Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Chapter 33, Problem 57PQ
(a)
To determine
The values of
(b)
To determine
The frequency at which current lags the voltage by
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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 33 Solutions
Physics for Scientists and Engineers: Foundations and Connections
Ch. 33.1 - Prob. 33.1CECh. 33.1 - Prob. 33.2CECh. 33.2 - Prob. 33.3CECh. 33.3 - Prob. 33.4CECh. 33.4 - Prob. 33.5CECh. 33.5 - Prob. 33.6CECh. 33.7 - Prob. 33.7CECh. 33 - Prob. 1PQCh. 33 - Prob. 2PQCh. 33 - Prob. 3PQ
Ch. 33 - Prob. 4PQCh. 33 - Prob. 5PQCh. 33 - Prob. 6PQCh. 33 - Prob. 7PQCh. 33 - Prob. 8PQCh. 33 - Prob. 9PQCh. 33 - Prob. 10PQCh. 33 - Prob. 11PQCh. 33 - At one instant, a current of 6.0 A flows through...Ch. 33 - Prob. 13PQCh. 33 - Prob. 14PQCh. 33 - Prob. 15PQCh. 33 - In Figure 33.9A (page 1052), the switch is closed...Ch. 33 - Prob. 17PQCh. 33 - Prob. 18PQCh. 33 - Prob. 19PQCh. 33 - Prob. 20PQCh. 33 - Prob. 21PQCh. 33 - Prob. 22PQCh. 33 - In the LC circuit in Figure 33.11, the inductance...Ch. 33 - A 2.0-F capacitor is charged to a potential...Ch. 33 - Prob. 26PQCh. 33 - Prob. 27PQCh. 33 - Prob. 28PQCh. 33 - For an LC circuit, show that the total energy...Ch. 33 - Prob. 30PQCh. 33 - Prob. 31PQCh. 33 - Prob. 32PQCh. 33 - Prob. 33PQCh. 33 - Suppose you connect a small lightbulb across a DC...Ch. 33 - Prob. 35PQCh. 33 - Prob. 36PQCh. 33 - Prob. 37PQCh. 33 - Prob. 38PQCh. 33 - Prob. 39PQCh. 33 - Prob. 40PQCh. 33 - Prob. 41PQCh. 33 - Prob. 42PQCh. 33 - Prob. 43PQCh. 33 - In an ideal AC circuit with capacitance, there is...Ch. 33 - Prob. 45PQCh. 33 - Prob. 46PQCh. 33 - Prob. 47PQCh. 33 - Prob. 48PQCh. 33 - Prob. 49PQCh. 33 - An AC generator with an rms emf of 15.0 V is...Ch. 33 - Prob. 51PQCh. 33 - Prob. 52PQCh. 33 - Prob. 53PQCh. 33 - Prob. 54PQCh. 33 - Prob. 55PQCh. 33 - Prob. 56PQCh. 33 - Prob. 57PQCh. 33 - Prob. 58PQCh. 33 - Prob. 59PQCh. 33 - An AC source of angular frequency is connected to...Ch. 33 - An RLC series circuit is constructed with R =...Ch. 33 - Prob. 62PQCh. 33 - A series RLC circuit driven by a source with an...Ch. 33 - Prob. 64PQCh. 33 - Prob. 65PQCh. 33 - Prob. 66PQCh. 33 - Prob. 67PQCh. 33 - Prob. 68PQCh. 33 - Prob. 69PQCh. 33 - Prob. 70PQCh. 33 - Problems 71 and 72 paired. Figure P33.71 shows a...Ch. 33 - Prob. 72PQCh. 33 - Prob. 73PQCh. 33 - Prob. 74PQCh. 33 - Prob. 75PQCh. 33 - In a series RLC circuit with a maximum current of...Ch. 33 - Prob. 77PQCh. 33 - Two coaxial cables of length with radii a and b...Ch. 33 - Prob. 79PQCh. 33 - Prob. 80PQCh. 33 - Prob. 81PQCh. 33 - Prob. 82PQCh. 33 - Prob. 83PQCh. 33 - Prob. 84PQ
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