Magnetic Field in Rotating Cylinder w/ Non-Const. Angular Vel.

[cadmus]

Homework Statement

A hollow cylinder of length L and radius R, is madeout of a non-conducting material, is charged with a constant surface charge σ, and is rotating, along its axis of symmetry, with an angular velocity w(t) = αt.

Q:What is the magnetic field inside the cylinder?

Homework Equations

Maxwell correction for Ampere law.

The Attempt at a Solution

The answer in the manual is B = μαtRσ

Where μ is ofcurse μ zero. [ the magnetic constant ].

The manual's solution makes perfect sense if I knew that the circular electric field which is induced by the fact that the magnetic field is changing in time is constant.

because then i could say that that the displacement current density is zero.

Q: How can derive that the circular electric field, induced by the changing -in-time magnetic field, is not changing with time?


Thanks in advance

 

[Planck const]

What is I?
its the relation betwwen the charge (you know it from sigma) and the period (you know it from w)
The charge inside the cylinder its Sigma*A(r) (and not all A!)
What is the integral of B*ds ?
its the product of B and the scale circuits that thir radius its r (r<R)

You need to replace those sizes into Amper equation.. and get B(r)

 

[cadmus]

Thanks for the reply.

But it did not address my question,

I would like to know why in this problem there is a certainty that the Electric field is not changing with time ?

id est, look at Ampere's Law after Maxwell correction:

$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\$

and of course, the integral form of this equation:

$\oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 I_S + \mu_0 \varepsilon_0 \frac {\partial \Phi_{E,S}}{\partial t}$

I could, with rather ease, derive the manual's solution if I knew that $\frac{\partial \mathbf{E}} {\partial t} \$ is zero.

Any notions about why E is constant in time?

Thanks in advance

 

[Planck const]

The answer is simple. (I will call sigma -> rho)
if dE/dt (partial derivative) = 0

==> (Gauss law)

d$\rho$/dt = 0

==> (math)

$\rho$ is constant in time. (stationary current)

And you can see in the problem data, that they didn't say anything about the function $\rho$ .

 

[cadmus]

Again, thanks for the reply.

The $\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}$ formula applies only to E filed that are not circular. [ I mean in order to derive the total electric field inside the cylinder you will have to find the E in the theta direction as well]

According to Faraday's Law:
$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$

Which means that a changing in time magnetic field induces a circular E field.

How can I infer that the circular E field is not changing by time? [ prior to calculating the magnetic field - because then i just use faraday law to see that]