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

\(B=0.1T\)

\(q=3C\)

\(m=0.01kg\)

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.

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