W09D2:
Faraday’s Law
Today’s Reading Assignment Course Notes: Sections 10.1-10.4
1
Announcements
Math Review Week Ten Tuesday from 9-11 pm in 26-152
PS 7 due Week Ten Tuesday at 9 pm in boxes outside 32-082 or
26-152
Next Reading Assignment:
W09D3 Course Notes: Sections 9.10.2; 10.7, 10.9-10
2
Outline
Faraday’s Law
Applications of Faraday’s Law
Problem Solving
Experiment 3: Faraday’s Law
3
Faraday’s Law of Induction
dΦ B
∝ I induced
dt
Changing magnetic flux
induces a current
4
Demonstration:
Induction
and Simulation of Induction
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%203&show=0
http://public.mitx.mit.edu/gwt-teal/FaradaysLaw2.html
5
Demo: Electromagnetic Induction
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%203&show=0
6
Demonstration: Magnet Falling
Through Plastic Tube and
Aluminum Tube
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%2016&show=0
7
Magnetic Flux Thru Wire Loop
Analogous to Electric Flux (Gauss’ Law)

(1) Uniform B
 
Φ B = B⊥ A = B Ac o sθ = B ⋅ A

(2) Non-Uniform B
 
Φ B = ∫∫ B ⋅ d A
S
8
Electromotive Force (emf)
Imagine a charge q moving around
a closed path

under the action of a force F . Then the work done
per unit charge is called the electromotive force

(emf).
ε
F 
= 
⋅ds
∫
q
closd path
Has the same dimensions as electric potential so SI
unit is the volt.
If the closed path is a circuit with resistance R then
the electromotive force will cause a current to flow
according to
ε = IR
9
Motional emf
Pull a loop with speed v
Through uniform magnetic
field (shaded area). At
one instant in time,
electromotive force is
ε = ∫
loop

Fmag
 
qv × B 

⋅ds= ∫
⋅ d s = vBw
q
q
left leg
10
Changing Magnetic Flux and
Motional emf
Magnetic flux through
loop is decreasing
−
ΔΦ mag
BwΔs
=
Δt
Δt
BwvΔt
=
= vBw
Δt
=ε
11
Concept Question: Loop in Uniform Field
While a rectangular wire loop is
pulled upward though a uniform
magnetic field B field penetrating its
bottom half, as shown, there is
1.  a current in the loop.
2.  no current in the loop.
3.  I do not understand the concepts of current and
magnetic field.
4.  I understand the concepts of current and magnetic field
but am not sure of the answer.
12
Concept Q.: Loop in Uniform Field
While a rectangular wire loop is
pulled sideways though a uniform
magnetic field B field penetrating its
bottom half, as shown, there is
1.  a current in the loop.
2.  no current in the loop.
3.  I do not understand the concepts of current and
magnetic field.
4.  I understand the concepts of current and magnetic field
but am not sure of the answer.
13
Ways to Induce emf
ε
d
=−
BAcosθ
dt
(
)
Quantities which can vary with time:
•  Magnitude of B
•  Area A enclosed by the loop
•  Angle between B and normal vector to loop
14
Problem: Changing Area
Conducting rod pulled along two conducting rails in a
uniform magnetic field B at constant velocity v
a)  Calculate the magnitude of the
derivative of the magnetic flux with
respect to time.
b)  Calculate the magnitude of EMF due to
the magnetic force. Does your result
agree with your result from part a)?
c)  What is the magnitude of induced
current?
15
Generalization: Faraday’s Law
Loop is at rest, and magnet is moving resulting
in a changing magnetic flux through loop and
an induced emf and current in the loop
ε
dΦ B
=−
dt
16
Induced Electric Field and
Electromotive Force (emf)
 
ε = ∫ E ⋅ d s
closd path
An induced electric field appears in the current loop
resulting in an electric force on the charges in the
conductor producing an induced current
ε = IR
17
Faraday’s Law of Induction
If C is a stationary closed curve and S is a
surface spanning C then
 
d
C∫ E ⋅ d s = − dt


∫∫ B ⋅ d A
S
The changing magnetic flux through S induces
a non-electrostatic electric field whose line
integral around C is non-zero
ε
dΦ B
=−
dt
18
Sign Conventions: Right Hand Rule
 
 
d
E⋅ds = −
B⋅dA

∫
∫∫
dt open surface
closed path
By the right hand rule (RHR)
clockwise integration direction
for line integral of electric field
(emf) requires that unit normal
points into plane of figure for
magnetic flux surface integral
Magnetic flux positive into page,
negative out of page
19
Sign Conventions: Right Hand Rule
 
 
d
E⋅ds = −
B⋅dA

∫
∫∫
dt open surface
closed path
By the right hand rule (RHR)
counterclockwise
integration direction for line
integral of electric field (emf)
requires that unit normal
points out of plane of figure for
magnetic flux surface integral
Magnetic flux positive out of
page, negative into page
20
Problem: Calculating Electric Field
 
d
C∫ E ⋅ d s = − dt


∫∫ B ⋅ d A
S
Consider a uniform magnetic field
which points into the page and is
confined to a circular region with
radius R. Suppose the magnitude
increases with time, i.e. dB/dt > 0.
Find the magnitude and direction of
the electric field in the regions
(i) r < R, and (ii) r > R. (iii) Plot the
magnitude of the electric field as a
21
function r.
Minus Sign? Lenz’s Law
ε
dΦ B
=−
dt
Induced EMF is in direction that opposes the
change in flux that caused it
22
For a Closed Conducting Path
dΦ B
I ind R = −
dt
Induced current is source of induced magnetic
flux that opposes the change in magnetic flux
that caused it
23
Concept Question: Loop
The magnetic field through
a wire loop is pointed
upwards and increasing
with time. The induced
current in the coil is
1.  Clockwise as seen from the top
2.  Counterclockwise
24
Direction of Induced Current
For a counterclockwise integration direction for the line
integral hence a upwards unit normal for the surface integral
!
dΦ B
ε =−
>0
dt
⇒ I ind counterclockwise
!
dΦ B
ε =−
<0
dt
⇒ I ind is clockwise
25
Concept Question: Moving Loop
A circuit in the form of a
rectangular piece of wire is
pulled away from a long wire
carrying current I in the
direction shown in the sketch.
The induced current in the
rectangular circuit is
1.  Clockwise
2.  Counterclockwise
3.  Neither, the current is zero
26
Lenz’s Law
Induced current, torque, or force is always
directed to oppose the change that caused it
27
Induced Current Loop Acts Like Magnet
Force between magnet and induced current loop is
like force between magnets
28
Concept Question: Faraday’s Law:
Loop
A coil moves up from
underneath a magnet
with its north pole
pointing upward. The
current in the coil and
the force on the coil:
1.
2.
3.
4.
Current clockwise; force up
Current counterclockwise; force up
Current clockwise; force down
Current counterclockwise; force down
29
Demonstration: Jumping Ring
An aluminum ring
jumps into the air
when the solenoid
beneath it is
energized
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%2022&show=0
30
What is Going On?
This is a dramatic example of
Faraday’s Law and Lenz’s
Law: When current is turned on
through the solenoid the created
magnetic field tries to permeate
the conducting aluminum ring,
currents are induced in the ring
to try to keep this from
happening, and the ring is
repelled upwards.
31
Experiment 3:
Faraday’s Law of Induction
http://public.mitx.mit.edu/gwt-teal/FaradaysLaw2.html
32