Download Lecture 17 - UConn Physics

Physics 1202: Lecture 17
Today’s Agenda
• Announcements:
– Lectures posted on:
www.phys.uconn.edu/~rcote/
– HW assignments, etc.
• Homework #5:
– Due this Friday
• Midterm 1:
– Answers today
– New average = 63%
R
C
e
~
L
w
f
f
f
Suppose:
Phasors for L,C,R
i

0
w
wt
i
0
i
w
wt
i
0
i
w
i
wt
2
eR= -RI
eC= -Q/C
C
Phasors: LCR
w
R
e
~
L
f
eL= -L DI / Dt
f
f
• The phasor diagram has been relabeled in terms of the
reactances defined from:
The unknowns (im,f) can now be solved for
graphically since the vector sum of the voltages
VL + VC + VR must sum to the driving emf e.
Phasors:LCR
f
f
f

f

Phasors:Tips
y
• This phasor diagram was drawn as a
snapshot of time t=0 with the voltages
being given as the projections along the
y-axis.
• Sometimes, in working problems, it is
easier to draw the diagram at a time when
the current is along the x-axis (when i=0).
f
f
f
x
i mX L
em
f
i mR
imXC
From this diagram, we can also create a
triangle which allows us to calculate the
impedance Z:
Z
| XL-XC |
| f
R
“Full Phasor Diagram”
“ Impedance Triangle”
Resonance
• For fixed R,C,L the current im will be a maximum at the
resonant frequency w0 which makes the impedance Z purely
resistive.
ie:
reaches a maximum when:
XL=XC
the frequency at which this condition is obtained is given from:

• Note that this resonant frequency is identical to the natural
frequency of the LC circuit by itself!
• At this frequency, the current and the driving voltage are in
phase!
Resonance
The current in an LCR circuit depends on the values
of the elements and on the driving frequency through
the relation
Z
| XL-XC |
| f
R
“ Impedance Triangle”
Suppose you plot the current
versus w, the source voltage
frequency, you would get:
em / R0
R=Ro
im
0
R=2Ro
0
1
wx
2w
2o
Power in LCR Circuit
• The power supplied by the emf in a series LCR circuit
depends on the frequency w. It will turn out that the maximum
power is supplied at the resonant frequency w0.
• The instantaneous power (for some frequency, w) delivered at
time t is given by:
Remember what
this stands for
• The most useful quantity to consider here is not the
instantaneous power but rather the average power delivered
in a cycle.
• To evaluate the average on the right, we first expand the
sin(wt-f) term.
Power in LCR Circuit
•
Expanding,
•
Taking the averages,
+1
(Integral of Product of even and odd function = 0)
•
Generally:
0
-10
•
sinwtcoswt
wt
2p
Putting it all back together again,
1/2
0
+1
sin2wt
0
-1
0
wt
2p
Power in LCR Circuit
• The power can be expressed in term of i max:

• This result is often rewritten in terms of rms values:

• Power delivered depends on the phase, f,the “power
factor”
• phase depends on the values of L, C, R, and w
Fields from Circuits?
• We have been focusing on what happens within the circuits we have
been studying (eg currents, voltages, etc.)
• What’s happening outside the circuits??
– We know that:
» charges create electric fields and
» moving charges (currents) create magnetic fields.
– Can we detect these fields?
– Demos:
» We saw a bulb connected to a loop glow when the loop came
near a solenoidal magnet.
» Light spreads out and makes interference patterns.
Do we understand this?
f( x
f( x )
x
x
z
y
Maxwell’s Equations
• These equations describe all of Electricity and
Magnetism.
• They are consistent with modern ideas such as
relativity.
• They describe light ! (electromagnetic wave)
E & B in Electromagnetic Wave
• Plane Harmonic Wave:
where:
y
x
2
z
Note: the direction of propagation
where
is given by the cross product
are the unit vectors in the (E,B) directions.
Nothing special about (Ey,Bz); eg could have (Ey,-Bx)
Note cyclical relation:
Lecture 17, ACT 1
• Suppose the electric field in an e-m wave is given by:
– In what direction is this wave traveling ?
(a) + z direction
(c) +y direction
(b) -z direction
(d) -y direction
Lecture 17, ACT 2
• Suppose the electric field in an e-m wave is given
by:
• Which of the following expressions describes
the magnetic field associated with this wave?
(a) Bx = -(Eo/c)cos(kz + wt)
(b) Bx = +(Eo/c)cos(kz - wt)
(c) Bx = +(Eo/c)sin(kz - wt)
Generating E-M Waves
• Static charges produce a constant Electric
Field but no Magnetic Field.
• Moving charges (currents) produce both a
possibly changing electric field and a static
magnetic field.
• Accelerated charges produce EM radiation
(oscillating electric and magnetic fields).
• Antennas are often used to produce EM
waves in a controlled manner.
•
A Dipole Antenna
V(t)=Vocos(wt)
+
+
-
E
E
+
+
-
• time t=0
x
z
y
• time t=p/2w
• time t=p/w
one half cycle
later
dipole radiation pattern
proportional to sin(wt)
• oscillating electric dipole generates e-m radiation that is
polarized in the direction of the dipole
• radiation pattern is doughnut shaped & outward traveling
– zero amplitude directly above and below dipole
– maximum amplitude in-plane
Receiving E-M Radiation
receiving antenna
y
Speaker
x
z
One way to receive an EM signal is to use the same sort
of antenna.
• Receiving antenna has charges which are
accelerated by the E field of the EM wave.
• The acceleration of charges is the same thing as an
EMF. Thus a voltage signal is created.
Lecture 17, ACT 3
• Consider an EM wave with the E field
POLARIZED to lie perpendicular to the ground.
y
x
z
In which orientation should you turn your receiving
dipole antenna in order to best receive this signal?
a) Along S
b) Along B
C) Along E
Loop Antennas
Magnetic Dipole Antennas
• The electric dipole antenna makes use of the
basic electric force on a charged particle
• Note that you can calculate the related
magnetic field using Ampere’s Law.
• We can also make an antenna that produces
magnetic fields that look like a magnetic dipole,
i.e. a loop of wire.
• This loop can receive signals by exploiting
DF B
e
=
Faraday’s Law.
Dt
DB
e = -A
Dt
For a changing B field
through a fixed loop of
area A: FB= A B
Lecture 17, ACT 4
• Consider an EM wave with the E field
POLARIZED to lie perpendicular to the ground.
y
x
z
In which orientation should you turn your receiving
loop antenna in order to best receive this signal?
a) â Along S
b) â Along B
C) â Along E
Review of Waves from 1201
• The one-dimensional wave equation:
has a general solution of the form:
where h1 represents a wave traveling in the +x direction and h2
represents a wave traveling in the -x direction.
• A specific solution for harmonic waves traveling in the +x
direction is:
h l
A
x
A = amplitude
l = wavelength
f = frequency
v = speed
k = wave number
E & B in Electromagnetic Wave
• Plane Harmonic Wave:
where:
y
x
z
• From general properties of waves :

Velocity of Electromagnetic Waves
• The wave equation for Ex:
(derived from Maxwell’s Eqn)
• Therefore, we now know the velocity of
electromagnetic waves in free space:
• Putting in the measured values for m0 & e0, we get:
• This value is identical to the measured speed of light!
– We identify light as an electromagnetic wave.
The EM Spectrum
• These EM waves can take on any wavelength from
angstroms to miles (and beyond).
• We give these waves different names depending on the
wavelength.
10-14
10-10
10-6
10-2 1 102
Wavelength [m]
106
1010
Lecture 17, ACT 5
• Consider your favorite radio station. I will
assume that it is at 100 on your FM dial.
That means that it transmits radio waves
with a frequency f=100 MHz.
• What is the wavelength of the signal ?
A) 3 cm
B) 3 m
C) ~0.5 m
D) ~500 m
The EM Spectrum
• Each wavelength shows different details
The EM Spectrum
• Each wavelength shows different details
Energy in EM Waves / review
• Electromagnetic waves contain energy which is stored in E
and B fields:
=
• Therefore, the total energy density in an e-m wave = u, where
• The Intensity of a wave is defined as the average power
transmitted per unit area = average energy density times wave
velocity:
Momentum in EM Waves
• Electromagnetic waves contain momentum:
• The momentum transferred to a surface depends on the
area of the surface. Thus Pressure is a more useful quantity.
• If a surface completely absorbs the incident light, the
momentum gained by the surface p
• We use the above expression plus Newton’s Second Law in the
form F=Dp/Dt to derive the following expression for the Pressure,

• If the surface completely reflects
the light, conservation of
momentum indicates the light
pressure will be double that for
the surface that absorbs.