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

Magnetic fields are wondrous things, bound by geometric relationships to the moving currents that generate them. Learn these links and the things they govern, from transformers to electric motors.

\(B=0.1T\)

\(q=3C\)

\(m=0.01kg\)

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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