Faraday’s Law - University Of IllinoisFaraday’s Discovery True no matter how we change the flux....
Transcript of Faraday’s Law - University Of IllinoisFaraday’s Discovery True no matter how we change the flux....
Physics 212 Lecture 17, Slide 1
Physics 212 Lecture 17
Faraday’s Law
Physics 212 Lecture 17, Slide 2
Motional EMF
Change Area of loop Change magnetic field
through loop Change orientation of loop relative to B
B B dA
In each case the flux of the magnetic field through the circuit changes with time and an EMF is produced.
EMF Bd
dt
Physics 212 Lecture 17, Slide 3
B
Rotate the loop,change flux, generate emf.
Physics 212 Lecture 17, Slide 4
Move loop to a place wherethe B field is different, change flux, generate emf.
B2
v
B1
Checkpoint 1a
Physics 212 Lecture 17, Slide 5
• The flux is NOT changing • B does not change • the area does not change • the orientation of B and A does not change
• Motional emf is ZERO • v X B = 0 • no charge separation • no E field • no emf
A copper loop is placed in a uniform magnetic field as shown. You are looking from the right.
Suppose the loop is moving to the right. The current induced in the loop is:
A. zero B. clockwise C. counterclockwise
Physics 212 Lecture 17, Slide 6
Current changes direction every time the loop becomes perpendicular with the B field
emf ~ d/dt
(B dA = max) d/dt (B dA ) = 0
X
O B dA X
O
B dA
Checkpoint 1c Now suppose that the loop is spun around a vertical axis as shown, and that it makes one
complete revolution every second.
The current induced in the loop:
A. Is zero
B. Changes direction once per second
C. Changes direction twice per second
Bd
emfdt
Faraday’s Discovery
True no matter how we change the flux. In fact, the circuit may be stationary (no motional EMF) and only the B-field changes with time. An EMF is still produced. This implies that:
Bdemf E dl
dt
Faraday’s Law
Physics 212 Lecture 17, Slide 8
B(t) decreasing
Change the B field in time so flux changes. Induce an emf nnd therefore an Electric field. This emf tries to oppose the change in flux. (Lenz’s Law)
dt
ddEemf B
Induces an E field even if there is no circuit there!
Physics 212 Lecture 17, Slide 9
Checkpoint 1b
• Motional emf is ZERO • Circuit is stationary !
• HOWEVER: The flux is changing • B decreases in time • current induced to oppose the flux change
• clockwise current tries to restore B that was removed
X X X X X X X X
X X X X X X X X
X X X X X X X X
X X X X X X X X
X X X X X X X X
Looking from right
Clockwise current tries to restore B
Checkpoint 1b A copper loop is placed in a uniform magnetic field as shown. You are looking from the right.
Now suppose the that loop is stationary and that the magnetic field is
decreasing in time. The current induced in the loop is:
A. zero B. clockwise C. counterclockwise
Physics 212 Lecture 17, Slide 10
(copper is not ferromagnetic)
This one is hard ! B field increases upward as loop falls
Clockwise current (viewed from top) is induced
Ftotal < mg
a < g
X
O
B
B
Like poles repel
F
Checkpoint 2 A horizontal copper ring is dropped from rest directly above the north pole of a permanent magnet
Will the acceleration a of the falling ring in the presence of the magnet
be any different than it would have been under the influence of just
gravity (i.e. g)?
A. a > g B. a = g C. a < g
Physics 212 Lecture 17, Slide 11
Main Field produces horizontal forces “Fringe” Field produces vertical force
I
Looking down
I
B B
IL X B points UP
Ftotal < mg
a < g
HOW IT
WORKS
(copper is not ferromagnetic)
This one is hard ! B field increases upward as loop falls
Clockwise current (viewed from top) is induced
Checkpoint 2 A horizontal copper ring is dropped from rest directly above the north pole of a permanent magnet
Will the acceleration a of the falling ring in the presence of the magnet
be any different than it would have been under the influence of just
gravity (i.e. g)?
A. a > g B. a = g C. a < g
Physics 212 Lecture 17, Slide 12
Calculation
• Conceptual Analysis – Once loop enters B field region, flux will be changing in time – Faraday’s Law then says emf will be induced
• Strategic Analysis – Find the emf – Find the current in the loop – Find the force on the current
y
x
v0
a
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
B A rectangular loop (height = a, length = b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction. What is the direction and the magnitude of the force on the loop when half of it is in the field?
Physics 212 Lecture 17, Slide 13
What is the magnitude of the emf induced in the loop just after it enters the field?
(A) e = Babv02 (B) e = ½ Bav0 (C) e = ½ Bbv0 (D) e = Bav0 (E) e = Bbv0
In a time dt it moves by v0dt
Change in Flux = dB = BdA = Bav0dt
Calculation A rectangular loop (height = a, length = b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction.
y
x
v0
a
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
B
y
x
v0
a
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
B
The area in field changes by dA = v0dt a
a
dt
demf B
oB Bav
dt
d
Physics 212 Lecture 17, Slide 14
What is the direction of the current induced in the loop just after it enters the field?
(A) clockwise (B) counterclockwise (C) no current is induced
Flux is increasing into the screen
emf is induced in direction to oppose the change in flux that produced it
Induced emf produces flux out of screen
Calculation y
x
v0
a
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
B A rectangular loop (height = a, length = b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction.
y
x
v0
a
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
B
dt
demf B
Physics 212 Lecture 17, Slide 15
What is the direction of the net force on the loop just after it enters the field?
(A) +y (B) -y (C) +x (D) -x
x
y
v0 a
b x x x x x x x
x x x x x x x
B
I • Force on top and bottom segments cancel (red arrows)
Calculation y
x
v0
a
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
B A rectangular loop (height = a, length = b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction.
• Force on right segment is directed in –x direction.
dt
demf B
Force on a current in a magnetic field: BLIF
Physics 212 Lecture 17, Slide 16 x
y
v0 a
b x x x x x x x
x x x x x x x
B
I F
What is the magnitude of the net force on the loop just after it enters the field?
(A) (B) (C) (D)
e = Bav0
R
vaBaB
R
BavF oo
22
Calculation y
x
v0
a
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
B A rectangular loop (height = a, length = b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction.
ILB
F IL B
dt
demf B
RaBvF o4 RBvaF o2 RvBaF o /222 RvBaF o /22
BLIF
since ILBF BL
R
Bav
RI o
e
Physics 212 Lecture 17, Slide 17
Follow-Up A rectangular loop (sides = a,b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction. What is the velocity of the loop when half of it is in the field?
y
x
v0
a
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
B
t = dt: e = Bav0
Which of these plots best represents the velocity as a function of time as the loop moves form entering the field to halfway through ? (A) (B) (C) D) (E)
This is not obvious, but we know v must decrease Why?
v0 a
b x x x x x x x
x x x x x x x
B
I Fright
Fright points to left Acceleration negative Speed must decrease
X X X
Physics 212 Lecture 17, Slide 18
Follow-Up
2 2 /dv
F a B v R mdt
A rectangular loop (sides = a,b, resistance = R, mass = m) coasts with a constant velocity v0 in + x direction as shown. At t =0, the loop enters a region of constant magnetic field B directed in the –z direction. What is the velocity of the loop when half of it is in the field?
Which of these plots best represents the velocity as a function of time as the loop moves form entering the field to halfway through ?
y
x
v0
a
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
B
e = Bav0
(A) (D) • Why (D), not (A)? – F is not constant, depends on v
Challenge: Look at energy Claim: The decrease in kinetic energy of loop is equal to the energy dissipated as heat in the resistor. Can you verify??
dt
dvm
R
vBaF
22
where
toevv
mR
Ba 22