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Transcript
Announcements
• Read Sec. 1.8 and 1.9 for Monday
• HWK Set 2 available.
• We will start counting clicker points on
Monday. Bring those clickers!
• Mon. office hours will run 3-5 PM in the
back of the Physics HelpRoom.
• Homework #2 due Wednesday at 10AM
Electricity and magnetism have many
cases where relative motion appears:
Coulomb’s law (electric field due to a point charge)
G
E=
1
qr
4πε 0 r
2
Charge might be static
in one inertial frame,…
Biot-Savart law (magnetic field due to a segment
of current) G μ IΔsG × r
but it moves in all others!
B= 0
And creates both E and B
4π r 2
in them.
Where ε 0 and μ 0 are universal constants that do
not depend on a reference frame.
It’s the LAWS we expect to be the same
from frame to frame, NOT E and B.
Coulomb’s law (electric field due to a point charge)
G
E=
1
qr
4πε 0 r
2
Charge might be static
in one inertial frame,…
Biot-Savart law (magnetic field due to a segment
of current) G μ IΔsG × r
but it moves in all others!
B= 0
And creates both E and B
4π r 2
in them.
Where ε 0 and μ 0 are universal constants that do
not depend on a reference frame.
Maxwell’s equations
Predict that light travels
as a wave with speed
c=
1
ε 0 μ0
= 3×108
m/s
Question: this is the
speed relative to what?
(That’s a good question!)
Waves
• A sound wave propagates through air, with velocity
relative to the air.
• A water wave propagates in water, with velocity relative
to the water.
• “The wave” propagates through a crowd in a stadium,
with velocity relative to the crowd.
• An electromagnetic wave propagates through...
Answer (19th century physics): The “luminiferous ether.”
The ether
v
c
Suppose the earth moves through the fixed ether with speed v.
A light wave traveling at speed c with respect to the ether is
heading in the opposite direction. According to Galilean
relativity, what is the speed of the light wave as viewed from
the earth? (Further assume the earth is not accelerating).
a) c
b) c+v
c) c-v
Warning
We’re about to leave Galilean relativity
behind. Fasten your seatbelts.
Michelson and Morely
Performed a famous
experiment that
effectively measured the
speed of light in different
directions with respect to
the “ether wind.”
Result: No difference!
Speed of light is the
same in all directions.
Yes, but...
Q: What if the ether
Is “dragged along”
The surface of the
earth, like air flowing
around a tennis ball?
A: If so, this would
require a “viscosity”
of the ether, and
would require rewriting Maxwell’s
equations.
Remark: lots of effort tried to save
the idea of the ether, but none
held up.
Present View: There is no ether
Electromagnetic waves are special. A time-changing
electric field induces a magnetic field, and vice-versa. A
medium (“ether”) is not necessary.
Einstein’s relativity postulate #1
All the laws of physics (i.e., including
electromagnetism) are the same in all
inertial reference frames.
Einstein’s relativity postulate #2
The speed of light is the same in all
inertial frames of reference.
The speed of light is the same in
all inertial frames of reference.
There is no ether
v
c
Suppose the earth moves through space with speed v.
A light wave traveling at speed c with respect to faraway stars
is heading in the opposite direction. According to Einstein’s
relativity, what is the speed of the light wave as viewed from
the earth? (Further assume the earth is not accelerating).
a) c
b) c+v
c) c-v
Time to talk about time
• Measuring time in
different frames
• Synchronization of
clocks
• What time is it? How
do you know?
Question
The speed of light in vacuum is 3.0X108 m/s.
About how long does it take light to move
the length of your arm?
a) 1ms (10-3 s)
b) 1μs (10-6 s)
c) 1ns (10-9 s)
d) 1ps (10-12 s)
Useful detail
Speed of light c = 3.0 X 108 m/s
= 0.3 m/ns
= 300 m/μs
Where 1 ns = 10-9 s
(Note also, c ~ 1 foot/ns, but you didn’t
hear that from me!)
Comparing inertial frames
... -3
-2
-1
0
1
2
3 ...
Some observers and a ball are at rest in reference frame
S. At t=0, the observer at the origin in S flashes a light
pulse to be received at x = 3 m.
At Δt=10 ns, the light is received. Observers in S
measure a distance Δx = 3 m, so the speed of light in
frame S is
Δx 3m
u=
=
= 0.3m / ns
Δt 10ns
Comparing inertial frames
v
... -3
-2
-1
0
1
2
3 ...
... -3
-2
-1
0
1
2
3 ...
S’ is moving with respect to S at v = 0.2 m/ns.
At t=0, The observer at the origin in S flashes a light
pulse to be received at x = 3 m.
Ten nanoseconds later
v
... -3
-2
-1
... -3
0
1
2
3 ...
-2
-1
0
1
2
3 ...
S’ is moving with respect to S at v = 0.2 m/ns.
At Δt=10 ns, the light is received. In Galilean relativity,
how far does the observer in S’ think the light has
traveled?
a) 3 m
b) 2 m
c) 1 m
d) 0 m
Ten nanoseconds later
v
... -3
-2
-1
... -3
0
1
2
3 ...
-2
-1
0
1
2
3 ...
S’ is moving with respect to S at v = 0.2 m/ns.
At Δt=10 ns, the light is received. In Galilean relativity,
the observer in S’ would therefore measure the speed of
light as
Δx′ 1m
u=
=
= 0.1m / ns
Uh-oh!
Δt ′ 10ns
Uh-oh
If we are to believe Einstein’s second postulate (and
we do), then
In frame S
Δx
c=
Δt
In frame S’
Δx′
c=
Δt ′
Conclusion: Δt ≠ Δt ′ i.e., time passes at a different
rate in the two frames of reference!
Question:
In a given reference frame, the time of an event is
given by
a) The time the observer at the origin sees it.
b) The time that any observer anywhere in the
frame sees it.
c) The time according to the clock nearest the
event when it happens.
d) The time according to a properly synchronized
clock nearest the event when it happens.
Recall
We have argued that to describe a physical event,
we must specify both where it is (in some inertial
coordinate system) and what time it occurs
(according to some clock). But which clock?
More on reference frames
An observer at (0,0)
has a clock; events
there are covered.
An observer at
(3m,2m) had better
have a clock, too, if
you want to know
about events there.
And, the two clocks
had better show the
same time.
Back to reference frames
An observer at (0,0)
has a clock; events
there are covered.
And there had better
be clocks
everywhere, so you
don’t miss any event.
(Think of having
infinitely many cable
channels).
Synchronizing clocks
... -3
-2
-1
0
1
2
3 ...
At the origin, at three
o’clock, the clock
sends out a light
signal to tell
everybody it’s three
o’clock.
Time passes as the
signal gets to the
clock at x = 3m.
When the signal
arrives, the clock at
x=3m is set to 3:00…
Lightning strikes the top of Bear Mountain, instantly
generating a clap of thunder. At what time did the
lightning strike?
A. At the instant you hear the thunder.
B. At the instant you see the lightning.
C. Very slightly before you see the lightning.
D. Very slightly before you hear the thunder.
E. Very slightly after you see the lightning.
Well, that was dumb
... -3
-2
-1
0
1
2
3 ...
At the origin, at three
o’clock, the clock
sends out a light
signal to tell
everybody it’s three
o’clock.
Time passes as the
signal gets to the
clock at x = 3m.
If you do this, then the clock
at x = 3m is 10 ns slow,
because of the delay!
When the signal
arrives, the clock at
x=3m is set to 3:00.
Synchronizing clocks
... -3
-2
-1
0
1
2
3 ...
When the signal arrives, the
clock at x=3m is set to 3:00
plus the 10 ns delay.
At the origin, at three
o’clock, the clock
sends out a light
signal to tell
everybody it’s three
o’clock.
Time passes as the
signal gets to the
clock at x = 3m.
Simultaneity in one frame
... -3
-2
-1
0
1
2
3 ...
Using this procedure, it is now possible to say that
all the clocks in a given inertial reference frame
read the same time.
Even if I don’t go out there to check it myself.
Now I know when events really happen, even if I
don’t find out until later (due to finite speed of light).
Recap: The postulates of
special relativity
All the laws of physics (i.e., including
electromagnetism) are the same in all
inertial reference frames.
The speed of light is the same in all inertial
frames of reference.
Good place to pause!
Recall from electricity and
magnetism
Coulomb’s law (electric field due to a point charge)
G
E=
1
qr
4πε 0 r
2
Charge might be static
in one inertial frame,…
Biot-Savart law (magnetic field due to a segment
of current) G μ IΔsG × r
but it moves in all others!
B= 0
And creates both E and B
4π r 2
in them.
Where ε 0 and μ 0 are universal constants that do
not depend on a reference frame.
Maxwell’s equations
Predict that light travels
as a wave with speed
c=
1
ε 0 μ0
= 3×108
m/s
Question: this is the
speed relative to what?
(That’s a good question!)
Waves
• A sound wave propagates through air, with velocity
relative to the air.
• A water wave propagates in water, with velocity relative
to the water.
• “The wave” propagates through a crowd in a stadium,
with velocity relative to the crowd.
• An electromagnetic wave propagates through...
Answer (19th century physics): The “luminiferous ether.”
Suppose you’re in a spaceship traveling through the solar
system at at a constant speed of one-half impulse power,
v = 1.5 X 108 m/s. You fire a pulse of laser light out the
front of your vessel. (Speed of light = 3.0 X 108 m/s).
Q: How fast do you see the pulse leave your ship?
a) 1.5 X 108 m/s
c) 4.5 X 108 m/s
b) 3.0 X 108 m/s
d) none of these
Suppose you’re in a spaceship traveling through the solar
system at at a constant speed of one-half impulse power,
v = 1.5 X 108 m/s. You fire a pulse of laser light out the
front of your vessel. (Speed of light = 3.0 X 108 m/s).
Q: How fast does an inertial observer on Mars see the
pulse leave your ship?
a) 1.5 X 108 m/s
c) 4.5 X 108 m/s
b) 3.0 X 108 m/s
d) none of these
Einstein’s relativity postulate #2
The speed of light is the same in all
inertial frames of reference.
This was new in 1905 when Einstein
proposed it. Now it has been
experimentally tested lots of times.