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

In summary, the conversation discusses a problem involving a hollow cylinder made of non-conducting material that is charged and rotating. The question asks about the magnetic field inside the cylinder and the manual provides a solution involving the product of the angular velocity, radius, and surface charge. The conversation then delves into deriving the circular electric field induced by the changing magnetic field and how it is assumed to be constant in time. The conclusion is reached by applying Gauss and Faraday's laws and inferring that the circular electric field is not changing with time.

## 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?

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)

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?

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 $\rho$ .

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]

$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$

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]

## 1. What is the magnetic field in a rotating cylinder with non-constant angular velocity?

The magnetic field in a rotating cylinder with non-constant angular velocity is a complex and dynamic phenomenon. It is dependent on various factors such as the cylinder's geometry, the material properties of the cylinder and the surrounding medium, and the time-varying angular velocity of the cylinder.

## 2. How does the magnetic field change with the angular velocity of the cylinder?

The magnetic field in a rotating cylinder with non-constant angular velocity changes in both magnitude and direction as the angular velocity of the cylinder changes. This is due to the induced currents and eddy currents that are generated in the cylinder and the surrounding medium.

## 3. Can the magnetic field be calculated analytically for a rotating cylinder with non-constant angular velocity?

In most cases, the magnetic field in a rotating cylinder with non-constant angular velocity cannot be calculated analytically. This is because the problem is highly non-linear and requires complex mathematical models and numerical methods to accurately predict the magnetic field.

## 4. What is the practical application of studying the magnetic field in a rotating cylinder with non-constant angular velocity?

Studying the magnetic field in a rotating cylinder with non-constant angular velocity has several practical applications, such as in the design of electrical generators, motors, and other rotating machinery. It can also be used to understand the behavior of magnetic fluids and to optimize the performance of magnetic levitation systems.

## 5. How does the magnetic field in a rotating cylinder with non-constant angular velocity affect the stability of the system?

The magnetic field in a rotating cylinder with non-constant angular velocity can affect the stability of the system by inducing vibrations and instabilities in the cylinder and surrounding medium. This can be mitigated by carefully designing the geometry and material properties of the cylinder and the surrounding medium.

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