Download Inductance & Inductors

“Motional EMF”
A= I, B= ii, C=iii.
a) Clockwise
b) zero
What is the direction of the magnetic field produced by
this current loop inside the loop?
A] upward
B] downward
I is CW from above
B
Note the splay
in the field lines!
Viewed from above, current in the loop
A] will flow clockwise
B] will flow counterclockwise
C] will not flow at all
A
Consider vxB. Since the field lines splay,
vxB is CW from above.
The current in the loop causes a downward B field
inside the loop. The flux of this induced B field
Opposes the flux change that would otherwise occur.
“Lenz’s Law”
Induced current is CW, seen from above, by RHR.
Viewed from above, current in the loop
A] will flow clockwise
B] will flow counterclockwise
C] will not flow at all
The flux through the loop is
downward and decreasing. To oppose
this change, we need the current in
the loop to flow CLOCKWISE, seen
from above.
(Again, if the loop is moving, you can
use vxB to find the direction of the
force. But you have to remember that
the field lines are getting farther apart
farther from the magnet.)
We saw that, when we move the loop down, a CW current flows,
owing to the B field acting on the (downward moving) charges.
If, instead, we move the magnet UP, keeping the loop still,
current:
A] will flow clockwise (viewed from above)
B] will flow counterclockwise
C] will not flow all, since B fields don’t act on stationary charges.
While it’s true that B fields don’t act on stationary charges, it
really shouldn’t matter which object is moved!
Let’s do the experiment!
Changing magnetic fields DO result in the motion of
stationary charges.
Thus, changing magnetic fields must “induce” electric
fields. (It is perhaps better to say that changing
magnetic fields are “coupled to” electric fields….)
The four cases
What can we say about the induced E field?

d
E  dl  
dt
dm
 B  dA   dt
Like with Ampere’s law, we can only
find E from this easily when there is A LOT of symmetry.

E must (by symmetry) be the same everywhere, or zero,
in our loop.
(In cases without symmetry, there is still an E field; we
just don’t have the math tools to find it.)
Examples where we can “easily” find E from a
changing magnetic field.
1) Circular loop, N pole of magnet, with change of
flux specified.
2) A very long (infinite) solenoid, with ramping
current (and B field)
A loop of copper wire is shown.
Moving the magnet up:
A] causes increasing upward B flux
B] causes decreasing upward B flux
C] causes decreasing downward B flux
D] causes increasing downward B flux
E] has no effect on the flux through the loop
A loop of copper wire is shown.
Moving the magnet up
-causes increasing upward B flux.
In what direction should the B field caused
by the induced current be?
A] up
B] down
A loop of copper wire is shown.
Moving the magnet up
-causes increasing upward B flux.
The loop current should oppose the
flux change. So the field from the
loop current should be DOWN.
What direction does the current
flow, viewed from above?
A] CW
B] CCW
A loop of copper wire is shown.
Moving the magnet up
-causes increasing upward B flux.
The loop current should oppose the
flux change. So the field from the
loop current should be DOWN.
The induced current must flow CW,
seen from above, by the RHR.
What is the direction
of the induced current
in the ring, as seen
from above?
A] CW
B] CCW
C] There is no
induced current
A
Considering the
magnetic field of
the solenoid as a
magnet:
The top of the
solenoid is a
A] N
B] S
pole
A
Considering the
magnetic field of
the loop as a little
magnet: the North
pole of the loop is
A] up
B] down
B
The magnetic force
between the
solenoid and the
coil should be
A] attractive
B] repulsive
C] zero
B
Generators
At this instant, current
through the light bulb will
flow:
A] left to right in side view
B] right to left in side view
C] not at all
A
Generators
At this instant, current through
the light bulb will flow:
A] top to bottom in side view
B] bottom to top in side view
C] not at all
C
Let’s do the math for a rotating loop in a uniform B field.
B
More about motional EMF
A square loop is pulled through a constant B field.
What is the magnitude of the motional emf?
A] 0
B] vBL
C] 2vBL
D] vBL2
A
More about motional EMF
Although there is magnetic flux through the loop,
the amount is NOT changing with time. So emf = 0.
Here, the left side is not moving, so there is no magnetic force on the initially
stationary charges on that side. There is magnetic force on the charges on
the right side, pushing positive charges up.
Each charge acquires an energy = qvBL = force x distance.
That energy is then lost as the charge “slides downhill”, through the lightbulb,
heating it.
Where does the energy come from that lights the bulb?
It cannot come from the B-field, as that is unchanging while the bar is sliding.
Answer: you must use force to pull the right side at constant v.
How much force?
More on Solenoids
Long solenoids have spatially uniform B
inside (from Ampere’s law)
If the current is increased linearly with
time, the B field will increase linearly with
time. In this case, the field is out of the
page (top view) and increasing with time.
If this is done, what will be the
direction of the induced E field at
point b, distance r from the axis?
On the top view:
A] up
B] down C] left
E] out of the page
D] right
B
More on Solenoids
Consider a loop at a radius r. The flux is
upward and increasing, so the induced
EMF must be such as to cause a
downward B field if a loop of wire were
there.
So the EMF must be CW (from
above) and E must point
downward (top view.)
Let’s do the math….
More on Solenoids
What is the induced E
field at point a, on the
solenoid axis?
A] 0
B] not zero
A
More on Solenoids
What is the induced E
field at point c, outside
the solenoid, where B
is essentially zero?
.c
A] 0
B] not zero, upward
(top view)
C] not zero, downward
(top view)
D] not zero,
leftward(top view)
B
Now consider 4 loops, all with the
same area. The B field is
increasing with time. What is true?
A] loops a,b,c have the same ,
but d has less.
B] loops a and c have  =0, but b
and d have same nonzero .
C] loop a has  =0, but b,c,d have
the same, nonzero .
D] loop c has  =0, loop a has a
little, b has more, and d the most.
E] loop c has  =0, but loops a,b,d
all have the same nonzero .
E
Mutual Inductance & Inductors
Electric Toothbrush Recharges via mutual induction
Resistor: opposes
“motion” (current),
like friction
Inductor: opposes
change in motion,
like inertia
Do not confuse “cause” and
“effect”.
The voltage drop from a to b (in
part c) is an effect of the
increasing current.
Something has to “overcome”
(meaning provide) this voltage
drop if the current is to increase.
Think of the “back EMF”
this way: if you replace the
inductor with a battery, the
battery (by itself) would
drive current that would
oppose the change.
Immediately after closing the switch,
where is the potential higher?
A] A
B] B
C] Potential at A& B is the same
A
A very long time after closing the
switch, where is the potential
higher?
A] A
B] B
C] Potential at A& B is the same
C
After the switch has been closed a
long time and a steady state
reached, the switch is opened.
Where is the potential higher?
A] A
B] B
C] Potential at A& B is the same
B
Archuleta
C Herrera
Es Martinez
Middleton
Sinyenko
Warren
Wildau
Wed Dec 1, 2010
Which curve shows the current after switch s1 is closed? B
Let’s calculate…
Which curve shows the voltage
drop across the inductor after
S1 is closed? C
Which curve shows the voltage
drop across the resistor after
S1 is closed? B
After reaching a steady current,
S1 is opened and S2 is closed,
simultaneously.
What curve shows the voltage
drop (from a to b) across the
resistor vs time? C
After reaching a steady current,
S1 is opened and S2 is closed,
simultaneously.
What curve shows the voltage
drop (from b to c) across the
inductor vs time? D
A little while (t=L/2R) after the switch is closed,
what is the voltage around the circuit? C
What is the voltage around the circuit a long time
after the switch is closed? B
Switch s1 is closed. Just after, what is the current through the
resistor? A
A] 0
B] E/R
C] E/(RL)
D] E/(RC)
Just after closing switch S1, what is the voltage drop
across the inductor? B
A] 0
B] E
C] E/2
D] E/L
A long time after closing switch S1, what is the charge
on the capacitor? B
A] 0
B] CE
C] E/C
D]E/(RC)
Ampere’s
law
This completes
“Maxwell’s Equations”
LC Oscillations
A] +
B] -
C] 0
A] electric
B] magnetic
A] +
B] -
C] 0
Inductor exerts electric force on the charges (induced EMF is
an electric field.)
Immediately after
closing switch, what is
current through
inductor?
A] 0
B] 1/2 ampere
C] 1 ampere
D] 2 amperes
Immediately after
closing switch, what is
voltage across
capacitor?
A] 0 V
B] 20 V
C] 40 V
D] 160 V
No current through inductor, no voltage across capacitor!
Immediately after
closing switch, what is
current through battery?
A] 0
B] 1/2 ampere
C] 1 ampere
D] 2 amperes
No current through inductor, no voltage across capacitor!
Immediately after
closing switch, what is
di/dt for the inductor?
(Be careful!)
A] 0
B] 20 kA/s
C] 40 kA/s
D] 0.5 kA/s
What would be the answer if there were no cap in parallel?
C
A long time after
closing the switch,
what is the current
through the capacitor?
A] 0
B] 1/2 ampere
C] 1 ampere
D] 2 amperes
A long time after
closing the switch,
what is the current
through the inductor?
A] 0
B] 1/2 ampere
C] 1 ampere
D] 2 amperes
Critical damping
gives the
fastest
return to
equilibrium
Also applies to car
shocks & springs!