So far we have found that
Electric charges create E-fields
Moving electric charges create B-fields
v kQ
E = 2 rˆ
r
v μ qvv × rˆ
B= 0
4π r 2
Electric Fields can also be created by a changing Magnetic Field.
This is described by Faraday’s Law.
Magnetic Fields can also be created by a changing Electric Field.
This will help complete the set of equations called
Maxwell’s equations of electricity and magnetism.
But this is not the whole story!
1
Recall Electric Flux
Faraday’s Law of Induction
ε =−
2
ΦE =
dΦ B
dt
v v
E
∫ ⋅ dA
surf
Magnetic Flux is quite similar except for Magnetic Fields.
Two new quantities introduced.
ΦB =
1. Magnetic Flux = ΦB
2. Electro Motive Force (EMF) = ε
v v
B
∫ ⋅ dA
surf
3
Be careful to see how the integration is done…
4
Clicker Question
We have a square imaginary loop with side of length a. There
is a Magnetic Field that is uniform throughout the region as
shown. How does the Magnetic Flux magnitude |ΦB| change if
we halve the Magnetic Field strength and double the sides of
the square loop?
A) Flux goes to ½ original value
B) Flux goes to ¼ original value
C) Flux goes to 2 times original value
X B
D) Flux does not change
E) None of the above
v v Q
Φ E = ∫ E • da = inside
ε0
In Gauss’ Law, the Electric Flux is evaluated over a closed
surface! (i.e. the 2 dimensional surface of a 3 dimensional
closed object)
In Faraday’s Law, we will evaluate the Magnetic Flux over a
surface, but not a closed surface.
5
6
1
Clicker Question
Electro Motive Force (EMF)
At what angle of orientation of the surface is the flux through
the surface half of the maximum possible flux?
θ = 0 degrees
θ = 30 degrees
θ = 45 degrees
θ = 60 degrees
None of the above
A)
B)
C)
D)
E)
B
θ
This is often a confusing concept for multiple reasons.
ε = (roughly speaking) a Voltage difference capable of
generating power.
Batteries have an EMF.
Resistors do not have an EMF, even though there is a Voltage
difference across them ΔV=iR.
7
Technically, the EMF around a closed Loop is defined as:
8
EMF is not a force! It does not have units of Newtons.
v v
ε = ∫ E ⋅ dl
It has units of Voltage (electric potential).
Loop
Sometimes we distinguish a battery as a chemical EMF and
that from changing Magnetic Flux as motional EMF.
But, recall that Voltage is defined as:
v v
ΔV = − ∫ E ⋅ dl
The motional EMF is not always zero around a closed loop !
We will talk about this more later.
Was not this integral around a closed loop always zero?
9
10
v
v
d
E
⋅
d
l
= −
∫
dt
Loop
Faraday’s Law
ε = −
v
∫E
Loop
dΦ
dt
v
d
⋅ dl = −
dt
B
v
v
∫
v
v
B ⋅ dA
surf
Bexternal
∫ B ⋅ dA
surf
R
How do the loop and the surface relate?
11
v
d
E (2π R ) = −
dt
[Bv
ext
(π R 2 )
]
12
2
Wire Loop
B
V
Demonstration
If B = constant,
Then Magnetic Flux = constant
dΦB/dt = 0
EMF = 0 and Voltmeter reads 0.
If B =increasing,
Then Magnetic Flux = changing
|dΦB/dt| > 0
EMF > 0 and Voltmeter reads > 0.
What if the wire loop were not there?
13
14
Clicker Question
Many different ways to change the Magnetic Flux
A loop of wire is moving rapidly through a uniform magnetic
field as shown. Is a non-zero EMF induced in the loop?
1. Change the strength of the Magnetic Field
2. Change the area of the loop
A) No, there in no EMF
B) Yes, there is an EMF
3. Change the orientation of the loop (A vector) and the B-field
F = BA cos q
and the angle θ is changing.
15
Clicker Question
A loop of wire is spinning rapidly about a stationary axis in
uniform magnetic field as shown.
True or False: There is a non-zero EMF induced in the loop.
A) False, zero EMF
B) True, there is an induced EMF
17
16
The (-) negative sign in Faraday’s Law is more of a reminder.
ε = −
dΦ
dt
B
Lenz’s Law: The induced EMF tends to induce a
current in the direction which opposes the
changes in Magnetic Flux.
18
3
If we move the magnet closer to the loop, what is the direction
of the induced EMF and thus the direction of the current in the
loop?
v v
Φ B = B ⋅ dA
1. The Magnetic
surf
Flux is increasing.
∫
2. Therefore
dΦ B
>0
dt
i
Induced current must create B-field that
fights the change!
B
dΦ B
>0
dt
An induced B-field down through the
loop will create a slight decrease in ΦB.
3. Induced current must create Bfield that fights the change!
Thus, we have determined the direction
of the induced current and EMF.
19
Think of a sentence…
“The Magnetic Flux ΦB Up Through the Loop is Increasing.
To Fight this Change, an Induced B-field is made that
Decreases the Flux Up Through the Loop.
20
Clicker Question
A bar magnet is positioned below a loop of wire. The magnet
is pulled down, away from the loop. As viewed from above, is
the induced current in the loop clockwise, counterclockwise, or
zero?
eyeball
B
i
This is accomplished by the current shown.”
A: clockwise
Note that the induced flux change must always be smaller
than the original flux change.
C: zero
B: counter-clockwise
Answer: Counterclockwise.
As the magnet is pulled away, the flux is decreasing.
To fight the decrease, the induced B-field should add
to the original B-field.
21
22
Clicker Question
A loop of wire is sitting in a uniform, constant magnet field as
shown. Suddenly, the loop is bent into a smaller area loop.
During the bending of the loop, the induced current in the loop
is ...
A: zero
B: clockwise
C: counterclockwise
B(in)
B(in)
Answer: Clockwise. The flux into the page is decreasing
as the loop area decreases. To fight the decrease, we
want the induced B to add to the original B. By the right
hand rule (version II) , a clockwise induced current will
make an induced B into the page, adding to the original B.
23
4