Physics 272
November 6
Fall 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_fall_272_uhm.html
Prof. Philip von Doetinchem
philipvd@hawaii.edu
Phys272 - Fall 14 - von Doetinchem - 1
Midterm 2
Phys272 - Fall 14 - von Doetinchem - 2
Maxwell's equations
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Gauss's law for electric fields (surface integral)
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Gauss's law for magnetism (surface integral)
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Ampere's law (line integral)
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Faraday's law (line integral)
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Mutual 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
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
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Mutual Inductance
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Flux through coil 2 changes:
powered
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Change in flux is proportional to current in coil 1
→ mutual inductance
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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
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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 unwanted
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
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Calculating mutual inductance
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Current in either coil causes
a flux through the other coil
No field outside of solenoid
Substantial induced emf
possible for rapid change
Coil 1 is powered
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Self-inductance and inductors
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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
Self-induced emf prevents rapid changes in
circuit
The greater the rate change of current
→ the greater the self-induced emf
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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
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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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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
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Inductors as circuit elements
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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
(ideal inductor: zero resistance → no energy
dissipation)
Similar to charging capacitor: trying to increase
current against magnetic field of inductor
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Magnetic-field energy
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When current decreases inductor acts 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 - Fall 14 - von Doetinchem - 15
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:
Valid for any magnetic-field configuration in a material with constant
permeability
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
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The R-L circuit
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Inductors in circuits:
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When we go through an inductor in the same direction as
the current → voltage drop (-Ldi/dt)
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When we go through an inductor in the opposite direction
to the current → voltage increase (Ldi/dt)
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The R-L circuit
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Solve differential equation
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The R-L circuit
Final current does
not depend on L
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For long times → current change gets
slower
At the instant when S1 is closed:
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No current drop over resistor, i=0:
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The greater L, the slower the increase
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Current increases over time and approaches
final current → di/dt goes to zero
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Current decay in an R-L circuit
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If S1 is closed for a while
→ current reached final value
→ then close S2 and open S1 at
the same time
Current goes to zero following an
exponential decrease
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R-L circuit
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Time constant for R-L circuit:
small L current increases/decreases faster
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Compare to R-C circuit:
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Energy consideration:
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The L-C circuit
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Start with a charged capacitor
Capacitor discharges through
inductor
Capacitor potential difference
always equals induced emf in
inductor
Current increases to a certain
maximum value
Increasing current in inductor establishes magnetic field
→ energy stored in electric field of capacitor is
transferred into magnetic field in inductor
Once the capacitor is fully discharged (potential zero)
the current in the circuit does not stop instantaneously
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The L-C circuit
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Instead it continues in the same
direction for a little bit
Capacitor is charging in opposite
direction → current drops
At the same time the magnetic
field in the inductor becomes
weaker
Inductor provides emf in direction of current and
supports charging of capacitor in opposite direction
Capacitor is fully charged in opposite direction
→ no current flow, no magnetic field in inductor, no
opposing emf
Process starts the other way around → Oscillation
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Electrical osciallations in an L-C circuit
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Formal way:
This is the same type of equation describing
mechanical harmonic oscillations
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Electrical osciallations in an L-C circuit
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Solution:
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Current in LC circuit:
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Energy in an L-C circuit
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Energy conservation gives same result:
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Additional Material
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Calculating mutual inductance
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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
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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)
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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
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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
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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 - Fall 14 - von Doetinchem - 39
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 - Fall 14 - von Doetinchem - 40
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
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Inductor example
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Sensitive electronics with R=175 operates at 36mA
Critical: current should not rise above 4.9mA within the
first 58s after turn on
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Use an inductor in series with component
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Required emf:
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Calculate required L from current increase formula in R-L
circuit:
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LC circuit example
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A 300V has charged a 25F capacitor connected to
a 10mH inductor
Capacitor charge after 1.2ms:
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