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.