physics 1202: lecture 17 today’s agenda announcements: –lectures posted on: rcote/ rcote/ –hw...
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Physics 1202: Lecture 17Today’s Agenda
• Announcements:– Lectures posted on:
www.phys.uconn.edu/~rcote/
– HW assignments, etc.
• Homework #5:Homework #5:– Due this FridayDue this Friday
• Midterm 1:– Answers today
– New average = 63%
LC
R
0
i
0
i
Phasors for L,C,Ri
t
i
t
i
t
Suppose:
0
i
Phasors: LCR
• The phasor diagram has been relabeled in terms of the reactances defined from:
LC
R
The unknowns (im,) can now be solved for graphically since the vector sum of the voltages VL + VC + VR must sum to the driving emf.
C= -Q/C
L= -L I / t
R= -RI
Phasors:LCR
Phasors:Tips• This phasor diagram was drawn as a snapshot of time t=0 with the voltages being given as the projections along the y-axis.
y
x
imR
imXL
imXC
m
“Full Phasor Diagram”
From this diagram, we can also create a triangle which allows us to calculate the impedance Z:
• 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).
“ Impedance Triangle”
Z
|
R
| XL-XC |
Resonance• For fixed R,C,L the current im will be a maximum at the
resonant frequency 0 which makes the impedance Z purely resistive.
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!
ie:
reaches a maximum when: XL=XC
ResonanceThe current in an LCR circuit depends on the values
of the elements and on the driving frequency through the relation
Suppose you plot the current versus , the source voltage frequency, you would get:
“ Impedance Triangle”
Z
|
R
| XL-XC |
1 2x
im
00
o
R=Ro
m / R0
R=2Ro
Power in LCR Circuit• The power supplied by the emf in a series LCR circuit
depends on the frequency . It will turn out that the maximum power is supplied at the resonant frequency 0.
• The instantaneous power (for some frequency, ) delivered at time t is given by:
• 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(t-) term.
Remember what this stands for
Power in LCR Circuit• Expanding,
• Taking the averages,
• Generally:
sin2t
t0
0
+1
-1
• Putting it all back together again,
01/2
(Integral of Product of even and odd function = 0)sintcost
t0
0
+1
-1
Power in LCR Circuit• The power can be expressed in term of i max:
• Power delivered depends on the phase, the“power factor”
• phase depends on the values of L, C, R, and
• This result is often rewritten in terms of rms values:
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
x
f( 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
z
Nothing special about (Ey,Bz); eg could have (Ey,-Bx)
Note: the direction of propagation is given by the cross product
where are the unit vectors in the (E,B) directions.
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 (b) -z direction
(c) +y 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 + t) (b) Bx = +(Eo/c)cos(kz - t) (c) Bx = +(Eo/c)sin(kz - t)
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(t)
x
zy
• time t=0
++
--
E
• time t=/2
E
• time t=/ one half cycle later
--
++
dipole radiation pattern
• 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
proportional to sin(t)
Receiving E-M Radiation
receiving antenna
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.
Speaker
y
x
z
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?
C) Along Ea) Along S b) Along B
Loop AntennasMagnetic 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 Faraday’s Law.
For a changing B field through a fixed loop of area A: = 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:
• A specific solution for harmonic waves traveling in the +x direction is:
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.
h
x
A
A = amplitude = wavelengthf = frequencyv = speedk = 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 0 & 0, 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.
Wavelength [m]10-14 10-10 10-6 10-2 1 102 106 1010
Gam
ma
Ray
s
Infr
ared
Mic
row
aves
Sh
ort
Wav
e R
adio
TV
an
d F
M R
adio
AM
Rad
io
Lo
ng
Rad
io W
aves
Ult
ravi
ole
t
Vis
ible
Lig
ht
X R
ays
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:
• The Intensity of a wave is defined as the average power transmitted per unit area = average energy density times wave velocity:
• Therefore, the total energy density in an e-m wave = u, where
=
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=p/t 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.