Download Φ21 Fall 2006 HW15 Solutions 1 Faraday`s Law and Induced EMF

Φ21 Fall 2006
1
HW15 Solutions
Faraday's Law and Induced EMF
To understand the terms in Faraday's law and to be
able to identify the magnitude and direction of induced emf. Faraday's
law states that induced emf is directly proportional to the time rate of
change of magnetic ux. Mathematically, it can be written as
Learning Goal:
E =−
∆ΦB
,
∆t
B
where E is the emf induced in a closed loop, and ∆Φ
∆t is the rate of change
of the magnetic ux through a surface bounded by the loop. For uniform
~ ·A
~ = BA cos(θ),
magnetic elds the magnetic ux is given by ΦB = B
Figure 1: A rectangular loop with
~ and the normal to width x and height y in a B
~ eld.
where theta is the angle between the magnetic eld B
the surface of area A.
To nd the direction of the induced emf, one can use Lenz's law:
The induced current's magnetic eld opposes the change in the magnetic ux that induced the
current.
For example, if the magnetic ux through a loop increases, the induced magnetic eld is directed opposite to
the "parent" magnetic eld, thus countering the increase in ux. If the ux decreases, the induced current's
magnetic eld has the same direction as the parent magnetic eld, thus countering the decrease in ux.
Recall that to relate the direction of the electric current and its magnetic eld, you can use the right-hand
rule: When the ngers on your right hand are curled in the direction of the current in a loop, your thumb
gives the direction of the magnetic eld generated by this current.
In this problem, we will consider a rectangular loop of wire with sides x and y placed in a region where a
~ exists (see Figure 1). The resistance of the loop is R.
uniform magnetic eld B
Initially, the eld is perpendicular to the plane of the loop and is directed out of the page. The loop can
rotate about either the vertical or horizontal axis, passing through the midpoints of the opposite sides, as
shown.
Which of the following changes would induce an electromotive force (emf) in the loop? When you
consider each option, assume that no other changes occur.
Part A.
~ increases.
The magnitude of B
~ decreases.
The magnitude of B
C. The loop rotates about the vertical axis (vertical dotted line) shown in the diagram.
D. The loop rotates about the horizontal axis (horizontal dotted line) shown in the diagram.
E. The loop moves to the right while remaining in the plane of the page.
F. The loop moves toward you, out of the page, while remaining parallel to itself.
A.
B.
The options which change the ux ΦB are ABCD. Changing the eld or rotating the loop changes
~ ·A
~ . Moving the loop does not change the ux, as the B
~ eld exists everywhere.
the dot product B
Solution:
Part B.
Find the ux ΦB through the loop.
Solution:
Part C.
The ux is
~ ·A
~ = BA cos θ = Bxy
ΦB = B
If the magnetic eld steadily decreases from B to zero during a time interval t, what is the magnitude
E of the induced emf?
1
Solution:
The EMF is the change in ux per unit time. Assuming that B decreases linearly,
E =−
Part D.
dΦB
−dB
Bxy
= xy
=
dt
dt
t
If the magnetic eld steadily decreases from B to zero during a time interval t, what is the magnitude
I of the induced current?
Solution:
The current is found from Ohm's Law.
I=
V
E
Bxy
=
=
R
R
Rt
Part E. If the magnetic eld steadily decreases from B to zero during a time interval t, what is the direction
of the induced current?
Since the ux is decreasing in magnitude, the induced current's magnetic eld must be in the same
~ , i.e. out of the page. By the right-hand-rule, the induced current is then counter-clockwise.
direction as B
Solution:
Which of the following changes would result in a clockwise emf in the loop? When you consider
each option, assume that no other changes occur.
Part F.
~ increases.
The magnitude of B
~
B. The magnitude of B decreases.
C. The loop rotates through 45 degrees about the vertical axis (vertical dotted line) shown in the diagram.
D. The loop rotates through 45 degrees about the horizontal axis (horizontal dotted line) shown in the
diagram.
E. The loop moves to the right while remaining in the plane of the page.
F. The loop moves toward you, out of the page, while remaining parallel to itself.
A.
Since the decrease in ΦB caused a counter-clockwise current, we need to increase ΦB . In this
situation, only increasing B will increase the ux, so the answer is A.
Solution:
2
Problem K33.12
The loop in Figure 2 is being pushed into the 0.8 T magnetic eld at 10
m/s. The length L is 0.3 m. The resistance of the loop is 0.1 Ω.
Parts A and B.
in the loop?
What are the magnitude and direction of the current
The magnitude of the current is found from Ohm's Law and
Faraday's Law. The magnitude of the change in ux is
Solution:
¯
¯
¯ dΦB ¯
¯
¯ = BvL
|E| = ¯
dt ¯
This causes a current
I=
V
E
BvL
(0.8 T) (10 m/s) (0.3 m)
=
=
=
= 24 A
R
R
R
(0.1 Ω)
Figure 2: A wire loop moving in
~ eld.
The ux is increasing in the direction out of the page, so the induced a limited B
current must create a eld into the page and it must be clockwise.
2
3
Problem K33.4
What is the magnetic ux through the loop shown in the gure
~ point into the page, and note
below? (Assume that the area vectors dA
2
that the units of magnetic ux , Tm , are also called webers, Wb).
Part A.
~ eld
Separately calculate the ux due to each part of the B
and add the results together.
Solution:
ΦB
= Φleft + Φright
Figure 3: A wire loop with a non~ eld.
uniform B
= (0.2 m) (0.2 m) (2.0 T) + (0.2 m) (0.2 m) (−1.0 T)
= 0.04 Wb
4
Problem K33.11
A 1100-turn coil of wire that is 2.00 cm in diameter is in a magnetic eld that drops from 0.150 T to 0 T
in 9.00 ms. The axis of the coil is parallel to the eld. Part A. What is the emf of the coil? Don't forget
there is a factor equal to the number of turns.
Each turn of the coil contributes to the magnetic ux. Assuming the ux decreases linearly, the
derivative is just the change in ux over the change in time.
Solution:
¡
N B πR
dΦB
∆ΦB
N BA
=
=
=
dt
∆t
t
t
2
³
´
2
(1100) (0.15 T) π (0.01 m)
¢
=
3
(0.009 s)
= 5.76 V