Transcript Physics 121: Electricity & Magnetism

Physics 121: Electricity &
Magnetism – Lecture 13
E-M Oscillations and AC Current
Dale E. Gary
Wenda Cao
NJIT Physics Department
Electromagnetic Oscillations
2
q
1 2
UUB E 2Li
2 C
December 5, 2007
Oscillating Quantities


We will write oscillating quantities with a lower-case symbol, and the
corresponding amplitude of the oscillation with upper case.
Oscillating Quantity
Amplitude
Voltage
v
V
Current
i
I
Charge
q
Q
Examples:
q  Q cos(t   )
q2 Q2

cos 2 (t   )
2C 2C
di
d cos(t   )
I
dt
dt
December 5, 2007
Derivation of Oscillation Frequency
We have shown qualitatively that LC circuits act like an oscillator.
 We can discover the frequency of oscillation by looking at the
equations governing the total energy.
q2 1 2
U  UE UB 
 Li
2C 2
 Since the total energy is constant, the time derivative should be zero:
dU q dq
di

 Li  0
dt C dt
dt
dq
di d 2 q
d 2q q
i
 2 , so making these substitutions: L 2   0
 But
and
dt
dt dt
dt
C
 This is a second-order, homogeneous differential equation, whose
solution is q  Q cos(t   )
 i.e. the charge varies according to a cosine wave with amplitude Q and
2
frequency . Check by taking
dq
d
q
 Q sin( t   )
 Q 2 cos(t   )
2
two time derivatives of charge: dt
dt
 Plug into original equation:
1
1
d 2q q
Q
2
2


 L   0
L 2    LQ cos(t   )  cos(t   )  0
LC
C
dt
C
C

December 5, 2007
Which Current is Greatest?
1.
The expressions below give the charge on a
capacitor in an LC circuit. Choose the one that
will have the greatest maximum current?
A.
q = 2 cos 4t
q = 2 cos(4t+p/2)
q = 2 sin t
q = 4 cos 4t
q = 2 sin 5t
B.
C.
D.
E.
December 5, 2007
Time to Discharge Capacitor
2.
The three circuits below have identical inductors
and capacitors. Rank the circuits according to
the time taken to fully discharge the capacitor
during an oscillation, greatest first.
A.
I, II, III.
II, I, III.
III, I, II.
III, II, I.
II, III, I.
B.
C.
D.
E.
I.
II.
III.
December 5, 2007
Charge, Current & Energy
Oscillations




2
d
The solution to the equation L q  q  0 is q  Q cos(t   ) , which
dt 2 C
gives the charge oscillation.
From this, we can determine the corresponding oscillation of current:
dq
i
 Q sin( t   )
dt
1 2 1
q2 Q2
2 2
2
And energy
UE 

cos 2 (t   ) U B  Li  LQ  sin (t   )
2
2
2C 2C
Q2
But recall that   1 , so
.
UB 
sin 2 (t   )
LC
2C
That is why our graph for the energy oscillation
had the same amplitude for both UE and UB.
 Note that
Q2
Q2
2
2
UE UB 
[cos (t   )  sin (t   )] 
2C
2C Constant

December 5, 2007
Damped Oscillations
Recall that all circuits have at least a little
bit of resistance.
 In this general case, we really have an RLC
circuit, where the oscillations get smaller
with time. They are said to be “damped
oscillations.”


Then the power equation becomes
dU q dq
di

 Li  i 2 R
dt C dt
dt
Power lost due to resistive heating
dq
di d 2 q
 2
 As before, substituting i 
and
dt
dt dt
gives the differential equation for q
2
L
d q
dq q

R
 0
2
dt
dt C
Solution:
e  Rt / 2 L
Damped Oscillations
q  Qe  Rt / 2 L cos(t   )
    2  ( R / 2 L) 2
December 5, 2007
Resonant Frequency
3.
How does the resonant frequency  for an ideal
LC circuit (no resistance) compare with ’ for a
non-ideal one where resistance cannot be
ignored?
A.
The resonant frequency for the non-ideal circuit is
higher than for the ideal one (’ > ).
The resonant frequency for the non-ideal circuit is lower
than for the ideal one (’ < ).
The resistance in the circuit does not affect the
resonant frequency—they are the same (’ = ).
B.
C.
December 5, 2007
Alternating Current





The electric power out of a home or office power socket is in the form of
alternating current (AC), as opposed to the direct current (DC) of a battery.
Alternating current is used because it is easier to transport, and easier to
“transform” from one voltage to another using a transformer.
In the U.S., the frequency of oscillation of AC is 60 Hz. In most other
countries it is 50 Hz.
The figure at right shows one way to make
an alternating current by rotating a coil of
wire in a magnetic field. The slip rings and
brushes allow the coil to rotate without
twisting the connecting wires. Such a
device is called a generator.
It takes power to rotate the coil, but that
power can come from moving water (a
water turbine), or air (windmill), or a
   m sin d t i  I sin( d t   )
gasoline motor (as in your car), or steam
(as in a nuclear power plant).
December 5, 2007
RLC Circuits with AC Power

When an RLC circuit is driven with an AC
power source, the “driving” frequency d
is the frequency of the power source, while
the circuit can have a different “resonant”
frequency    1 / LC  ( R / 2 L) 2 .
Let’s look at three different circuits driven
by an AC EMF. The device connected to
the EMF is called the “load.”
 What we are interested in is how the
voltage oscillations across the load relate
to the current oscillations.
 We will find that the “phase” relationships
change, depending on the type of load
(resistive, capacitive, or inductive).

December 5, 2007
A Resistive Load
Phasor Diagram: shows the
instantaneous phase of either
voltage or current.
 For a resistor, the current
follows the voltage, so the
voltage and current are in
phase (  0).


If vR  VR sin d t

Then iR  I R sin d t 

VR
sin d t
R
December 5, 2007
Power in a Resistive Circuit
4.
A.
B.
C.
D.
E.
The plot below shows the current and voltage
oscillations in a purely resistive circuit. Below that are
four curves. Which color curve best represents the
power dissipated in the resistor?
The green curve (straight line).
The blue curve.
The black curve.
The red curve.
PR
None are correct.
t
December 5, 2007
A Capacitive Load
For a capacitive load, the voltage across the capacitor
is proportional to the charge
q Q
vC   sin d t
C C
 But the current is the time derivative of the charge
dq
iC 
 d CVC cos d t
dt
 In analogy to the resistance, which is the
proportionality constant between current and voltage,
we define the “capacitive reactance” as
1
XC 
d C
VC
 So that iC 
cos d t .
XC


The phase relationship is that   90º, and current
leads voltage.
December 5, 2007
An Inductive Load
For an inductive load, the voltage across the inductor
is proportional to the time derivative of the current
di
vL  L L
dt
 But the current is the time derivative of the charge

iL 
 VL 
VL

 cos d t
sin

t
dt


d

L
 d L 

Again in analogy to the resistance, which is the
proportionality constant between current and voltage,
we define the “inductive reactance” as X L  d L

So that iL  

The phase relationship is that   90º, and current
lags voltage.
VL
cos d .t
XL
December 5, 2007
Units of Reactance
XC 
1
d C
5.
We just learned that capacitive reactance is
and
X L . What
inductive reactance is
are the units of
dL
reactance?
A.
Seconds per coulomb.
Henry-seconds.
Ohms.
Volts per Amp.
The two reactances have different units.
B.
C.
D.
E.
December 5, 2007
Summary Table
Circuit
Element
Symbol
Resistance or
Reactance
Phase of
Current
Phase
Constant
Amplitude
Relation
Resistor
R
R
In phase
with vR
0º (0 rad)
VR=IRR
Capacitor
C
XC=1/dC
Leads vR
by 90º
90º (p/2)
VC=ICXC
Inductor
L
XL=dL
Lags vR by
90º
90º (p/2)
VL=ILXL
December 5, 2007
Summary



Energy in inductor: U B 
q2 1 2
LC circuits: total electric + magnetic energy is conserved U  U E  U B 
 Li
2C 2
LC circuit:
Charge equation Current equation Oscillation frequency
q  Q cos(t   )

1 2
Li Energy in magnetic field
2
i  Q sin( t   )

1
LC
LRC circuit:
Charge equation
Oscillation frequency
q  Qe  Rt / 2 L cos(t   )
    2  ( R / 2 L) 2
Reactances:
 Resistive, X R  R
V
iR  I R sin d t  R sin d t
R
XC 
capacitive,
V
iC  C cos d t
XC
1
d C
inductive X L  d L
V
iL   L cos d t
XL
December 5, 2007
Thoughts on Clickers
6.
How did you like using the clickers in this class?
A.
Great!
It had its moments.
I could take it or leave it.
I would rather leave it.
It was the worst!
B.
C.
D.
E.
December 5, 2007
Thoughts on Clickers
7.
Which answer describes the most important way that
the clicker aided you in learning the material?
A.
It helped me to think about the material presented just
before each question.
It broke up the lecture and kept me awake.
It tested my understanding.
It provided immediate feedback.
It showed me what others were thinking.
B.
C.
D.
E.
December 5, 2007
Thoughts on Clickers
8.
Which answer describes the second most important way
that the clicker aided you in learning the material?
A.
It helped me to think about the material presented just
before each question.
It broke up the lecture and kept me awake.
It tested my understanding.
It provided immediate feedback.
It showed me what others were thinking.
B.
C.
D.
E.
December 5, 2007
Thoughts on Clickers
9.
How would you react to clickers being used in other
classes at NJIT?
A.
I
I
I
I
I
B.
C.
D.
E.
think it would be excellent.
think it is a good idea.
wouldn’t mind.
would rather not.
definitely hope not.
December 5, 2007
Thoughts on Clickers
10.
What problems did you have with your clicker?
A.
I had no problems with my clicker.
It was too big or bulky, a pain to carry around.
I had trouble remembering to bring it to class.
My clicker had mechanical problems.
I lost or misplaced it (for all or part of the semester).
B.
C.
D.
E.
December 5, 2007
Thoughts on Clickers
11.
If you had the choice between using a clicker versus
having a lecture quiz where you had to fill in a scantron,
which would you prefer?
A.
I would prefer the clicker.
I would prefer the scantron quiz.
B.
December 5, 2007
Have a Nice Day
12.
Please click any button on your clicker as you turn your
clicker in. This will register your name as having turned
in your clicker.
December 5, 2007