Charge moving with a constant linear velocity....

[Ibix]

Yes, except this is only a theory and if such was true we would have a method of absolute speed measuring using excess charge body as a sensor...

No you would not. For the reason I've given twice, and vanhees71 and A.T. have given at least once each.

 

[Maciej Orman]

According to the link in my previous post, Rowland had a 20cm diameter ring charged to 10^-7C spinning at 50 revs/s. And was able to detect a magnetic field from it, which suggests epic experimental skills to me.

No, triboelectric charging method creates volume charge for which there is no method of measurement... Thus suggesting that the charge is much larger than estimated...

 

[Ibix]

No, triboelectric charging method creates volume charge for which there is no method of measurement... Thus suggesting that the charge is much larger than estimated...

Reference please.

 

[vanhees71]

Reference, please. Note that simply citing the Rowland disc is not sufficient, since the results are entirely explicable in terms of the linear speed of the charges (see p98-9 of https://books.google.co.uk/books?id=MbSUqqzFDocC&pg=PA98&lpg=PA98&dq=rowland+disc+experiment&source=bl&ots=0OzuvkesWW&sig=G_oqn3m8UNNQmnz97pz9_tQXVyQ&hl=en&sa=X&ved=0ahUKEwiI3dfXo_LVAhWGC8AKHbvyCWsQ6AEIDTAB#v=onepage&q=rowland disc experiment&f=false).

The point of Rowland's experiment (and other similar experiments in the late 19th century) was to proof that electric currents are due to moving charges ("electrons"). Nowadays that sounds trivial, because according to our present understanding all electric currents (and magnetizations) are due to elementary particles, but at this time the very nature of electric charges and currents was not that clear.

If I understand the excerpt from the textbook right, you have a charged ring of radius $R$. The corresponding current density is
$$\vec{j}(\vec{x})=\rho \vec{v}=\frac{q}{2 \pi R} \delta(\varrho-R) \delta(z) \omega \varrho \vec{e}_{\varphi},$$
where $(\varrho,\varphi,z)$ are standard cylinder coordinates. The corresponding magnetic field can be calculated using the vector potential (in Coulomb gauge),
$$\vec{A}(\vec{x})=\frac{1}{c} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \frac{\vec{j}(\vec{x}')}{4 \pi |\vec{x}-\vec{x}'|},$$
and finally the magnetic field as
$$\vec{B}=\vec{\nabla} \times \vec{A}.$$
The integral can be found in some good textbooks (like Jackson), leading to elliptic functions.