The Spinning Cylinder (Corrected)

A rotating charged hollow cylinder can be considered as a solenoid. Hence, the magnetic field inside it should be $\mu_0 \times$ the current per unit length just like a solenoid.

Charge per unit length of the cylinder in terms of surface charge density is $\sigma 2 \pi r$ .

A rotating charge can be considered as current equal to $\dfrac{q}{T}$ or $\dfrac{q \omega}{2 \pi}$ .

The current per unit length = $\dfrac{\text{Charge per unit length}}{\text{time period}}$

$= \dfrac{\sigma 2 \pi r\omega}{2\pi}$

$= \sigma r \omega$

Magnetic field = $\mu_0 \sigma r \omega.$

The corrent flowing through the cylinder among a piece of lenght "h" is:

$I=\sigma \omega r h$

Using the Ampère law for a closed,rectangular line with base "h" that extends in and above the cylinder we get:

$\int B ds= \mu i$ we divide the integral in four parts corrisponding at the sides of the rectangular line, 2 parts of the integral cancel due to the scalar product since two sides are perpendicular to the magnetic field, and a third cancel since is out of the cylinder, where there isn't any magnetic field. So we get:

$Bh=\mu \sigma \omega rh$ so

$B= \mu \sigma \omega r$

Assume that the cylinder is oriented along the z axis.

Differential surface area and differential charge:

$dA = 2 \pi r \, dz \\ dQ = \sigma \, dA = 2 \pi \sigma r \, dz$

Current associated with differential ring of height $dz$ :

$I = \frac{dQ}{T} = \frac{dQ}{2 \pi / \omega} = \sigma \omega r \, dz$

Parametrize the surface:

$x = r \cos\theta \\ y = r \sin\theta \\ z = z \\ \vec{dl} = dx \, \hat{\imath} + dy \, \hat{\jmath} + dz \, \hat{k} = -r \sin\theta \, d\theta \, \hat{\imath} + r \cos\theta \, d\theta \, \hat{\jmath} + 0 \hat{k}$

Displacement vector to origin:

$\vec{r} = -r \cos\theta \hat{\imath} - r \sin\theta \hat{\jmath} - z \hat{k}$

Cross-product of displacement vector with differential path vector:

$\vec{dl} \times \vec{r} = (-rz \, \cos\theta \hat{\imath} -rz \, \sin\theta \hat{\jmath} + r^2 \hat{k}) d\theta$

Biot-Savart integral for total B-field at the origin:

$\vec{B} = \frac{\mu_0}{4 \pi} \int_{-\infty}^{\infty} \int_{0}^{2 \pi} \frac{I \vec{dl} \times \vec{r}}{|\vec{r}|^3} \\ = \frac{\mu_0}{4 \pi} \int_{-\infty}^{\infty} \int_{0}^{2 \pi} \frac{\sigma \omega r \, (-rz \, \cos\theta \hat{\imath} -rz \, \sin\theta \hat{\jmath} + r^2 \hat{k})}{(r^2 + z^2)^{\frac{3}{2}}} d\theta \, dz$

Evaluating the $\theta$ integral and noting that the $x$ and $y$ components integrate to zero gives:

$\vec{B} = \frac{\mu_0 \sigma \omega r^3 \hat{k}}{2} \int_{-\infty}^{\infty} \frac{1}{(r^2 + z^2)^{\frac{3}{2}}} dz \\ = \frac{\mu_0 \sigma \omega r^3 \hat{k}}{2} \frac{2}{r^2} = \boxed{\mu_0 \sigma \omega r \, \hat{k}}$

The proportionality constant out in front is thus 1