Physics 122B Electricity and Magnetism

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Physics 122B Electricity and Magnetism
Lecture 22 (Knight 33.5 to 33.7) Faradays Law

• Martin Savage

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Lecture 21 Announcements

• Lecture HW has been posted and is due on
  Wednesday at 10 PM.
• Next Friday we will have Exam 3 in this room.
  It will consist of multiple-choice questions on
  the Laboratory (25 pts) and Lecture (35 pts).
  Bring a Scatron sheet, a double-sided page of
  notes, and a calculator with good batteries.

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Question
What is the ranking of the forces in the figure?
(a) F1F2F3F4 (b) F1ltF2F3gtF4 (c)
F1F3ltF2F4 (d) F1F4ltF2F3 (e)
F1ltF2ltF3F4
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Magnetic Flux

• The number of arrows passing through the
  loop depends on two factors
• (1) The density of arrows, which is proportional
  to B
• The effective areaAeff A cos q of the loop
• We use these ideas to define the magnetic
  flux

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Area Vector
Define the area vector A of a loop such that
it has the loop area as its magnitude and is
perpendicular to the plane of the loop. If a
current is present, the area vector points in
the direction given by the thumb of the right
hand when the fingers curl in the direction of
current flow. If the area is part of a closed
surface, the area vector points outside the
enclosed volume. With this definition
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Example A Circular Loop Rotating in a Magnetic
Field
The figure shows a 10 cm diameter loop
rotating in a uniform 0.050 T magnetic field.
What is the magnitude of the flux through the
loop when the angle is q00, 300, 600, and 900?
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Magnetic Fluxin a Nonuniform Field
So far, we have assumed that the loop is in
a uniform field. What if that is not the case?
The solution is to break up the area into
infinitesimal pieces, each so small that the
field within it is essentially constant. Then
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Example Magnetic Flux froma Long Straight Wire
The near edge of a 1.0 cm x 4.0 cm
rectangular loop is 1.0 cm from a long straight
wire that carries a current of 1.0 A, as shown in
the figure. What is the magnetic flux
through the loop?
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Lenzs Law (1)
Heinrich Friedrich Emil Lenz (1804-1865)
In 1834, Heinrich Lenz announced a rule for
determining the direction of an induced current,
which has come to be known as Lenzs Law.
Here is the statement of Lenzs Law There is an
induced current in a closed conducting loop if
and only if the magnetic flux through the loop is
changing. The direction of the induced current
is such that the induced magnetic field opposes
the change in the flux.
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Lenzs Law (2)
If the field of the bar magnet is already
inthe loop and the bar magnet is removed,
theinduced current is in the direction that
triesto keep the field constant. If the
loop is a superconductor, a persistentstanding
current is induced in the loop, and thefield
remains constant.
Superconductingloop
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Six Induced Current Scenarios
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Example Lenzs Law 1

-

-
The switch in the circuit shown has been
closed for a long time. What happens to the
lower loop when the switch is opened?
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Example Lenzs Law 2

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The figure shows two solenoids facing each
other. When the switch for coil 1 is closed,
does the current in coil 2 flow from right to
left or from left to right?
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Example A Rotating Loop
A loop of wire is initially in the xy plane
in a uniform magnetic field in the x direction.
It is suddenly rotated 900 about the y axis,
until it is in the yz plane. In what
direction will be the induced current in the loop?
Initially there is no flux through the coil.
After rotation the coil will be threaded by
magnetic flux in the x direction. The induced
current in the coil will oppose this change by
producing flux in the x direction. Let your
thumb point on the x direction, and your fingers
will curl clockwise. Therefore, the induced
current will be clockwise, as shown in the figure.
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Faradays Law
Consider the loop shown
This is Faradays Law. It can be stated as
follows An emf E is induced in a conducting
loop if the magnetic flux Fm through the loop
changes with time, so that E dFm/dt for the
loop. The emf will be in the direction that will
drive the induced current to oppose the flux
change, as given by Lenzs Law.
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Example Electromagnetic Induction in a Circular
Loop
The magnetic field shown in the figure
decreases from 1.0 T to 0.4 T in 1.2 s. A 6.0 cm
diameter loop with a resistance of 0.010 W is
perpendicular to the field. What is the size
and direction of the current induced in the loop?
I
The current direction is such as to reinforce the
diminishing B field. Therefore, the current I
will be clockwise.
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Example Electromagnetic Induction in a Solenoid
A 3.0 cm diameter loop with a resistance of
0.010 W is placed in the center of a solenoid.
The solenoid is 4.0 cm in diameter, 20 cm long,
and is wound with 1000 turns of square insulated
wire. The current through the solenoid wire as a
function of time is shown in (b). Find the
induced current in the loop.
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What does FaradaysLaw Tell Us?

• Faradays Law tells us that all induced
  currents are the associated with a changing
  magnetic flux. There are two fundamentally
  different ways to change the magnetic flux
  through a loop
• The loop can move, change size, or rotate,
  creating motional emf
• The magnetic field can change in magnitude or
  direction.
• We can write

new physics
motional emf
The second term says that an emf can be
created simply by changing a magnetic field, even
if nothing is moving.
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An Unanswered Question
A very long solenoid with no field outside
passes through a conducting loop. The current in
the solenoid is increased so that the B field
inside the solenoid increases. (B outside 0).
There is no B-field at the loop wire. Is a
current induced in the loop? YES! Since
the flux through the loop changes, an emf is
induced in the loop, even though the field that
produces the flux does not touch the loop.
How can this happen? Faraday would say that when
the number of lines of force in the solenoid
increases, they must come in from infinity and
must cut through the loop on their way in.
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Question
A conducting loop is half way into a
magnetic field. Suppose that the field begins to
increase rapidly in strength. Which
statement describes the behavior of the loop?

• The loop is pushed upward, toward the top of the
  page
• The loop is pushed downward, toward the bottom of
  the page
• The loop is pushed to the left, into the magnetic
  field
• The loop is pushed to the right, out of the
  magnetic field
• The tension in the wire increases, but the wire
  does not move.

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Induced Fields and Electromagnetic Waves

• There is still a puzzle piece
  missing.Faradays Law allows us to calculate an
  inducedcurrent, but what causes the current?
  Whatforce pushes the electrons around in the
  wire? If the wire is stationary, there can be no
  motional vxB magnetic force. Therefore, there
  must be an induced electric field. Thus, there
  are two ways to create an electric field
• A Coulomb electric field that is created by
  positive or negative charges
• A non-Coulomb electric field that is created by a
  changing magnetic field.

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Maxwells Theory
Maxwell produced a mathematical formulation
of Faradays lines of force picture. He
reasoned from this that if a changing magnetic
field produces an electric field, then a changing
electric field should be equivalent to a current
in producing a magnetic field. Otherwise,
there is a paradox. An Amperian loop near a
charging capacitor will predict a different
magnetic field, depending on whether the surface
enclosed by the loop passes throughthe current
(a) or through thecapacitor gap (b). If the
changing electric field is effectively a current
(called the displacement current) there is no
paradox.
 James Clerk Maxwell (1831-1879)
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Electromagnetic Waves
Maxwells formulation of electricity and
magnetism has an interesting consequence. The
equations can be manipulated to give a wave
equations for E and B of the form
This can be recognized as describing an
electromagnetic wave traveling through space with
a velocity of
This is quite a remarkable result. Somehow,
equations for charges and currents making
stationary electric and magnetic fields are
telling us about electromagnetic waves traveling
through space at the speed of light!
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Generators
The figure shows a coil with N turns rotating
in a magnetic field, with the coil connected to
an external circuit by slip rings that transmit
current independent of rotation. The flux
through the coil is
Therefore, the device produces emf and
current that will vary sinusoidally, alternately
positive and negative. This is called an
alternating current generator, producing what we
call AC voltage.
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Example An AC Generator
A coil with area 2.0 m2 rotates in a 0.10 T
magnetic field at a frequency of 60 Hz. How many
turns are needed to generate an AC emf with a
peak voltage of 160 V?
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Transformers
When a coil wound around aniron core is
driven by an AC voltageV1cos wt, it produces an
oscillatingmagnetic field that will induce
anemf V2cos wt in a secondary coilwound on the
same core. This iscalled a transformer.
The input emf V1 induces acurrent I1 in the
primary coil thatis proportional to 1/N1. The
flux inthe iron is proportional to this, andit
induces an emf V2 in the secondary coil that is
proportional to N2. Therefore, V2 V1(N2/N1).
From conservation of energy, assuming no losses
in the core, V1I1 V2I2. Therefore, the
currents in the primary and secondary are related
by the relation I1 I2(N2/N1). A
transformer with N2gtgtN1 is called a step-up
transformer, which boosts the secondary voltage.
A transformer with N2ltltN1 is called a step-down
transformer, and it drops the secondary voltage.
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The Tesla Coil
A special case of a step-up transformer is
the Tesla coil. It uses no magnetic material,
but has a very high N2/N1 ratio and uses
high-frequency electrical current to induce very
high voltages and very high frequencies in the
secondary. There is a phenomenon called
the skin effect that causes high frequency AC
currents to reside mainly on the outer surfaces
of conductors. Because of the skin effect, one
does not feel (much) the electrical discharges
from a Tesla coil.
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Metal Detectors
Metal detectors like those used at airports
can detect any metal objects, not just magnetic
materials like iron. They operate by induced
currents. A transmitter coil sends high
frequency alternating currents that will induce
current flow in conductors in its field. Because
of Lenzs Law, the induced current opposes the
field from the transmitter, so that net field is
reduced. A receiver coil detects the reduction
in the magnetic fields from the transmitter and
registers the presence of metal.
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(Self-) Inductance
We define the inductance L of a coil of wire
producing flux Fm as
The unit of inductance is the henry 1
henry 1 H 1 T m2/A 1 Wb/A
The circuit diagram symbol used to represent
inductance is
Example The inductance of a long solenoid
with N turns of cross sectional area A and length
l is
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Example Length of an Inductor
An inductor is made by tightly winding 0.30
mm diameter wire around a 4.0 mm diameter
cylinder. What length cylinder has an
inductance of 10 mH?
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Potential Across an Inductor
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Potential Across an Inductor (2)
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Lecture 21 Announcements

• Lecture HW has been posted and is due on
  Wednesday at 10 PM.
• Next Friday we will have Exam 3 in this room.
  It will consist of multiple-choice questions on
  the Laboratory (25 pts) and Lecture (35 pts).
  Bring a Scatron sheet, a double-sided page of
  notes, and a calculator with good batteries.