(II) A thin cylindrical shell of radius R1 is surrounded by...

(II) A thin cylindrical shell of radius $R_{1}$ is surrounded by a second concentric cylindrical shell of radius $R_{2}$ (Fig. 35$)$ . The inner shell has a total charge $+Q$ and the outer shell $-Q .$ Assuming the length $\ell$ of the shells is much greater than $R_{1}$ or $R_{2},$ determine the electric field as a function of $R$ (the perpendicular distance from the common axis of the cylin- ders) for $(a) 0<R<R_{1},(b) R_{1}<R<R_{2},$ and $(c) R>R_{2}$ . (d) What is the kinetic energy of an electron if it moves between (and concentric with the shells in a circular orbit of radius $\left(R_{1}+R_{2}\right) / 2 ?$ Neglect thickness of shells.

Key Concepts

Gauss's Law

Gauss's Law relates the net electric flux through a closed surface to the enclosed charge divided by the permittivity of free space. This principle is central to determining electric fields in systems with high symmetry, as it allows for the use of appropriate Gaussian surfaces to simplify complex problems.

Cylindrical Symmetry

Cylindrical symmetry implies that the physical properties of a system, such as charge distributions and resulting electric fields, remain unchanged under rotations about a central axis. This symmetry allows one to simplify the expressions for fields and potentials, making the analysis of coaxial cylindrical shells much more tractable in electrostatics.

Electric Field from Charged Cylindrical Surfaces

In systems involving charged cylindrical shells, the electric field can be derived by considering Gaussian surfaces that are co-axial with the shells. Depending on the region (inside, between, or outside the shells), the amount of charge enclosed changes, leading to different expressions for the electric field. This method is rooted in using Gauss's Law in a context of cylindrical symmetry.

Electric Potential and Energy Conversion

Electric potential relates to the work done in moving a charge within an electric field. In this context, computing the potential difference between two points facilitates the determination of the kinetic energy acquired by a charged particle moving in the field. This conversion from potential energy to kinetic energy is fundamental in understanding particle dynamics in electric fields.

Transcript

00:01 Okay, so we're doing chapter 22, problem 35 here.

00:04 So this says a thin cylindrical shell of radius r1 is surrounded by a second concentric cylindrical shell of radius r2.

00:15 The inner shell has total charge q and the outer shell total charge negative q.

00:26 Assuming the length l of the shells is much greater than the radius is, much greater than r1 or r2, determine the electric field as a function of r for different distances.

00:39 Okay.

00:40 So, since we're talking about cylindrical shells, we're going to be using a gaussian cylinder around our shells here to solve for this.

00:53 And we kind of talked about this in a solution for 33, so if you're confused, go back to that one first.

00:59 But we can see that gauss's law here, e .da.

01:05 If we make our surface a cylinder or surrounding the cylinders, we can see that the electric field is going to be perpendicular at all points.

01:13 And now we pull the electric field out times the surface area.

01:21 And this is equal to what galsazol says, which is the charge and closed over epsilon knot.

01:26 Meaning that a gaussian cylinder has an electric field that's radial and goes as q1.

01:38 Enclosed over 2 pi epsilon not r l so this is what we're going to use mainly and we might need to go back and look at different examples following 226 if you're confused at this point example 226 in the book okay so for part a we're trying to figure out the electric field for r less than r1 so for this case we should see immediately that the charge enclosed is zero so obviously the electric field is zero for this case.

02:16 Okay, so moving on to part b, now r is greater than r1, less than r2.

02:24 Sorry, that is just a, oops, that should just be r there.

02:31 Perfect, okay.

02:33 For this case, we got to figure out what our charge enclosed is.

02:41 Well, if we're drawing our gaussian surface here, here means we're enclosing all the charge on the inner cylinder.

02:49 So that means our q enclosed is positive q, because that's what was on the inner cylinder...

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