W9_parallel_resonance.eps

Week 8: Faraday’s Law and Induction 299

b) Assuming that the conducting wire it is made of has radiusaand resistivityρ, find its resistance

R.

c) Find the currentI(t) in the circuit assuming that the switchSis closed at timet= 0.

Problem 6.

Complete the toroidal solenoid example begun for you above (see figure 103). Find the self-
inductanceLof a toroidal solenoid ofNturns that has inner radiusa, outer radiusb, and height
h.

Problem 7.

Complete the coaxial-cable example begun for you above (see figure104). Find the high-frequency
self-inductanceper unit lengthof a coaxial cable with inner conductor radiusa, outer conductor
radiusb.

Problem 8.

R

N turns

A

ω

A magnetic braking system is drawn above. A wheel hasMpowerful permanent magnets
mounted around the rim. Each magnet produces a uniform fieldBacross a cross-sectional area
A. As the wheel spins at angular velocityω, the magnets cross in front of a coil withNturns in a
circuit with a resistanceR.Estimatethe braking power of the system as follows:

• Assume that each magnet produces a total fluxφ=BA.


• Assume that the flux of each magnet ramps uplinearlyfrom zero toφand back down to zero
  in the time required for the magnet to swing past a loop.


• From this, estimate the induced voltage and current during the ramp up and ramp down
  phases. Plot them as a function of time for several cycles, assuming constantω.


• Compute (and plot) the (effectively average)powerduring the ramp up and ramp down phases.


• Advanced!You should have gotten a power of the general form:

P=−Cω^2 =

dK

dt

for a constantCthat depends onM,N, etc (not given as you are supposed to derive this).