PHYSICS FOR SCI. & ENGR(LL W/WEBASSIGN)
10th Edition
ISBN: 9781337888721
Author: SERWAY
Publisher: CENGAGE L
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Chapter 37, Problem 38AP
Two wavelengths λ and λ + Δλ (with Δλ << λ) are incident on a diffraction grating. Show that the angular separation between the spectral lines in the mth-order spectrum is
where d is the slit spacing and m is the order number.
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Question B3
Consider the following FLRW spacetime:
t2
ds² = -dt² +
(dx²
+ dy²+ dz²),
t2
where t is a constant.
a)
State whether this universe is spatially open, closed or flat.
[2 marks]
b) Determine the Hubble factor H(t), and represent it in a (roughly drawn) plot as a function
of time t, starting at t = 0.
[3 marks]
c) Taking galaxy A to be located at (x, y, z) = (0,0,0), determine the proper distance to galaxy
B located at (x, y, z) = (L, 0, 0). Determine the recessional velocity of galaxy B with respect
to galaxy A.
d) The Friedmann equations are
2
k
8πG
а
4πG
+
a²
(p+3p).
3
a
3
[5 marks]
Use these equations to determine the energy density p(t) and the pressure p(t) for the
FLRW spacetime specified at the top of the page.
[5 marks]
e) Given the result of question B3.d, state whether the FLRW universe in question is (i)
radiation-dominated, (ii) matter-dominated, (iii) cosmological-constant-dominated, or (iv)
none of the previous. Justify your answer.
f)
[5 marks]
A conformally…
SECTION B
Answer ONLY TWO questions in Section B
[Expect to use one single-sided A4 page for each Section-B sub question.]
Question B1
Consider the line element
where w is a constant.
ds²=-dt²+e2wt dx²,
a) Determine the components of the metric and of the inverse metric.
[2 marks]
b) Determine the Christoffel symbols. [See the Appendix of this document.]
[10 marks]
c)
Write down the geodesic equations.
[5 marks]
d) Show that e2wt it is a constant of geodesic motion.
[4 marks]
e)
Solve the geodesic equations for null geodesics.
[4 marks]
Page 2
SECTION A
Answer ALL questions in Section A
[Expect to use one single-sided A4 page for each Section-A sub question.]
Question A1
SPA6308 (2024)
Consider Minkowski spacetime in Cartesian coordinates th
=
(t, x, y, z), such that
ds² = dt² + dx² + dy² + dz².
(a) Consider the vector with components V" = (1,-1,0,0). Determine V and V. V.
(b) Consider now the coordinate system x' (u, v, y, z) such that
u =t-x,
v=t+x.
[2 marks]
Write down the line element, the metric, the Christoffel symbols and the Riemann curvature
tensor in the new coordinates. [See the Appendix of this document.]
[5 marks]
(c) Determine V", that is, write the object in question A1.a in the coordinate system x'. Verify
explicitly that V. V is invariant under the coordinate transformation.
Question A2
[5 marks]
Suppose that A, is a covector field, and consider the object
Fv=AAμ.
(a) Show explicitly that F is a tensor, that is, show that it transforms appropriately under a
coordinate transformation.
[5 marks]
(b)…
Chapter 37 Solutions
PHYSICS FOR SCI. & ENGR(LL W/WEBASSIGN)
Ch. 37.2 - Suppose the slit width in Figure 37.4 is made half...Ch. 37.3 - Cats eyes have pupils that can be modeled as...Ch. 37.3 - Suppose you are observing a binary star with a...Ch. 37.4 - Ultraviolet light of wavelength 350 nm is incident...Ch. 37.6 - A polarizer for microwaves can be made as a grid...Ch. 37.6 - You are walking down a long hallway that has many...Ch. 37 - Heliumneon laser light ( = 632.8 nm) is sent...Ch. 37 - From Equation 37.2, find an expression for the...Ch. 37 - Light of wavelength 540 nm passes through a slit...Ch. 37 - In Figure 37.7, show mathematically how many...
Ch. 37 - Assume light of wavelength 650 nm passes through...Ch. 37 - What If? Suppose light strikes a single slit of...Ch. 37 - A diffraction pattern is formed on a screen 120 cm...Ch. 37 - Coherent light of wavelength 501.5 nm is sent...Ch. 37 - The objective lens of a certain refracting...Ch. 37 - Yellow light of wavelength 589 nm is used to view...Ch. 37 - What is the approximate size of the smallest...Ch. 37 - A heliumneon laser emits light that has a...Ch. 37 - To increase the resolving power of a microscope,...Ch. 37 - Prob. 14PCh. 37 - Impressionist painter Georges Seurat created...Ch. 37 - Narrow, parallel, glowing gas-filled tubes in a...Ch. 37 - Consider an array of parallel wires with uniform...Ch. 37 - Three discrete spectral lines occur at angles of...Ch. 37 - A grating with 250 grooves/mm is used with an...Ch. 37 - Show that whenever white light is passed through a...Ch. 37 - Light from an argon laser strikes a diffraction...Ch. 37 - A wide beam of laser light with a wavelength of...Ch. 37 - You are working as a demonstration assistant for a...Ch. 37 - Prob. 24PCh. 37 - Prob. 25PCh. 37 - Prob. 26PCh. 37 - Prob. 27PCh. 37 - Why is the following situation impossible? A...Ch. 37 - The critical angle for total internal reflection...Ch. 37 - For a particular transparent medium surrounded by...Ch. 37 - Prob. 31PCh. 37 - An unpolarized beam of light is incident on a...Ch. 37 - In a single-slit diffraction pattern, assuming...Ch. 37 - Laser light with a wavelength of 632.8 nm is...Ch. 37 - Prob. 35APCh. 37 - Two motorcycles separated laterally by 2.30 m are...Ch. 37 - The Very Large Array (VLA) is a set of 27 radio...Ch. 37 - Two wavelengths and + (with ) are incident on...Ch. 37 - Review. A beam of 541-nm light is incident on a...Ch. 37 - Prob. 40APCh. 37 - Prob. 41APCh. 37 - Prob. 42APCh. 37 - A pinhole camera has a small circular aperture of...Ch. 37 - Prob. 44APCh. 37 - Prob. 45APCh. 37 - (a) Light traveling in a medium of index of...Ch. 37 - The intensity of light in a diffraction pattern of...Ch. 37 - Prob. 48APCh. 37 - Two closely spaced wavelengths of light are...Ch. 37 - A spy satellite can consist of a large-diameter...Ch. 37 - Prob. 51CPCh. 37 - In Figure P37.52, suppose the transmission axes of...Ch. 37 - Consider a light wave passing through a slit and...
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