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Hour Exam 3

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Title: Hour Exam 3


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

3
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
4
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.

5
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
6
  • 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

7
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)

8
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
9
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

10
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)
11
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.

12
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.

13
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
14
Apply Amperes Law
15
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.
16
Review of Waves from Physics 111
17
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).

18
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?

19
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

20
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!!

21
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.

22
E B in Electromagnetic Wave
  • Plane Harmonic Wave

where
23
Lecture 21, ACT 3
  • Suppose the electric field in an e-m wave is
    given by

24
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
25
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)
26
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

27
The fields must be perpendicular to each other
and to the direction of propagation.
28
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).
29
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.

30
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

31
What you said
Magnitude of field is determined only by value of
z (and t) !!!
32
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

33
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)

34
4-step Plane Wave Derivation
  • Step 4 Combine results from steps 2 and 3 to
    eliminate By

35
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
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