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```					Faraday’s Law

Brief review of magnetic fields
Magnetic Flux
Lenz’s Law

Wednesday, June 17, Session 1
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N. J. Smith, MTSU, June 2009

Faraday’s law connects electric fields and magnetic fields.
Understanding Faraday’s law lets us build electric motors,
electric generators.
Faraday’s law is one of Maxwell’s equations –
understanding of electromagnetic radiation

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N. J. Smith, MTSU, June 2009
Magnetic Fields

S                     N

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N. J. Smith, MTSU, June 2009
The direction of a magnetic field line is the same direction that a
compass would point if placed in the field line.

Conversely, a compass points in the direction of the magnetic
field line, at any point in space.

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N. J. Smith, MTSU, June 2009
The magnetic field is a vector field:
It has a magnitude (e.g. how strongly does it attract or
repel another magnet), measured in Tesla (T).
It has a direction.

At any point in space, the
direction of the magnetic field
vector is tangent to the magnetic
field line.
S                    N

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N. J. Smith, MTSU, June 2009
Electric currents produce magnetic
fields

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Introducing Flux
The flux of a quantity is the amount of it that passes through
a given area. Flux is given the symbol .

Examples:

The amount of water coming out a hose        Number of individual threads of a
that has cross-sectional area A.             rope through an area A.

The flux of water would be measured in
(kg/s)m2.                                                                        7
N. J. Smith, MTSU, June 2009
Magnetic Flux
Consider a uniform magnetic field, with strength B and
directed out of the page, as shown. The magnetic flux,
B, through a single loop with area A, is

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Examples:
What is the magnetic flux through the area shown in each of the
following situations?

1. B = 55 T, l = 5 cm, w                        2. B = 6 T, r = 14 cm
= 18 cm

l
r
w

Solution                                       Solution

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Examples (cont.):

3. The magnetic field has constant                Solution
strength, but makes an angle of 42
from the vertical, as shown.
B = 102 T, l = 0.5 m, w = 0.2 m

42

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N. J. Smith, MTSU, June 2009
As we have just seen, only the component of the magnetic field
perpendicular to the area contributes to the magnetic flux. This
prompts a more general definition of the magnetic flux:

where  is the angle between the magnetic field and the
perpendicular line from the surface.



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N. J. Smith, MTSU, June 2009
One last refinement for the
magnetic flux
In the real world, we often measure the magnetic flux through a loop
or coil of wire. If there is only one loop of wire, our current definition
for magnetic flux is OK:

If there are two loops together, we have effectively doubled the area
for the magnetic field to pass through, so we have

In general, for N loops, the magnetic flux is given by

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N. J. Smith, MTSU, June 2009
PhET Experiment

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What can we learn from the simulation?

When the magnet moves, a voltage appears (the
voltmeter needle moved).

When the magnet moves, the light bulb lights up, which
means there is a current through the globe, and therefore
through the entire circuit.

There is no voltage or current when the magnet is
stationary.

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N. J. Smith, MTSU, June 2009

The observations from the previous slide can be summed up in

A voltage is induced in the loop of wire when the number of
magnetic field lines through the loop is changing.

A more rigorous statement of Faraday’s law is given in terms of
the magnetic flux:

The magnitude of the induced voltage in a conducting loop is equal
to the change in magnetic flux through the loop.

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Examples:

Describe the change in flux in each of the situations below.

The magnetic field magnitude is                 The radius r is decreasing.
increasing with time.

signifies the magnetic field pointing into the page.

Solution                                        Solution
The magnetic flux, due to the field             The flux, due to the magnetic field
pointing up through the area of the loop,       through the loop into the page, is
is increasing, since the strength of the        decreasing, since the area of the
magnetic field itself (B) is increasing.        loop is decreasing.
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N. J. Smith, MTSU, June 2009
Examples:

A magnetic field passes through a             In the previous problem, the wire has
loop of wire that has an area of 3 cm2.       a resistance of 10 . What is the
The magnitude of the magnetic field           magnitude of the induced current in
decreases from 0.5 T to 0 T over a            the wire?
period of 4 minutes. What is the
induced voltage in the wire?
Solution
Solution

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N. J. Smith, MTSU, June 2009
Lenz’s Law
An induced voltage in the conducting loop will create an electric
current, Iind. The question is, what is the direction of the current
around the loop?

The direction of the induced voltage is such that it will induce a
current that will create a magnetic field, Bind, that opposes the
change in the magnetic flux.

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N. J. Smith, MTSU, June 2009
In each of the situations, which direction will the induced magnetic field and
induced current be?

The magnetic field magnitude is                The radius r is decreasing.
increasing with time.

signifies the magnetic field pointing into the page.

(green arrows)
(green arrows)

(blue arrow)

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N. J. Smith, MTSU, June 2009
Magnet moving into a loop

As the magnet moves to the right, the
magnetic flux through the loop
increases, because more field lines
pass through the loop.
S                   N
According to Lenz’s law, the induced
magnetic field will act to oppose the
increase in magnetic flux. It must
therefore point to the left.

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A practical example: an electric
generator
Lets put everything we’ve learned so far together, and learn how
Faraday’s law and Lenz’s law allow us to generate electricity.

N                                         S

If we change any of N, B,  or A, we can change B.
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N. J. Smith, MTSU, June 2009
A practical example: an electric
generator
Lets put everything we’ve learned so far together, and learn how
Faraday’s law and Lenz’s law allow us to generate electricity.

N                                           S

Faraday’s law tells us a current will be induced in the loop to oppose
the change in flux. The induced magnetic field is shown in green.        22
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Maxwell’s Equations

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