[SOLVED] Physics 396 homework set 3

1. Consider one inertial reference frame with coordinates (t
0
, x0
, y0
, z0
) moving at a constant
speed v relative to another inertial reference frame with coordinates (t, x, y, z).The
coordinate transformations relating the coordinates of these two frames are of the form
t
0 = (t vx/c2
)
x0 = (x vt)
y0 = y
z0 = z (1)
where (1 v2/c2)1/2 and are known as Lorentz boosts.a) By first calculating dierentials of Eqs. (1), explicitly show that these Lorentz boosts
leave the line element unchanged.In PHYS 230, you first learned three rules before arriving at the general Lorentz boosts.
Explicitly show that the Lorentz boosts reduce to the following rules for special cases
b) Moving clocks run slow by the factor p1 v2/c2.
c) Moving objects are contracted by the factor p1 v2/c2.d) Two clocks synchronized in their own frame are NOT synchronized in other frames.
The front (leading) clock reads an earlier time (lags) the chasing clock by
t
0 = vD/c2 (2)
where D is the rest distance between them.2. Maxwells equations constitute a set of coupled, first-order, linear partial dierential
equations that classically describe E~ and B~ fields as arising from charges and currents.
These equations can be decoupled by taking the curl of two of them and then employing
the remaining two.In regions of space where there are no charges or currents, these
equations take the form
r2
E~ 1
c2
@2
@t2E~ = 0 (3)
r2
B~ 1
c2
@2
@t2B~ = 0, (4)
where E~ = E~ (x, y, z, t), B~ = B~ (x, y, z, t), and
r2
@2
@x2 +
@2
@y2 +
@2
@z2 . (5)It is noted that Eqs. (3) and (4) correspond to wave equations in 3D and predict the
existence of electromagnetic waves that move at a constant speed c.
Consider one inertial reference frame with coordinates (t
0
, x0
, y0
, z0
) moving at a constant
speed v relative to another inertial reference frame with coordinates (t, x, y, z).Show that
Eqs. (3) and (4) are invariant under the Lorentz boosts given in Eq. (1). This means
that both the S and S0 observers will have equivalent wave equations in their own frame
of reference and that both observers will predict that the electromagnetic waves propagate
at the same speed c.3. a) Construct a spacetime diagram at rest relative to you with coordinates (ct, x).
Indicate the positions of two events (A, B) by dots. The line connecting the two
events should be labeled ` and forming an angle with the x axis.b) Now construct the line element s2 in terms of ` and .
c) Setting ` = 1, plot s2 for angles 0 < < /2 in Excel or an equivalent (or better)
plotting software.d) What is the range of values for this s2? What angle has the smallest magnitude for
this distance between points that defines this Minkowski spacetime? For what angles
do we find the maximum and minimum values of s2?4. Two spaceships A and B are moving in opposite directions, each measures the speed of
the other to be (12/13)c. Relative to our frame of reference, spaceship A moves to the
right and spaceship B moves to the left.Each spaceship contains two clocks with one at
the nose and one at the tail, synchronized with one another in each ships frame. For
spaceship A, label the clocks as NA and TA corresponding to the nose and tail clock of
A, respectively.Likewise, for spaceship B label the nose and tail clocks as NB and TB.
The rest length of spaceship A is 598 m and the rest length of spaceship B is 299 m. Just
as the nose of B reaches the nose of A, both ships set their nose clocks to read t = 0.
Analyze this problem from the rest frame of spaceship A.a) Sketch both spaceships from As point of view at the moment when the noses meet.
Use a ruler to make sure that your drawn lengths are consistent with your calculated
values.b) At this moment described in a), what are the readings of all four ship clocks?
c) How long does it take the nose of B to reach the tail of A?d) When the nose of B reaches the tail of A, again sketch both spaceships from As point
of view.
e) At this moment described in d), what are the readings of all four ship clocks?5. Spaceship A leaves Earth for a distant star and travels at a constant speed of vA = (5/13)c.
Assume that the Earth and the star are mutually at rest and that their clocks have
been previously synchronized.Both spaceship As clock and the Earths clock read
t = 0 at the start of this journey. After 7.00 years have passed on Earths clock, spaceship
B departs Earth in route for the same distant star traveling at a constant speed of
vB = (12/13)c. Remarkably, both spaceship A and B arrive at the star at precisely the
same time, according to the stars clock.a) Draw a set of three pictures in the rest frame of the Earth and star for the three
important events, that is, at the time of spaceship As departure, at the time of
spaceship Bs departure, and at the time of arrival of spaceships A and B.b) According to Earths clock, how long does it take the spaceships to arrive at this star?c) What is this rest length distance from Earth to the star?
d) At the moment when spaceship B leaves Earth, what does the spaceship As clock
read?e) At the moment when the spaceships arrive at the star, how much time passed by on
spaceship As clock?
f) How long did spaceship Bs journey take to reach this star, according to spaceship Bs
clock?g) What is the distance from the Earth to the star, according to spaceship A?
h) What is the distance from the Earth to the star, according to spaceship B?6. Reconsider Problem 5 that featured two spaceships, A and B, that both departed Earth
for a distant star traveling at speeds (5/13)c and (12/13)c, respectively. Spaceship B
departed Earth 7.00 years after Spaceship A, according to Earths clocks. Let the Earth
rest frame be the S frame, Spaceship As rest frame be the S0
, and Spaceship Bs rest
frame be the S00 frame.a) Draw a spacetime diagram in Earths frame. This spacetime diagram should take up
most of a full sheet of paper with a scale of 1 cyr : 2 cm. Be sure to label each axis
and include a proper scale.b) Draw the world lines of Spaceship A and Spaceship B on this spacetime diagram,
labeling them on the diagram as such.c) Draw the (x0
, ct0
) and (x00, ct00) axes on this spacetime diagram, labeling them as such.d) Indicate the locations of the three important events on this spacetime diagram with a
dot, labeling them as E1, E2, and E3.e) Draw the world line of the distant star as a dashed line.