W9_parallel_resonance.eps

272 Week 8: Faraday’s Law and Induction

R

B (in)

L

V 0

Figure 99: A conducting rod sits on conducting, frictionless rails anda switch is closed att= 0 to
send current through the loop thus formed. A magnetic field (into the page) exerts a force on the
rod.

and battery with potential differenceV 0. A uniform magnetic field of magnitudeBpoints into the
page as shown.

We would like to find a number of things in this problem:

a) The voltage in the loop as a function ofv, the (eventual) velocity of the rod.

b) The current in the loop as a function of this voltage.

c) The force on the rod as a function of this current.

d) Theterminalvelocity of the rod, after the switch has been closed for a long time.

e) The equation of motion of the rod as determined by the force.

f) The velocity of the rod as a function of time.

This list lays out a very nice solution strategy. Using Faraday’s Law

Vind=−ddtφm=−

dBLx

dt

=−BLv (584)

(where the minus sign is Lenz’s Law and must be interpreted accordingly). Note that the induced
voltage is zero until the rod is moving, then decreases in thedirection that will cause currents that
experience forces that oppose the motion.

Using Kirchoff’s rule for the loop:

V 0 −BLv−IR= 0 (585)

We can then solve for the current in the loop:

I=

V 0 −BLv

R

(586)

and will circulateclockwisein the loop initially whenvis small.

This lets us easily compute the force on the loop:

F=BLI=

BLV 0 −B^2 L^2 v

R

(587)