Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Chapter 23, Problem 53PQ
A metal sphere with charge +8.00 nC is attached to the left-hand end of a nonconducting rod of length L = 2.00 m. A second sphere with charge +2.00 nC is fixed to the right-hand end of the rod (Fig. P23.53). At what position d along the rod can a charged bead be placed for the bead to be in equilibrium?
FIGURE P23.53
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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 23 Solutions
Physics for Scientists and Engineers: Foundations and Connections
Ch. 23.2 - Initially a glass rod and a piece of silk are...Ch. 23.3 - a. In Figure 23.8, why are there three plus signs...Ch. 23.3 - When wool is rubbed against amber, the wool...Ch. 23.3 - Prob. 23.4CECh. 23.4 - The following scenarios involve a metal ball and a...Ch. 23.4 - Prob. 23.6CECh. 23 - What is the difference between a contact force and...Ch. 23 - Many textbooks claim Franklin decided that moving...Ch. 23 - An object has a charge of 35 nC. How many excess...Ch. 23 - As part of a demonstration, a physics professor...
Ch. 23 - A single coulomb represents a large amount of...Ch. 23 - A sphere has a net charge of 8.05 nC, and a...Ch. 23 - A glass rod is initially neutral. After it is...Ch. 23 - After an initially neutral glass rod is rubbed...Ch. 23 - A 50.0-g piece of aluminum has a net charge of...Ch. 23 - Prob. 10PQCh. 23 - A silk scarf is rubbed against glass, and a wool...Ch. 23 - CASE STUDY A person in Franklins time may have...Ch. 23 - Prob. 13PQCh. 23 - Prob. 14PQCh. 23 - A charge of 36.3 nC is transferred to a neutral...Ch. 23 - Prob. 16PQCh. 23 - Prob. 17PQCh. 23 - An electrophorus is a device developed more than...Ch. 23 - Prob. 19PQCh. 23 - An electroscope is a device used to measure the...Ch. 23 - Two particles with charges of +5.50 nC and 8.95 nC...Ch. 23 - Particle A has a charge of 34.5 nC, and particle B...Ch. 23 - Prob. 23PQCh. 23 - Prob. 24PQCh. 23 - Particle A has charge qA and particle B has charge...Ch. 23 - Two charged particles are placed along the y axis....Ch. 23 - A 1.75-nC charged particle located at the origin...Ch. 23 - A 1.75-nC charged particle located at the origin...Ch. 23 - Two particles with charges q1 and q2 are separated...Ch. 23 - An electron with charge e and mass m moves in a...Ch. 23 - Two electrons in adjacent atomic shells are...Ch. 23 - Two small, identical metal balls with charges 5.0...Ch. 23 - Two identical spheres each have a mass of 5.0 g...Ch. 23 - One end of a light spring with force constant k =...Ch. 23 - Two 25.0-g copper spheres are placed 75.0 cm...Ch. 23 - Three charged particles lie along a single line....Ch. 23 - Given the arrangement of charged particles shown...Ch. 23 - Given the arrangement of charged particles in...Ch. 23 - Given the arrangement of charged particles in...Ch. 23 - Three charged metal spheres are arrayed in the xy...Ch. 23 - Charges A, B, and C are arrayed along the y axis,...Ch. 23 - Three identical conducting spheres are fixed along...Ch. 23 - Charges A, B, and C are arranged in the xy plane...Ch. 23 - Prob. 44PQCh. 23 - A particle with charge q is located at the origin,...Ch. 23 - Figure P23.46 shows four identical conducting...Ch. 23 - Prob. 47PQCh. 23 - Two metal spheres of identical mass m = 4.00 g are...Ch. 23 - Figure P23.49 shows two identical small, charged...Ch. 23 - Two small spherical conductors are suspended from...Ch. 23 - Four equally charged particles with charge q are...Ch. 23 - Four charged particles q, q, q, and q are Fixed...Ch. 23 - A metal sphere with charge +8.00 nC is attached to...Ch. 23 - Prob. 54PQCh. 23 - Three small metallic spheres with identical mass m...Ch. 23 - How does a negatively charged rubber balloon stick...Ch. 23 - How many electrons are in a 1.00-g electrically...Ch. 23 - Prob. 58PQCh. 23 - Prob. 59PQCh. 23 - Prob. 60PQCh. 23 - Three charged particles are arranged in the xy...Ch. 23 - A We saw in Figure 23.16 that a neutral metal can...Ch. 23 - Prob. 63PQCh. 23 - A Figure P23.65 shows two identical conducting...Ch. 23 - Two helium-filled, spherical balloons, each with...Ch. 23 - Two small metallic spheres, each with a mass of...Ch. 23 - A Two positively charged spheres with charges 4e...Ch. 23 - Prob. 69PQCh. 23 - Three charged spheres are at rest in a plane as...Ch. 23 - Prob. 71PQCh. 23 - Three particles with charges of 1.0 C, 1.0 C, and...Ch. 23 - A Two positively charged particles, each with...Ch. 23 - Prob. 74PQCh. 23 - Eight small conducting spheres with identical...Ch. 23 - Prob. 76PQCh. 23 - Prob. 77PQCh. 23 - Prob. 78PQCh. 23 - Prob. 79PQ
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