Symmetry Arguments-a finite cylindrical can

[jromega3]

Symmetry Arguments--a finite cylindrical can

Homework Statement

Consider a finite cylindrical can shape that has charge uniformly distributed on its surface. Symmetry does allow us to say some things about the electric field of this distribution
A) at points along the can's central axis
B) At points lying on the plane that cuts the can in half perpendicular to its central axis
C) at the can's exact center

What does symmetry tell us in these three cases?

Hint: be sure to consider both the direction of the electric field and on what variables it might or might not depend

Homework Equations

The Attempt at a Solution

Ok, first off, I'm not even sure what I'm supposed to be doing exactly, and is the shape just like a soda can, and the surface the outside not including the top/bottom, ie the body?

a and b to me are confusing. I could see them being zero, but at the same time, they depend on the point you're using on the plane, so I believe you'd have to use an integral, but that'd be a double integral and that seems to complex for this problem.
For A it seems as if it's completely symmetrical across the plane, and it seems to be true for B as well. Meaning there'd be no net charge on either. hmm

c) I'd imagine this has to be zero. Perfect symmetry in all directions and uniform distribution throughout.

So I guess I'm just looking for a start on where I should be heading for and what I should be trying to solve. Thanks, as always, in advance for the great help. Truly appreciated.

 

[Redbelly98]

Hmm, I interpreted a "can shape" as including the top and bottom. However, I don't think that will affect these questions.

(a) and (b) have more to do with what direction the field points toward. They don't want the actual value of the field, just its direction, so no integration is necessary.

You're correct about (c).

 

[thomate1]

Hello,

I would like to provide a response to the content regarding symmetry arguments for a finite cylindrical can.

Firstly, the shape of the can is not specified in the problem, so we can assume it to be a general cylindrical shape with a finite length and radius. The charge is uniformly distributed on the surface of the can, which means that the charge density is constant throughout the surface.

Now, let's consider the three cases mentioned in the problem:
A) At points along the can's central axis: In this case, symmetry tells us that the electric field will be pointing in the direction of the central axis at all points along it. This is because the charge distribution is symmetric about the axis, and the electric field lines will be perpendicular to the surface of the can at these points. The magnitude of the electric field will depend on the distance from the central axis, but not on the angle around the axis. In other words, the electric field will be independent of the azimuthal angle (the angle around the central axis).

B) At points lying on the plane that cuts the can in half perpendicular to its central axis: In this case, the electric field will be zero. This is because the charge distribution is symmetric about this plane, and the electric field lines will cancel out in pairs, resulting in a net electric field of zero. The electric field will depend on the distance from the plane, but not on the angle around the plane. So, the electric field will be independent of the polar angle (the angle from the central axis to the point on the plane).

C) At the can's exact center: In this case, the electric field will also be zero. This is because the charge distribution is symmetric about the center point, and the electric field lines will again cancel out in pairs, resulting in a net electric field of zero. The electric field will be independent of both the azimuthal and polar angles, as the center point is equidistant from all points on the surface of the can.

In summary, symmetry arguments tell us that the electric field will be zero at points lying on the plane perpendicular to the central axis and at the exact center of the can. At points along the central axis, the electric field will be pointing in the direction of the axis, and its magnitude will depend on the distance from the axis. These conclusions are based on the assumption of a uniformly charged surface on the can, and may change if the charge