Hour Exam 3

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(No Transcript)
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Today . . .

• Speed of light
• Maxwells Displacement Current
• Charge conservation
• Modification of Amperes Law
• Modified Maxwell equations ? waves!!

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Speed of LightAn aside with foreshadowing
Galileos hilltop experiment
Response time 0.1 s C 3km / 1s 30,000 m/s
1675 Roemer looks at eclipse of moons of
Jupiter, depending on where Earth is ? v is
finite (v 2.3108 m/s)
1860 Foucault (also Fizeau) - change mirror rot
ation speed - no signal unless mirror rotates 1/
8 in ?T
c ? 2.98108 m/s
2007 Physics 212 ? Back to Galileo
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Finite Speed of Light Applications

• Global Positioning System (GPS) time of flight
  to several satellites allows triangulation of
  location to within meters.

• LIDAR (Light Detection and Ranging) optical
  remote sensing technology that measures
  properties of scattered light to find range
  and/or other information of a distant target.
• geology and geography (e.g., measure
  pre-earthquake shifts)
  
• atmospheric science (wind speed, particle
  concentrations, etc.)
  
• remote sensing (e.g., monitor distance to moon at
  mm-resolution)
  
• law enforcement vehicle speed measurements

• Medicine Optical coherence tomography (OCT)
  micron-resolution imaging inside tissue

• Engineering Laser rangefinders used for 3-D
  object modeling and recognition. Beam is scanned
  to determine object shape (from time-of-flight
  data)

• High-speed electronics must account for the
  travel-time within ultrafast circuits.

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Our story thus far

• J. C. Maxwell (1860) realizes that the equations
  of electricity magnetism are inconsistent!

Also there is a lack of symmetry ?

In the case of no sources
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• 4) Consider a capacitor that is charging (i.e., a
  switch has just been closed that hooks it up to a
  battery). While the capacitor is charging, there
  is a current between the plates.
• True
• b) False
• 5) In the figure we have two (imaginary) loops, 1
  and 2. A wire carrying current I passes through
  each of them. Shown shaded are surfaces that are
  bounded by the loops (i.e., loop 1 bounds a flat
  disk, while loop 2 bounds a surface like a little
  pouch).
• For which loop is (the line
  integral of the magnetic field) the largest?
  
• Loop 1 b) Loop 2 c) the same

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Maxwells Displacement Current

• Consider applying Amperes Law to the current
  shown in the diagram.
  
• If the surface is chosen as 1, 2 or 4, the
  enclosed current I
  
• If the surface is chosen as 3, the enclosed
  current 0! (i.e., there is no current between
  the plates of the capacitor)

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Maxwells Displacement Current

• But where does the displacement current come
  from?!
  
• Although there is no actual charge moving
  between the plates, nevertheless, something is
  changing the electric field between them!
  
• The Electric Field E between the plates of the
  capacitor is determined by the charge Q on the
  plate of area A
  
• E Q/(Ae0) ? Q E Ae0
• Because there is current flowing through the
  wire, there must be a change in the charge on the
  plates

Recall defn of flux
Modified Amperes Law
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Points A and B lie inside a capacitor. At time t
0 the switch is closed.
7) After the switch is closed, there will be a
magnetic field at point A which is proportional
to the current in the circuit.

• True
• False

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Lecture 21, ACT 1

• Suppose that at time t the currents flowing into
  capacitors CI and CII 4CI are identical, and
  that CII has twice the radius (and 4 times the
  area) of CI , as shown.
• Compare the net displacement current for the two
  cases.

• Compare the magnetic fields at a radial distance
  r0 from the axes of CI and CII.

(a) BI(r0)
(c) BI(r0) BII(r0)
(b) BI(r0) BII(r0)
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Lecture 21, ACT 1

• Suppose that at time t the currents flowing into
  capacitors CI and CII 4CI are identical, and
  that CII has twice the radius (and 4 times the
  area) of CI , as shown.
• Compare the net displacement current for the two
  cases.

• Although there is no actual current flowing
  between the capacitor plates, the displacement is
  always equal to the real current.
  
• Therefore, since the two capacitors have the
  same real current, they must have the same total
  displacement current.

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Lecture 21, ACT 1

• Suppose that at time t the currents flowing into
  capacitors CI and CII 4CI are identical, and
  that CII has twice the radius (and 4 times the
  area) of CI , as shown.

• Compare the magnetic fields at a radial distance
  r0 from the axes of CI and CII.

(a) BI(r0)
(b) BI(r0) BII(r0)
(c) BI(r0) BII(r0)

• You could solve this using the expression for B
  in terms of the flux
  
• of E. However, it is simpler to answer by
  pretending that the displacement current were
  uniformly distributed over the entire
  
• area of the capacitor, and simply using B m0
  Ienclosed/2 p r0 .
  
• For CI the entire displacement current is
  enclosed (by an imaginary Amperean loop for CII
  only 1/4 of the total displacement current is
  enclosed. Therefore, BII(r0) BI(r0)/4.

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Points A and B lie inside a capacitor. At time t
0 the switch is closed.
8) Compare the magnitudes of the magnetic fields
at points A and B
a) BA BB
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Apply Amperes Law
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On to Waves!!

• Note the symmetry now of Maxwells Equations in
  free space, meaning when no charges or currents
  are present
  

• Combining these equations (see Appendix A) leads
  to wave equations for E and B, e.g.,
  
• Do you remember the wave equation???

h is the variable that is changing in space (x)
and time (t). v is the velocity of the wave.
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Review of Waves from Physics 111
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Movies from 111

• Transverse Wave
• Note how the wave pattern definitely moves to
  the right.
  
• However any particular point (look at the blue
  one) just moves transversely (i.e., up and down)
  to the direction of the wave.

• Wave Velocity
• The wave velocity is defined as the wavelength
  divided by the time it takes a wavelength (green)
  to pass by a fixed point (blue).

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Lecture 21, ACT 2

• Snapshots of a wave with angular frequency w are
  shown at 3 times
  
• Which of the following expressions describes this
  wave?

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Lecture 21, ACT 2

• Snapshots of a wave with angular frequency w are
  shown at 3 times
  
• Which of the following expressions describes this
  wave?

(a) y sin(kx-wt)
(b) y sin(kxwt)
(c) y cos(kxwt)

• The t 0 snapshot Þ at t 0, y sinkx
• At t p/2w and x0, (a) Þ y sin(-p/2) -1
• At t p/2w and x 0, (b) Þ y sin(p/2) 1

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Lecture 21, ACT 2

• Snapshots of a wave with angular frequency w are
  shown at 3 times
  
• Which of the following expressions describes this
  wave?

(a) y sin(kx-wt)
(b) y sin(kxwt)
(c) y cos(kxwt)

• In what direction is this wave traveling?

(b) -x direction
(a) x direction

• We claim this wave moves in the -x direction.
• The orange dot marks a point of constant phase.

• It clearly moves in the -x direction as time
  increases!!

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Velocity of Electromagnetic Waves

• We derived the wave equation for Ex (Maxwell did
  it first, in 1865!)

• Comparing to the general wave equation
• we have the velocity of electromagnetic waves
  in free space

• This value is essentially identical to the speed
  of light measured by Foucault in 1860!
  
• Maxwell identified light as an electromagnetic
  wave.

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E B in Electromagnetic Wave

• Plane Harmonic Wave

where
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Lecture 21, ACT 3

• Suppose the electric field in an e-m wave is
  given by

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Lecture 21, ACT 3

• Suppose the electric field in an e-m wave is
  given by
  
• In what direction is this wave traveling ?

(b) - z direction
(a) z direction
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Lecture 21, ACT 3

• Suppose the electric field in an e-m wave is
  given by
  
• In what direction is this wave traveling ?

(b) - z direction
(a) z direction

• Which of the following expressions describes the
  magnetic field associated with this wave?

(a) Bx -(Eo/c) cos(kz w t)
(b) Bx (Eo/c) cos(kz -w t) )
(c) Bx (Eo/c) sin(kz -w t)
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10) An electromagnetic wave is travelling along
the x-axis, with its electric field oscillating
along the y-axis. In what direction does the
magnetic field oscillate?

• along the x-axis
• along the z-axis
• along the y-axis

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The fields must be perpendicular to each other
and to the direction of propagation.
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Properties of electromagnetic waves (e.g., light)
Speed in vacuum, always 3108 m/s, no matter how
fast the source is moving (there is no
aether!). In material, the speed can be
reduced, usually only by 1.5, but in 1999 to 17
m/s!
In reality, light is often somewhat localized
transversely (e.g., a laser) or spreading in a
spherical wave (e.g., a star).
A plane wave can often be a good approximation
(e.g., the wavefronts hitting us from the sun are
nearly flat).
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Plane Waves

• For any given value of z, the magnitude of the
  electric field is uniform everywhere in the x-y
  plane with that z value.

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Shown is an EM wave at an instant in time.
Points A, B, and C lie in the same x-y plane.
3) Compare the magnitudes of the electric field
at points A and B.

• Ea
• Ea Eb
• Ea Eb

4) Compare the magnitudes of the electric field
at points A and C.

• Ea
• Ea Ec
• Ea Ec

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What you said
Magnitude of field is determined only by value of
z (and t) !!!
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Summary

• Repaired Amperes Law
• Maxwells Displacement Current

• Combined Faradays Law and Amperes Law
• time varying B-field induces E-field
• time varying E-field induces B-field

• Electromagnetic waves that travel at c 3 x 108
  m/s

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Appendix 4-step Plane Wave Derivation

• Step 1 Assume we have a plane wave propagating
  in z (i.e., E, B not functions of x or y)
  

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4-step Plane Wave Derivation

• Step 4 Combine results from steps 2 and 3 to
  eliminate By
  

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How is B related to E?

• We derived the wave eqn for Ex

• How are Ex and By related in phase and magnitude?

• Consider the harmonic solution
• 
  where