# Magnetic field inside a cylinder which is rotating in a non-constant angular velocity

Tags: angular, cylinder, field, inside, magnetic, nonconstant, rotating, velocity
 P: 4 1. The problem statement, all variables and given/known data 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? 2. Relevant equations Maxwell correction for Ampere law. 3. 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
 P: 15 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
 P: 4 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
P: 15

## Magnetic field inside a cylinder which is rotating in a non-constant angular velocity

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 $\rho$ .
 P: 4 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]

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