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Right-Hand Rule
 Right-hand Rule 1 gives
direction of Magnetic
Field due to current
 Right-hand Rule 2 gives
direction of Force on a
moving positive charge
Section 20.1
Example: Two Parallel Wires
 What is the force on
I
1
wire 1 due to wire 2?
 Is the force attractive or
r
L
repulsive?
2
I
Magnetic Force on a Current
 A current is a collection of moving
charges
 The direction of the force is given by the
right-hand rule 2
Section 20.4
Ampère’s Law
 Ampère’s Law can be
used to calculate the
magnetic field when there
is symmetry
 Similar to Gauss’ Law for
electric fields
 Relates the magnetic field
along a path to the electric
current enclosed by the
path
Section 20.7
Ampère’s Law, cont.
 The magnetic field along a
closed path is related to the
current enclosed by that path
 μo is the permeability of free
space
 μo = 4 π x 10-7 T . m / A
 If B varies in magnitude or
direction along the path,
Ampère’s Law is not useful
Section 20.7
Magnetic Field of a Long Straight Wire
 Ampère’s Law can be used to
find the magnetic field near a
long, straight wire
 Choose a circular, closed path
 B|| is the same all along the path
 If the circular path has a radius
r, then the total path length is 2
πr
 Applying Ampère’s Law gives
μo I
B
2π r
Section 20.7
Example: Two Parallel Wires
 What is the force on
I
1
wire 1 due to wire 2?
 Is the force attractive or
r
L
repulsive?
2
I
Field from a Current Loop
 It is not possible to find
a simple path along
which the magnetic field
is constant
 Ampère’s Law cannot be
easily applied
 From other techniques,
μo I
B
2R
Section 20.7
Field Inside a Solenoid
 By stacking many loops close
together, the field along the
axis is much larger than for a
single loop
 A helical winding of wire is
called a solenoid
 More practical than stacking
single loops
 For a long solenoid, there is
practically no field outside
Section 20.7
Example: Long Solenoid
What is the magnetic
field, B, inside a “long”
solenoid?
Torque on a Current Loop
 A magnetic field can produce a torque on a
current loop
 The directions of the forces can be found from
right-hand rule 2
Section 20.5
Torque, cont.
 On two sides, the current is parallel or antiparallel to the field, so the
force is zero on those sides
 The forces on sides 1 and 3 are in opposite directions and produce a
torque on the loop
 When the angle between the loop and the field is θ, the torque is
τ = I L2 Bext sin θ
 For different shapes, this becomes
Section 20.5
τ = I A Bext sin θ
Magnetic Moment, μ
 For a current loop, the
magnetic moment, μ, is I A
 The direction of the magnetic
moment is either along the axis
of the bar magnet or
perpendicular to the current
loop
 The strength of the torque
depends on the magnitude of
the magnetic moment
 τ = μ B sin θ
Section 20.5
Electric Motor
 A magnetic field can produce a torque on a current loop
 In a practical motor, a solenoid is used instead of a single loop
 Additional set-up is needed to keep the shaft rotating
 Electric generators are motors in reverse
 A generator produces an electric current by rotating a coil between
the poles of the magnet
Section 20.10
Magnetism Summary
 Magnetic Fields: (T)
 North and South poles
 Ampere’s Law



Field from straight wire
 Right-hand rule 1
Field from current loop
Field from Solenoids
 Magnetic Forces: (N)
 Force on moving charge



Right-hand rule 2
Mass spectrometer
Hall Effect
 Force on current wire


Torque on current loop
Magnetic moment
Chapter 21
Magnetic Induction
Electromagnetism
 Electric and magnetic phenomena were connected
by Ørsted in 1820
 He discovered an electric current in a wire can exert a
force on a compass needle
 He concluded that a changing electric field will
produce a magnetic field
 Can a magnetic field produce an electric field?
 Experiments were done by Michael Faraday
Section 21.1
Faraday’s Experiment
 If the bar magnet was in
motion, a current was
observed
 If the magnet is stationary,
the current and the
electric field are both zero
 Same results occurred by
moving the loop instead of
the magnet
Section 21.1
Faraday’s Experiment, cont.
 A solenoid is positioned inside a loop of wire
 When the current through the solenoid is constant, there is no
current in the wire
 When the switch is opened or closed, current flows in the wire
Section 21.1
Conclusions from Experiments
 Faraday’s experiments show that an electric current
is produced in the wire loop only when the magnetic
field through the loop is changing
 A changing magnetic field produces an
electric field
 An electric field produced in this way is called an
induced electric field
 The phenomena is called electromagnetic induction
Section 21.1
Magnetic Induction
 Moving electric charges created a magnetic field
 Moving charges change the electric field
 A changing magnetic field created an electric field
 This effect is called magnetic induction
 This links electricity and magnetism in a fundamental
way
 Magnetic induction is also the key to many practical
applications
 WiTricity:
http://www.ted.com/talks/eric_giler_demos_wireless_electricity.html
Magnetic Flux
 Faraday developed a quantitative theory of induction
now called Faraday’s Law
 The law shows how to calculate the induced electric
field in different situations
 Faraday’s Law uses the concept of magnetic flux
 Magnetic flux is similar to the concept of electric flux
 The magnetic flux is
Section 21.2
Magnetic Flux, cont.
 If the field is perpendicular to the surface, ΦB = B A
 If the field makes an angle θ with the normal to the
surface, ΦB = B A cos θ
 If the field is parallel to the surface, ΦB = 0
Section 21.2
Magnetic Flux, final
 The magnetic flux can be defined for any surface
 A complicated surface can be broken into small
regions and the definition of flux applied
 The total flux is the sum of the fluxes through all the
individual pieces of the surface
 The unit of magnetic flux is the Weber (Wb)
 1 Wb = 1 T . m2
Section 21.2
Faraday’s Law
 Faraday’s Law indicates how to calculate the
potential difference that produces the induced
current
 Written in terms of the emf (voltage) induced in the
wire loop
 B
ε
t
 The induced voltage is proportional to the rate of
change of the magnetic flux
 The negative sign is Lenz’s Law
Section 21.2
Applying Faraday’s Law, cont.
 The emf is produced by changes in the magnetic
flux through the circuit
 A constant flux does not produce an induced
voltage
 The flux can change due to
 Changes in the magnetic field
 Changes in the area
 Changes in the angle
 The voltmeter will indicate the direction of the
induced emf and induced current and electric field
 It is related to the electric field directly along and
inside the wire loop
Section 21.2
Faraday’s Law, Summary
 Only changes in the magnetic flux matter
 Rapid changes in the flux produce larger
values of emf than do slow changes
 This dependency on frequency means the
induced emf plays an important role in AC
circuits
 The magnitude of the emf is proportional to
the rate of change of the flux
 B
ε
t
 If the rate is constant, then the emf is
constant
 In most cases, this isn’t possible and AC
currents result
Section 21.2
Flux Though a Changing Area
 What is the magnitude of
the voltage induced in the
wire loop?
 What is the power
dissipated through the
resistor?
Section 21.2
Conservation of Energy
 The mechanical power put into the bar equals the
electrical power delivered to the resistor
 Energy is converted from mechanical to electrical,
but the total energy remains the same
 Conservation of energy is obeyed by
electromagnetic phenomena
Section 21.2
Changing a Magnetic Flux, Summary
 A change in magnetic flux can be produced in four
ways
 If the magnitude of the magnetic field changes with
time
 If the area changes with time
 If the loop rotates so that the angle changes with time
 If the loop moves from one region of magnetic field to
another region of differing magnetic field
Section 21.2
Lenz’s Law
 Lenz’s Law gives an easy way to
determine the sign of the induced
emf
 Determine direction of induced
current
 Lenz’s Law states “the magnetic
field produced by an induced
current always opposes any
changes in the magnetic flux”
Section 21.3
Lenz’s Law, Example 1
 Consider a metal loop in which the magnetic field passes
upward through the loop and increases with time
 The magnetic field produced by the induced emf must
oppose the change in flux
 Therefore, the induced magnetic field must be downward
and the induced current will be clockwise
Section 21.3
Lenz’s Law, Example 2
 Consider a metal loop in which the magnetic field passes
upward through the loop and decreases with time
 The magnetic field produced by the induced emf must
oppose the change in flux
 Therefore, the induced magnetic field must be downward
and the induced current will be counterclockwise
Section 21.3
Example: Falling Magnet
 Suppose a bar magnet
falls downward through a
loop of wire, north pole
first.
v
N
v
S
 What is the direction of the
induced current?
 Suppose a bar magnet
falls upward through a
loop of wire, south pole
first.
 What is the direction of the
induced current?
Lenz’s Law and Conservation of
Energy
 Mathematically, Lenz’s Law is just the negative sign in
Faraday’s Law
 It is actually a consequence of conservation of energy
 Therefore, conservation of energy is contained in
Faraday’s Law
 Physicists believe all laws of physics must (eventually)
satisfy the principle of conservation of energy
Section 21.3
Electrical Generator
 Uses changing flux to induce current in wire
loop
 The wire loop has a fixed area, but is mounted on a
rotating shaft
 The angle between the field and the plane of the
loop changes as the loop rotates
 If the shaft rotates with a constant angular
velocity, the flux varies sinusoidally with time
 Sinusoidal voltage is source of AC circuits
Section 21.2
Applications of Induction
 A ground fault interrupter (GFI) is a safety device used in many household
circuits
 It uses Faraday’s Law along with an electromechanical relay
 The relay uses the current through a coil to exert a force on a magnetic
metal bar in a switch
 During normal operation, there is zero magnetic field in the relay
 If the current in the return coil is smaller, a non-zero magnetic field opens the
relay switch and the current turns off
Section 21.7
Electric Guitars
 An electric guitar uses
Faraday’s Law to sense
the motion of the strings
 The string passes near a
pickup coil wound around
a permanent magnet
 As the string vibrates, it
produces a changing
magnetic flux
 The resulting emf is sent
to an amplifier and the
signal can be played
through speakers
Section 21.7
Railgun
 Current produces a large magnetic
field which interacts with the
current to produce a large force on
the projectile
 Projectile speeds much greater
than conventional ballistics are
possible
 Proposed uses are
 Weapons
 Space Program
 Inertial Confinement Fusion
 http://www.youtube.com/watch?v=6BfU-wMwL2U