W9_parallel_resonance.eps

Week 8: Faraday’s Law and Induction 273

and the terminal velocity, which we determine from the observationthat the net force on the loop
(and hence current in the loop) must be zero at the terminal velocity:

vterminal=

V 0

BL (588)

Using the force equation we can easily write Newton’s second Law andturn it into an equation
of motion:

F=

BLV 0 −B^2 L^2 v

R

=M a=M

dv

dt

(589)

which we can rearrange into afirst order, linear, homogeneous, ordinary differential equation:

dv

dt

= BLV^0 −B

(^2) L (^2) v
M R
= −

B^2 L^2

M R

(

v−

V 0

BL

)

( dv

v−BLV^0

) = −B

(^2) L 2
M R
dt
∫
( dv
v−BLV^0

) = −

∫

B^2 L^2

M R

dt

ln

(

v−

V 0

BL

)

= −

B^2 L^2

M Rt+C

v−

V 0

BL = e

−BMR^2 L^2 t∗eC

v(t) =

V 0

BL

(

1 −e−

BMR^2 L^2 t)

(590)

where we’ve used our initial condition,v(0) = 0, to set the constant of integration. Note well that
this curve represents anexponential approach to the terminal velocity.

With this in hand we can easily integrate over time again to getx(t), or differentiate it to get
a(t). We can compute the power being delivered to the circuit by the voltage and show that it equals
the rate at which energy is burned in the resistor plus the rate thatwork is being done on the rod.
We can answeranythingasked about the rod – the motion is now completely known.

8.6: Inductance

I 1

1

2

3

4

B field lines

Figure 100: A set of current loops indexed byi= 1, 2 , 3 ..., fixed in space and carrying currentsIi.
TheB-field produced by (say) currentI 1 swirls around the current and passes through both loop 1
and the other loops in the figure, creating bothself inductanceandmutual inductance.

We have seen that changing the current inonewire causes the magnetic field associated with
that current to change in time. That, in turn, will usually cause the magnetic flux through other