Title: Hour Exam 3
1(No Transcript)
2Today . . .
- Speed of light
- Maxwells Displacement Current
- Charge conservation
- Modification of Amperes Law
- Modified Maxwell equations ? waves!!
3Speed 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
4Finite 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.
5Our 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
7Maxwells 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)
8Maxwells 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
9Points 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.
10Lecture 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)
11Lecture 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.
12Lecture 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.
13Points 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
14Apply Amperes Law
15On 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.
16Review of Waves from Physics 111
17Movies 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).
18Lecture 21, ACT 2
- Snapshots of a wave with angular frequency w are
shown at 3 times
- Which of the following expressions describes this
wave?
19Lecture 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
20Lecture 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!!
21Velocity 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.
22E B in Electromagnetic Wave
where
23Lecture 21, ACT 3
- Suppose the electric field in an e-m wave is
given by
24Lecture 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
25Lecture 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)
2610) 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
27The fields must be perpendicular to each other
and to the direction of propagation.
28Properties 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).
29Plane 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.
30Shown 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.
4) Compare the magnitudes of the electric field
at points A and C.
31What you said
Magnitude of field is determined only by value of
z (and t) !!!
32Summary
- 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
33Appendix 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)
344-step Plane Wave Derivation
- Step 4 Combine results from steps 2 and 3 to
eliminate By
35How 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