Physics 272
April 1
Spring 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html
Prof. Philip von Doetinchem
philipvd@hawaii.edu
Phys272 - Spring 14 - von Doetinchem - 164
Summary
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Gauss's law for electric fields (surface integral)
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Electric field is related to total charge in an enclosed
surface
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Electric charges are sources of magnetic fields
Gauss's law for magnetism (surface integral)
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No magnetic monopoles exist
→ magnetic flux through closed surface is always zero
Phys272 - Spring 14 - von Doetinchem - 165
Summary
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Ampere's law (line integral)
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Conducting and displacement current act as sources of
magnetic fields
Faraday's law (line integral)
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A changing magnetic field or magnetic flux induces an
electric field
Phys272 - Spring 14 - von Doetinchem - 166
Inductance
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Coiled pieces of wire behave differently in a circuit
than just straight wire
A changing current in a coil induces an emf in an
adjacent coil
A changing current also induces a n emf in that
same coil
If a coil carries a current
→ energy is released when the current decreases
→ automotive ignition system
→ this energy was stored in the magnetic field
Phys272 - Spring 14 - von Doetinchem - 167
Mutual Inductance
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We already studied:
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Magnetic interaction between two wires carrying steady
currents
→ current in one wire causes a magnetic field
→ exert force on second wire
Now: changing currents
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If current in coil 1 changes
→ magnetic flux in coil 2
changes
→ according to Faraday's
law: emf is induced
Phys272 - Spring 14 - von Doetinchem - 168
Mutual Inductance
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Flux through coil 2 changes:
Change in flux is proportional to current in coil 1
→ mutual inductance
Phys272 - Spring 14 - von Doetinchem - 169
Mutual Inductance
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Flux through coil 2 is directly proportional to current
in coil 1 if both coils are in vacuum
Mutual inductance only depends on geometry of coils
and the orientation of the coils to each other
Mutual inductance is also the same for the case when
coil 2 carries the current and induces a current in coil 1:
Careful:
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only time varying currents induce emf in second coil
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Induced emf is directly proportional to rate of change of
the current
Phys272 - Spring 14 - von Doetinchem - 170
Mutual Inductance
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Unit for mutual inductance
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Typical values:
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millihenry (mH) or microhenry (H)
Source: http://en.wikipedia.org/wiki/Joseph_Henry
Joseph Henry
(1797-1878)
Circuit design requires to suppress unwante mutual
induction between nearby circuits
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e.g., place coils far apart from each other
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Can also be very useful: transformers
Phys272 - Spring 14 - von Doetinchem - 171
Calculating mutual inductance
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Current in either coil causes a flux through the other coil
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No field outside of solenoid
Coil 1 is powered
Phys272 - Spring 14 - von Doetinchem - 172
Calculating mutual inductance
Phys272 - Spring 14 - von Doetinchem - 173
Emf due to mutual inductance
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Substantial induced emf due to rapid change
Emf is constant for this case because the current
change happens at a constant rate
Phys272 - Spring 14 - von Doetinchem - 174
Self-inductance and inductors
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So far: two circuits:
Current in circuit 1 changes → flux through circuit 2
changes → emf is induced in circuit 2
Related effect occurs also in single isolated circuit
Current in circuit sets up magnetic flux through the
same circuit
This flux changes when the current in the circuit
changes
Any circuit carrying a varying current has an emf
induced by the variation of its own magnetic field
→ self-induced emf
Phys272 - Spring 14 - von Doetinchem - 175
Self-inductance and inductors
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Effect occurs in any circuit, but is enhanced for coils
with many turns
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Self-inductance L is defined as:
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current changes in circuit → magnetic flux changes
self-induced emf after applying Faraday's law
Phys272 - Spring 14 - von Doetinchem - 176
Inductors as circuit elements
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Circuit devices with a specific inductance are called
inductors:
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Important component of modern electronics
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Purpose:
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oppose variations in the current through the circuit
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DC: maintain a steady current, despite fluctuations in
applied emf
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AC: suppress variations that are too fast
Phys272 - Spring 14 - von Doetinchem - 177
Inductors as circuit elements
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We have to modify Kirchhoff's rule when using inductors
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The magnetically induced electric field in the coils of the inductor is
not conservative
Use Faraday's law:
In case of an open inductor i(t)=0,
potential difference changes with
current flow
Potential difference across an inductor depends on the rate of
change of the current
Phys272 - Spring 14 - von Doetinchem - 178
Inductors as circuit elements
Phys272 - Spring 14 - von Doetinchem - 179
Applications of inductors
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Inductors are useful to keep currents stable:
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When currents grow too high
→ induced emf reduces current
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When currents go too low
→ induced emf sustains current, prevents shut offs
Self-inductance depends on
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size, shape, turns
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magnetic properties of the enclosed material (0→)
(ferromagnetic material makes a big difference)
Phys272 - Spring 14 - von Doetinchem - 180
Applications of inductors
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Example: traffic light sensor
automobiles contain steel
→ driving a car over a current-carrying coil
embedded in the street
→ circuitry detects inductance change
→ green light will be triggered
Source: http://de.wikipedia.org/wiki/Induktionsschleife
Phys272 - Spring 14 - von Doetinchem - 181
Calculating self-inductance
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What is the self-inductance of a
toroidal solenoid?
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Cross section A, assume B uniform,
N turns
Phys272 - Spring 14 - von Doetinchem - 182
Calculating self-induced emf
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Current increases from 0 to 6A in 3s
Current is increasing
→ magnetic field is increasing
→ flux is increasing
self-induced emf opposes the incoming current
Phys272 - Spring 14 - von Doetinchem - 183
Magnetic-field energy
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Inductor carrying a current has energy stored in it
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This requires to input energy
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Energy input needed to establish final current in an
inductor (zero resistance → no energy dissipation)
Phys272 - Spring 14 - von Doetinchem - 186
Magnetic-field energy
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When current decreases inductor as a source
supplying 1/2LI2 to the external circuit
Example:
very sudden decrease in current by unplugging
device from wall socket
→ induced emf is large
→ ionizes the air and creates arc
Resistor vs. inductor:
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Energy is dissipated in resistor for any type of current,
steady or varying in time
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Ideal inductor stores energy if current is increasing and
releases energy when current is decreasing (no
dissipation), steady current → no change in energy
Phys272 - Spring 14 - von Doetinchem - 187
Magnetic energy density
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Energy is stored in magnetic field in inductor like the energy
is stored in the electric field of a capacitor
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Ideal toroidal solenoid (assume uniform magnetic field)
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Volume of toroidal solenoid:
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Energy density:
Phys272 - Spring 14 - von Doetinchem - 188
Magnetic energy density
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Energy is stored in magnetic field in inductor like the
energy is stored in the electric field of a capacitor
Energy density in terms of magnetic field:
This is also the correct expression for any magneticfield configuration in a material with constant
permeability
Phys272 - Spring 14 - von Doetinchem - 189
Magnetic energy density
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Example: ignition system in gasoline-powered cars
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Primary coil produces a strong magnetic field with N=250
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Secondary coil surrounds primary coil with N=25,000
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Interrupt current in primary coil
→ magnetic field drops
→ very high emf is induced in the second coil and
causes spark plug to fire
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energy stored in magnetic field causes a short powerful
pulse
Phys272 - Spring 14 - von Doetinchem - 190
Coronal mass ejection
Source: NASA Goddard Space Flight Center
https://www.youtube.com/watch?v=YtqYKRcemRs
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Magnetic field of the sun is constantly changing
Release of stored magnetic energy can cause
coronal mass ejection of billion tons of material
Phys272 - Spring 14 - von Doetinchem - 191