Download Physics 272

Physics 272
October 21
Fall 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_fall_272_uhm.html
Prof. Philip von Doetinchem
[email protected]
Phys272 - Fall 14 - von Doetinchem - 121
Ampere's law
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Magnetic field calculation so far based on adding
contribution of small line segments
→ analogous to electric field calculation with adding
up electric fields of
individual charges
Phys272 - Fall 14 - von Doetinchem - 122
Ampere's law
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Integrate along a closed path around a wire:
Phys272 - Fall 14 - von Doetinchem - 123
Ampere's law
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Integrate along a closed path, but not totally
enclosing the wire:
Important: does not
mean that the
magnetic field is
zero at every spot
along the path
●
If the current is not enclosed the line integral around
a closed path is zero
Phys272 - Fall 14 - von Doetinchem - 124
Ampere's law: general statement
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The scalar product always takes the parallel
projection for the integration
→ as long as the closed path integration encloses
the current we get:
This also true if multiple conductors are present
within the closed integration path
(magnetic field is the sum of the individual magnetic
field of the different conductors)
This law is true for steady currents
Phys272 - Fall 14 - von Doetinchem - 125
Important differences electric field vs. magnetic field
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Electric field:
–
Line integral along a closed path is zero
–
Electric force is conservative
→ force does zero net work on charge that returns to
start point
Magnetic field:
–
Magnetic force on a moving charge is always
perpendicular
→ line integral over magnetic field is not related to the
work done by magnetic force
–
Magnetic force is not conservative
→ force does not only depend on position, but also on
velocity
Phys272 - Fall 14 - von Doetinchem - 126
Field of a long, straight, current-carrying conductor
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Similar to the example before, but reversed
situation: magnetic field unknown, current known
Identify symmetry:
–
magnetic field is tangent to a circle around the conductor
–
Magnetic field magnitude is the same everywhere on the
circle
Phys272 - Fall 14 - von Doetinchem - 127
Field of a solenoid
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Helical winding of wire on a
cylinder
Each winding can be treated as
circular loop
All turns carry the same current
Field is very uniform in the
middle
External field near the middle is
very small
→ Assumption: magnetic field
zero outside and uniform inside
●
Field is the strongest in the
center
Phys272 - Fall 14 - von Doetinchem - 129
Field of a solenoid
Phys272 - Fall 14 - von Doetinchem - 130
Paramagnetism
●
Moving electrons in atoms
cause current loops
→ currents are typically
completely random in material
→ in some materials the current
loops can be oriented in an
external magnetic field
(material is magnetized
Source: http://de.wikipedia.org/wiki/Paramagnetismus
→ atomic magnetic field adds to the external magnetic field
●
When you place the material in a magnetic field
→ field exerts torque
→ tries to align magnetic moments
●
Magnetic field of a current loop is proportional to the magnetic dipole moment
→ Magnetization:
(total magnetic
moment per unit
volume)
Phys272 - Fall 14 - von Doetinchem - 131
Paramagnetism
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If a magnetized material completely surrounds a current-carrying wire:
Materials that can be magnetized
are called paramagnetic
Magnetic field at any point in such
a material is enhanced by a
dimensionless factor with respect
to vacuum
→ relative permeability: Km
Change of magnetic dipole moment
in material:
●
Two competing effects:
–
Alignment of magnetic dipole
moments in external field
–
Random thermal motion →
randomizes orientation of dipole
moments
→ increasing temperature
decreases magnetic susceptibility
→ paramagnetic bodies feel
stronger attraction to magnets at
cold temperatures
Phys272 - Fall 14 - von Doetinchem - 132
Ferromagnetism
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Examples: iron, nickel, cobalt, ...
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Strong interactions of atomic magnetic dipole moments
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magnetic domains: Complete regions with lined up/parallel magnetic moments
(also present without any external magnetic field)
Domains can be aligned with external field
Permeability is much higher than for paramagnetic materials (1,000-100,000x)
→ ferromagnetic materials are much stronger attracted by a magnet
→ for instance: magnets pick up iron nails, but no aluminum cans
→ use in electromagnets, transformers, generators,...
Phys272 - Fall 14 - von Doetinchem - 133
Diamagnetism
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Some atomic materials
have a zero total magnetic
moment when no magnetic
field is present
http://www.youtube.com/watch?v=IFv4VOrWecI
BUT: magnetic effects can
be caused by external magnetic fields altering the electron
motions inside the atom (diamagnetic)
→ additional current loops are created
→ additional field is in the opposite
direction of external field
(electromagnetic induction)
●
Diamagnetic susceptibility depends
on how easy it is to induce a net
magnetic moment in an atom with
no magnetic moment in the absence
of external fields
→ effect is independent of the initial
orientation of the atom
→ not affected much by temperature
→ weaken the external magnetic field
Phys272 - Fall 14 - von Doetinchem - 134
Electromagnetic induction
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Demo 1:
–
magnet moved in
→ magnetic flux through
solenoid changes
→ induced current
appears
–
The faster the magnet
the higher the induced
current
http://www.youtube.com/watch?v=hajIIGHPeuU
–
If solenoid is approached
first with the other magnetic pole, the direction of the induced
current changes
–
When magnet is moved away from the solenoid the direction of the
current changes again.
Demo 2:
–
Same as demo 1, but using a different coil and a digital multimeter.
Phys272 - Fall 14 - von Doetinchem - 135
Electromagnetic induction
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Demo 3:
–
two solenoids: one large
one connected in a
simple circuit and a
second, smaller one,
connected to an ammeter
–
When switch is closed
→ a DC current is
established in the circuit
→ steady magnetic field is
http://www.youtube.com/watch?v=hajIIGHPeuU
produced in the large
solenoid
→ no induced current in the small solenoid as the magnetic flux through it
does not change
–
when switch is switched on or off
→ an induced current is produced
→ for a short period of time the current changes
→ magnetic field is produced by the large solenoid changes as well
→ induced current in the small solenoid.
Phys272 - Fall 14 - von Doetinchem - 136
Changing magnetic flux
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The key component is the changing magnetic flux
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Flux changes caused by
●
–
magnetic field changes with time
–
coil moves through a non-uniform magnetic field
The changing flux causes an induced electromotive
force
–
Proportional to the rate of change of magnetic flux
through the coil
–
Direction of the induced emf depends on if the flux is
increasing or decreasing
–
No flux change = no induced emf
Phys272 - Fall 14 - von Doetinchem - 137
Faraday's law
●
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Induction is a very important
effect that is widely used
Electric generators produces emf
by varying magnetic flux through
coils of wire
Source: http://de.wikipedia.org/wiki/Elektrischer_Generator
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Basic concept: changing magnetic flux through a circuit
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Faraday's law of induction:
–
The induced electromotive force in a closed loop equals the
negative of the time rate of change of magnetic flux through the
loop.
Phys272 - Fall 14 - von Doetinchem - 138
Emf and current induced in a loop
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Uniform magnetic field
between poles of
electromagnet, but
magnitude is increasing
by 0.020T per second
Coil with area of
120cm2 is in this field,
total resistance 5
Phys272 - Fall 14 - von Doetinchem - 139
Direction of induced electromagnetic fields
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For increasing external magnetic field the induced magnetic field is in the opposite direction and
works against the external field
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For decreasing external magnetic field the induced magnetic field is in the same direction
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Sign rules for the direction of induced emf:
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Define positive direction of area
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Determine the sign of the magnetic flux from the area and the magnetic field
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If flux is increasing → induced emf is negative
If flux is decreasing → induced emf is positive
–
Right hand rule:
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align area vector with thumb
Positive emf → current is in the same direction as curled fingers
Negative emf → current is in the opposite direction of curled fingers
Not magnetic flux, but
changing magnetic flux
causes induction effects
Phys272 - Fall 14 - von Doetinchem - 140
Lenz's law
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Alternative method for determining
the direction of induced current or
emf
Lenz's law can be derived from
Faraday's law
The direction of any magnetic
induction effect is such as to
oppose the cause of the effect
Cause can be
–
Changing flux due to varying magnetic field
–
Changing flux due to motion of conductors
Source: http://de.wikipedia.org/wiki/Emil_Lenz
Heinrich F. E. Lenz
1804-1865
Think about it like: induced current tries keeping the
system in the state it was before the flux change happened.
Phys272 - Fall 14 - von Doetinchem - 141
A simple alternator
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An alternator is a device that generates emf
Phys272 - Fall 14 - von Doetinchem - 142
A simple alternator
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Emf is sinusoidal with time → alternating current
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Plane perpendicular to magnetic field: maximum(minimum) flux
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Plane (anti)parallel: zero flux
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Fastest change when plane (anti)parallel
when angular speed is doubled the rate of change of the flux
doubles and this causes the induced emf and induced current
to double
→ torque required is proportional to the current in the loop, so
the torque also doubles
Careful:
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Electromotive force is not created out of nowhere
–
Energy must be conserved and energy has to be supplied to
make the loop spin → energy conversion
Phys272 - Fall 14 - von Doetinchem - 143
Slidewire generator
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Look at individual charge in slidewire:
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Feels magnetic force
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Separates charges
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Builds up electric field
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Equilibrium between electric force and
magnetic force
(→ also see Hall effect)
No magnetic forces act on the charges in
the stationary U part, but sliding rod
creates potential difference (source of emf)
→ establishes current
Phys272 - Fall 14 - von Doetinchem - 147
Motional emf in the slidewire generator
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Suppose the moving rod in the slidewire configuration
is 0.1m long and 2.5m/s fast with a resistance of 0.03
and a uniform magnetic field of 0.6T:
–
Induced current in the loop:
–
Force on the rod
Phys272 - Fall 14 - von Doetinchem - 148
Motional electromotive force
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Origin of electromotive force is of non-electrostatic
nature (similar to battery → chemical)
Charges are brought to a higher potential
Concept can be generalized to conductors of any
shape and in any field (can be non-uniform, but not
varying with time)
–
take the perpendicular projection of the velocity with
respect to the magnetic field (cross product)
–
Use the parallel projection of the former along a line
element of the conductor (scalar product)
Phys272 - Fall 14 - von Doetinchem - 149
Additional material
Phys272 - Fall 14 - von Doetinchem - 150
Bar on inclined plane
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Bar on inclined plane
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Bar on inclined plane
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Bar on inclined plane
Phys272 - Fall 14 - von Doetinchem - 154
Field of a long cylindrical conductor
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Cylindrical conductor with radius R carries a current
I. Current is uniformly distributed over the crosssectional area of the conductor.
conductor
Phys272 - Fall 14 - von Doetinchem - 155
Field of a long cylindrical conductor
Phys272 - Fall 14 - von Doetinchem - 156
Field of a toroidal solenoid
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Tightly wound turns
in donut shape
magnetic field concentric
with toroid axis
Each turn is
perpendicular to the
circular axis of the
toroid
Ideal toroid confines
magnetic field to the
space between windings
Phys272 - Fall 14 - von Doetinchem - 157
Field of a toroidal solenoid
Phys272 - Fall 14 - von Doetinchem - 158
Field of a toroidal solenoid
Phys272 - Fall 14 - von Doetinchem - 159
A ferromagnetic material
●
Consider a cube (side length 2cm) shaped
permanent magnet with magnetization of 8x10 5 A/m
Phys272 - Fall 14 - von Doetinchem - 160
More complicated wire calculation
3
3
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More complicated wire calculation
2
2
Phys272 - Fall 14 - von Doetinchem - 162
More complicated wire calculation
B2,x
B2,y=
Phys272 - Fall 14 - von Doetinchem - 163
More complicated wire calculation
B3,z=
Phys272 - Fall 14 - von Doetinchem - 164
More complicated wire calculation
2,x
Phys272 - Fall 14 - von Doetinchem - 165
Bar on inclined plane
Phys272 - Fall 14 - von Doetinchem - 166
Bar on inclined plane
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Bar on inclined plane
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Bar on inclined plane
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