Electricity and Magnetism
# Magnetic Flux, Induction, and Ampere's Circuital Law

$B$ thin charged ring is rotating around it's own axis. Mass of ring is $m$ and charge is $q$. Find the velocity of precession ring's axis around magnetic field line passing through the centar of ring.

In homogenous magnetic field of induction$B=0.1T$

$q=3C$

$m=0.01kg$

One day, I was out bicycling when a thunderstorm cropped up. While frantically pedaling home at a speed of $10~\mbox{km/hr}$, a wind blew a power line down and it brushed against the inside of my front wheel, putting a positive $1~\mbox{C}$ charge on the wheel. I kept pedaling. If my bicycle wheels have radius $0.3~\mbox{m}$, what is the magnitude of the generated magnetic field **in Teslas** that I on my bike would measure at a distance of $0.1~\mbox{m}$ perpendicular from the center of my front wheel?

**Details and assumptions**

- Treat the wheels as thin conducting hoops.
- You may assume the charge has spread out over the wheel but not left it yet.
- My bike wheel was not slipping on the ground as I traveled.
- The vacuum permeability is $\mu_0=4\pi \times 10^{-7}~\mbox{H/m}$.

You probably know Gauss Law very well. It states that if you take any closed surface, the electric flux through this surface is proportional to the total charge enclosed. Mathematically: $\Phi_{E}:=\oint \vec{E} \cdot d\vec{A}=\frac{Q_{enc}}{\epsilon_{0}}.$ What about the magnetic flux? It turns out that $\Phi_{M}:=\oint \vec{B} \cdot d\vec{A}=0 \quad (\textrm{always!}).$ This is a Law of Nature, equivalent to one of Maxwell's equations and it reflects the experimental fact that there are no magnetic charges . In particular, $\Phi_{M}=0$ implies that not every magnetic field configuration can be realized in nature. For example, one can show that it is impossible to have a magnetic field that increases along the z-axis having only a z-component.

Consider an axially symmetric field with z-component (the field is symmetric about the z-axis) given by $B_{z}=B_{0}+ b z$ where $B_{0}= 2~ \mu \mbox{T}$ and $b=1 ~\mu \mbox{T/m}$.

Show that in addition to the z-component, this field must have a radial component $B_{r}$. Find $|B_{r}|$ **in Teslas** at a point located $50~\textrm{cm}$ away from the z-axis.

**in Volts** induced in the circuit.