Physics%20122B%20%20Electricity%20and%20Magnetism

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Physics 122B Electricity and Magnetism
Lecture 23 (Knight 33.8-33.10) Inductance and LR
Circuits

• Martin Savage

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Lecture 23 Announcements

• Lecture HW has been posted and is due on
  Wednesday at 10 PM.
• Midterm examination 3 is this coming
  Friday---questions from lecture, tutorial AND Lab.

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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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Inductors and Sparks
An inductor responds to an interruption in
current flow by developing a very large potential
(L dI/dt) across its terminals. This potential
is typically dropped across the object (usually a
switch) that is effecting the change in current.
Therefore, breaking a closed circuit with an
inductor often results in a spark across the
switch contacts. Over time, this can erode and
destroy the switch. Spark plugs in
automobiles are driven by inductances that are
electronically interrupted.
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ExampleLarge Voltage across an Inductor
A 1.0 A current passes through a 10 mH
inductor coil. What potential difference is
induced across the coil if the current drops to
zero in 5 ms?
( 1.0 A )
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Question
The potential at a is higher than the
potential at b. Which of the following
statements about the inductor current I is
consistent with this observation?

1. I is from a to b and steady.
2. I is from a to b and increasing
3. I is from a to b and decreasing
4. I is from b to a and steady
5. I is from b to a and increasing

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Energy in Inductors and Magnetic Fields
A magnetic field stores considerable energy.
Therefore, an inductor, which operates by
creating a magnetic field, stores energy. Lets
consider how much energy UL is stored in an
inductor L carrying current I
A solenoid of length l , area A, and N turns has
L m0N2A/l, so
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Energy Density in the Fields
This is the magnetic analog of the energy stored
in an electric field
Magnetic
Electric
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The LR Circuit
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Example Exponential Decayin an LR Circuit
The switch shown has been in position a for
a long time. It changes to position b at t0.
a. What is the current in the circuit at t 5
ms? b. At what time has the current decayed
to 1 of its original value?
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Plumbers RL Analogy
Constriction
Valve
P1
P2
Pump
Flywheel
The plumbers analogy of an RL circuit is a
pump (battery) pumping water in a closed loop of
pipe that includes a valve (switch), a
constriction (resistor), and rotating flywheel
turned by the flow. When the valve starts the
flow, the flywheel begins to spin until a steady
flow is achieved and the pressure difference
across the flywheel (P2-P3) goes to zero.
P3
Pump Battery Valve Switch Constriction
Resistor Flywheel Inductor Pressure
PotentialWater Flow Current
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Capacitors Early and Late
Initially, when a switch closes there is a
potential difference of 0 across an uncharged
capacitor. After a long time, the capacitor
reaches its maximum charge and there is no
current flow through the capacitor. Therefore,
at t0 the capacitor behaves like a short circuit
(R0), and at t8 the capacitor behaves like at
open circuit (R8). Example
100 V
100 V
12.5 A
100 V
10 A
2.5 A
0 V
at t0
at t8
Circuit
Calculate initial currents. IB 100 V/8 W
12.5 A
Calculate final potentials.
From Lecture 16
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Inductors Early and Late
Initially, when a switch closes an inductor
appears to have an infinite resistance and has a
maximum potential drop across it. Ultimately,
the inductor reaches a steady current flow with
no potential drop across it. Therefore, at t0
the inductor behaves like at open circuit (R8),
and at t8 the inductor behaves like a short
circuit (R0). This behavior is opposite that of
a conductor. Example
at t0
at t8
Circuit
Calculate initial potentials.
Calculate final currents.
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Question
Rank the order of the time constants in these
circuits.

• t1t2t3 (b) t1ltt2ltt3 (c) t2ltt1ltt3
  
• (d) t1gtt2gtt3 (e) t2gtt1gtt3

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Lecture 23 Announcements

• Lecture HW has been posted and is due on
  Wednesday at 10 PM.
• Midterm examination 3 is this coming
  Friday---questions from lecture, tutorial AND Lab.