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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.

Level 4-5

         

Consider a uniformly charged circular disk of radius \(R\) having charge \(Q\) rotating about a fixed point on its circumference with an angular velocity \(\omega\) . The magnetic field created at this point due to the motion is \(k \times \dfrac{\mu_{0} Q \omega}{R}\). Find the value of \(k\) .

In homogenous magnetic field of induction \(B\) 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.

\(B=0.1T\)

\(q=3C\)

\(m=0.01kg\)

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.

A wire is bent into the shape of an planar Archimedean spiral which in polar coordinates is described by the equation \[ r= b \theta. \] The spiral has \(N=100\) turns and outer radius \(R=10~\mbox{cm}\) (\(R\) is the distance from point O to point T). Note that in the figure below we show a spiral having only 3 turns. The circuit is placed in a homogeneous magnetic field perpendicular to the plane of the spiral. The time dependence of the magnetic field induction is given by \[ B=B_{0}\cos(\omega t)\] where \(B_{0}=1 ~\mu \mbox{T}\) and \(\omega=2\times 10^{6}~\mbox{s}^{-1}\). Determine, the amplitude of the emf in Volts induced in the circuit.

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