute themselves on the surface to cancel any internal fields. This behavior, combined with the boundary conditions at the conductor’s surface, is crucial when solving electrostatic problems involving conducting tubes or shells.

Charge Distributions

Analyzing charge distributions involves understanding how charges are arranged on different objects, such as a central line charge and a surrounding conducting tube. In such problems, the net electric field is determined by the superposition of the fields generated by each separate charge distribution, which may include positive and negative linear charge densities.

Superposition Principle

The superposition principle states that the net electric field created by multiple charge distributions is the vector sum of the fields produced by each distribution independently. This principle is critical when dealing with systems that consist of both a line charge and a conducting tube, as it allows one to analyze each contribution separately and then combine the effects to understand the overall field configuration.

Gauss's Law

Gauss’s Law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the charge enclosed by that surface. It is particularly useful in cases with high symmetry, such as cylindrical symmetry, where it simplifies the computation of the electric field produced by line charges or uniform charge distributions.

Cylindrical Symmetry

Cylindrical symmetry occurs when a system’s geometry and charge distribution remain invariant under rotations about a central axis and translations along that axis. This symmetry is exploited in problems involving line charges and conducting tubes, as it allows for straightforward application of Gauss's Law and simplifies the analysis of the electric field distribution.

Transcript

00:02 Okay, so for this example, it's very similar to our last problem, but in this case, instead of having our hollow cylinder have a positive lambda or positive alpha, our charge density, our linear charge density, it's actually a negative.

00:25 So we're going to see how that changes things.

00:27 Okay, so the first thing that we do is write down gauss's law.

00:35 So here we go.

00:38 And again, we just have a situation where we have a holo cylinder in a radius a, out of radius b, with a linear charge density of alpha.

00:49 That's just charged the length.

00:50 And on the inside of it, we have a thin wire with a linear charge density of positive alpha.

00:59 Okay.

01:00 So, gauss's law says that the integral of e .d .a, which is the flux, closed line integral, actually, equals the total charge and closed divided by epsilon zero.

01:14 Now, this is cylindrical symmetry, which means that our da element is 2 pi times the radius that you're looking at, times the length that you go along the wire.

01:28 We can also rewrite q as the linear charge density, which is charge per length, times the length that we're looking at.

01:40 That would be what we call q enclosed, charge enclosed, in our gaussian surface.

01:49 So anyways, we can rewrite this as the following.

01:57 Alpha l over epsilon.

02:00 This is just total charged and closed.

02:02 Alpha times l divided by upsilon equals the electric field times 2 pi rl.

02:10 This is our area element in cylindrical coordinates.

02:14 E is the electric field that we are looking for.

02:19 So this is just galsh's law.

02:24 Okay? we arranging, we see a general relationship between the electric field, the linear charge density, the length along our segment that we're going, the length by the way is this length r is going radially outward l is horizontal so here we go and in general we can write it like this and this holds so this is our general formula for the electric field for this problem using cylindrical coordinates okay so now let's look at the electric field at different values for the radius.

03:13 All right, so four values of r less than a, that means that we are on the inner, on the inside of this hollow cylinder.

03:25 The only thing on the inside of that is our linear charge.

03:30 So we just have this wire of charge with linear charge density alpha.

03:37 So that just means that the total electric field simply plug it in.

03:42 Alpha divided by 2 pi r that's a distance you're at times epsilon zero that's a constant pretty easy what about if we are now inside of our hollow cylinder so in this blue region it's a color of blue so what if we are inside this region well we draw another gaussian surface but we also know that it is a conductor so the electric field must be zero inside of there every single time so we have the electric field is zero inside of this hollow cylinder because e equals zero inside conductors all right now let's look outside completely outside of this hollow cylinder okay well this time we need to use the principle of super position.

04:52 So you just add things together, basically.

04:56 So we are taking the electric field.

05:02 We are taking the electric field due to our charge, our linear charge...

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