Transcript phys1444-lec15

PHYS 1444
Lecture #15
Thursday August 2, 2012
Ian Howley
•
•
Inductance
L/LR/LRC Circuits
Thursday August 2, 2012
Dr. B will assign final (?)
HW today(?)
It is due next Thursday(?)
PHYS 1444 Ian Howley
1
Self Inductance
• The concept of inductance applies to a single isolated coil of
N turns. How does this happen?
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When a changing current passes through a coil
A changing magnetic flux is produced inside the coil
The changing magnetic flux in turn induces an emf in the same coil
This emf opposes the change in flux. Whose law is this?
• Lenz’s law
• What would this do?
– When the current through the coil is increasing?
• The increasing magnetic flux induces an emf that opposes the original current
• This tends to impede the increased current
– When the current through the coil is decreasing?
• The decreasing flux induces an emf in the same direction as the current
Thursday August 2, 2012
PHYS 1444 Ian Howley
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Self Inductance
• Since the magnetic flux FB passing through an N turn
coil is proportional to current I in the coil, N F B  L I
• We define self-inductance, L:
N FB
Self Inductance
L
I
•The induced emf in a coil of self-inductance L is
dF
dI
B



N
 L
–
dt
dt
–What is the unit for self-inductance? 1H  1V  s
A 1  s
•What does magnitude of L depend on?
–Geometry and the presence of a ferromagnetic material
•Self inductance can be defined for any circuit or part
of a circuit
Thursday August 2, 2012
PHYS 1444 Ian Howley
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Inductor
• An electrical circuit always contains some inductance but it is often
negligible
– If a circuit contains a coil of many turns, it could have a large inductance
• A coil that has significant inductance, L, is called an inductor and is
express with the symbol
– Precision resistors are normally wire wound
• Would have both resistance and inductance
• The inductance can be minimized by winding the wire back on itself in opposite
direction to cancel magnetic flux
• This is called a “non-inductive winding”
• For an AC current, the greater the inductance the less the AC current
– An inductor thus acts like a resistor to impede the flow of alternating current (not
to DC, though. Why?)
– The quality of an inductor is indicated by the term reactance or impedance (see
section 30-7)
Thursday August 2, 2012
PHYS 1444 Ian Howley
4
Energy Stored in a Magnetic Field
• The work done to the system is the same as the
energy stored in the inductor when it is carrying
current I
–
1 2
U  LI
2
Energy Stored in a magnetic
field inside an inductor
–This is compared to the energy stored in a capacitor, C,
1
when the potential difference across it is V U  CV 2
2
–Just like the energy stored in a capacitor is considered to
reside in the electric field between its plates
–The energy in an inductor can be considered to be stored
in its magnetic field
Thursday August 2, 2012
PHYS 1444 Ian Howley
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LR Circuits
• This can be shown w/ Kirchhoff rule loop rules
– The emfs in the circuit are the battery voltage V0
and the emf =-L(dI/dt) (opposes the I after the bat. switched on)
– The sum of the potential changes through the circuit is
V0    IR  V0  L dI dt  IR  0
– Where I is the current at any instance
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–
By rearranging the terms, we obtain a differential eq.
L dI dt  IR  V0
We can integrate just as in RC circuit
1  V0  IR  t
So the solution is  R ln  V   L
0


Where t=L/R
dI

I 0 V  IR
0

I

I  V0 1  e t t

dt
t 0 L

t

R  I max 1  e t t
• This is the time constant t of the LR circuit and is the time required for the
current I to reach 0.63 of the maximum
Thursday August 2, 2012
PHYS 1444 Ian Howley
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
Discharge of LR Circuits
• If the switch is flipped away from the battery
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The differential equation becomes
L dI dt  IR  0
dI

I 0 IR
I
dt
t 0 L
So the integration is 

Which results in the solution
I  I0
R
 t
e L
t
ln
I
R
 t
I0
L
 I 0 e t t
The current decays exponentially to zero with the time
constant t=L/R
– So there always is a reaction time when a system with a
coil, such as an electromagnet, is turned on or off.
– The current in LR circuit behaves in a similar manner as
for the RC circuit, except that in steady state RC current is
0, and the time constant is inversely proportional to R in
7
LR circuit unlike the RC circuit
30-5 LC Circuits and Electromagnetic
Oscillations
An LC circuit is a charged capacitor shorted through an
inductor.
30-5 LC Circuits and Electromagnetic
Oscillations
Summing the potential drops around the circuit gives a
differential equation for Q:
This is the equation for simple harmonic motion, and
has solutions
..
30-5 LC Circuits and Electromagnetic
Oscillations
Substituting shows that the equation can only be true
for all times if the frequency is given by
The current is sinusoidal as well:
30-5 LC Circuits and Electromagnetic
Oscillations
The charge and current are both sinusoidal, but with
different phases.
30-5 LC Circuits and Electromagnetic
Oscillations
The total energy in the circuit is constant; it oscillates
between the capacitor and the inductor:
30-6 LC Oscillations with Resistance
(LRC Circuit)
Any real (nonsuperconducting) circuit will have
resistance.
30-6 LC Oscillations with Resistance
(LRC Circuit)
Now the voltage drops around the circuit give
The solutions to this equation are damped harmonic
oscillations. The system will be underdamped for R2 < 4L/C,
and overdamped for R2 > 4L/C. Critical damping will occur
when R2 = 4L/C.
30-6 LC Oscillations with Resistance
(LRC Circuit)
This figure shows the three cases of underdamping,
overdamping, and critical damping.
30-6 LC Oscillations with Resistance
(LRC Circuit)
The angular frequency for underdamped oscillations is
given by
.
The charge in the circuit as a function of time is
.