Calculate Electric Field and Magnetic Field

[noamsgl]

Homework Statement

Given a long cylinder with radius R, is charged with a homogeneous charge density ρ and is rotating with frequency w around it's symmetry axis.

A. Find the Electric Field E in every point in space.
B. Find the Magnetic Field B in the cylinder with distance r from the axis of symmetry (r<R)

ρ,w,R ------> E=?, B=?

Homework Equations

Angular Velocity = w*r
[PLAIN]https://www.physicsforums.com/latex_images/25/2542612-8.png

Some equations with charge density and velocity... I=J*L or something.

The Attempt at a Solution

I am clueless about the answer to A and the Electric field, although I am sure it is more of a logic problem than some long equation/calculation. I'm just not sure what to write though.

I have tried and am optimistic about my approach to B and the Magnetic field.
I might be totally off-track but this is what I tried.
I guessed that this problem is analogous to many many long coils stacked inside one-another, and that I want to find the current as a function of a small radius r (r<R).

I wanted to find the electric current I(r) and then plug it into Ampere's law, what I got so far is this:

dI= J*L*dr = ρvL*dr=ρ*w*r*L*dr

Now I need to integrate all these little 'dI's from r to R (right?).
So I tried it and this is where I think I made a mistake.

I_enc = /int[r][R] ρ*w*L*r*dr = ρ*w*L*pi*r²

Then I took that and plugged it into Ampere's law, but apparently my answer is incorrect.

Thanks a ton to whoever spends the time helping me. Please help me understand why this is incorrect and not just provide an answer, and also give me some tips on the Electric field. Many thanks.

 

[anathema]

Thank you for posting your question. Let's start by looking at part A, finding the electric field E in every point in space. The key here is to use Gauss's law, which states that the electric flux through a closed surface is equal to the charge enclosed by that surface divided by the permittivity of the medium. In this case, we can use a cylindrical Gaussian surface with radius r and length L, centered on the axis of symmetry of the cylinder. The electric field will be radial, pointing outward from the center of the cylinder.

Using Gauss's law, we can write:

∮E⃗⋅dA⃗=qenc/ε0

Where qenc is the charge enclosed by the Gaussian surface and ε0 is the permittivity of free space. Since the cylinder has a homogeneous charge density ρ, the charge enclosed by the Gaussian surface will be:

qenc=ρπr2L

Substituting this into the equation above and taking the integral over the surface, we get:

E⃗=ρr/2ε0

This is the electric field at any point inside the cylinder, as well as outside the cylinder (since the charge is distributed uniformly).

Moving on to part B, finding the magnetic field B at a distance r from the axis of symmetry (r<R). To solve this, we can use Ampere's law, which relates the magnetic field to the current enclosed by a closed loop. In this case, we can use a circular loop with radius r, centered on the axis of symmetry.

Using Ampere's law, we can write:

∮B⃗⋅dl⃗=μ0Ienc

Where Ienc is the current enclosed by the loop and μ0 is the permeability of free space. The current enclosed by the loop can be calculated by using the equation you mentioned in your attempt, I=JL. However, we need to be careful with the units. The current density J is given in units of A/m2, so we need to multiply it by the area of the loop (πr2) to get the total current enclosed.

Substituting this into the equation above and taking the integral over the loop, we get:

B⃗=μ0ρwr/2

This is the magnetic field at a distance r from the axis of symmetry. Note that this result is independent