PHYSICS:F/SCI.+ENGRS.,V.1
10th Edition
ISBN: 9781337553575
Author: SERWAY
Publisher: CENGAGE L
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Chapter 22, Problem 37AP
Two small spheres hang in equilibrium at the bottom ends of threads, 40.0 cm long, that have their top ends tied to the same fixed point. One sphere has mass 2.40 g and charge +300 nC. The other sphere has the same mass and charge +200 nC. Find the distance between the centers of the spheres.
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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 22 Solutions
PHYSICS:F/SCI.+ENGRS.,V.1
Ch. 22.1 - Three objects are brought close to each other, two...Ch. 22.2 - Three objects are brought close to one another,...Ch. 22.3 - Object A has a charge of +2 C, and object B has a...Ch. 22.4 - A test charge of +3 C is at a point P where an...Ch. 22.5 - Rank the magnitudes of the electric field at...Ch. 22 - Find to three significant digits the charge and...Ch. 22 - (a) Find the magnitude of the electric force...Ch. 22 - In a thundercloud, there may be electric charges...Ch. 22 - Nobel laureate Richard Feynman (19181088) once...Ch. 22 - A 7.50-nC point charge is located 1.80 m from a...
Ch. 22 - This afternoon, you have a physics symposium...Ch. 22 - Two small beads having positive charges q1 = 3q...Ch. 22 - Two small beads having charges q1 and q2 of the...Ch. 22 - Review. In the Bohr theory of the hydrogen atom,...Ch. 22 - Three point charges lie along a straight line as...Ch. 22 - A point charge +2Q is at the origin and a point...Ch. 22 - Particle A of charge 3.00 104 C is at the origin,...Ch. 22 - Review. Two identical particles, each having...Ch. 22 - Why is the following situation impossible? Two...Ch. 22 - What are the magnitude and direction of the...Ch. 22 - Consider n equal positively charged particles each...Ch. 22 - Two equal positively charged particles are at...Ch. 22 - Two charged particles are located on the x axis....Ch. 22 - Three point charges are located on a circular arc...Ch. 22 - Two 2.00-C point charges are located on the x...Ch. 22 - Three point charges are arranged as shown in...Ch. 22 - Consider the electric dipole shown in Figure...Ch. 22 - Three equal positive charges q are at the corners...Ch. 22 - A proton accelerates from rest in a uniform...Ch. 22 - A proton moves at 4.50 105 m/s in the horizontal...Ch. 22 - Protons are projected with an initial speed vi =...Ch. 22 - You are still fascinated by the process of inkjet...Ch. 22 - You are working on a research project in which you...Ch. 22 - Consider an infinite number of identical...Ch. 22 - A particle with charge 3.00 nC is at the origin,...Ch. 22 - A small block of mass m and charge Q is placed on...Ch. 22 - A small sphere of charge q1 = 0.800 C hangs from...Ch. 22 - A charged cork ball of mass 1.00 g is suspended on...Ch. 22 - A charged cork ball of mass m is suspended on a...Ch. 22 - Three charged particles are aligned along the x...Ch. 22 - Two point charges qA = 12.0 C and qB = 45.0 C and...Ch. 22 - Two small spheres hang in equilibrium at the...Ch. 22 - Four identical charged particles (q = +10.0 C) are...Ch. 22 - Review. Two identical blocks resting on a...Ch. 22 - Review. Two identical blocks resting on a...Ch. 22 - Three identical point charges, each of mass m =...Ch. 22 - Why is the following situation impossible? An...Ch. 22 - Two hard rubber spheres, each of mass m = 15.0 g,...Ch. 22 - Two identical beads each have a mass m and charge...Ch. 22 - Two small spheres of mass m are suspended from...Ch. 22 - You are working as an expert witness for an...Ch. 22 - Review. A 1.00-g cork ball with charge 2.00 C is...Ch. 22 - Eight charged panicles, each of magnitude q, are...Ch. 22 - Two particles, each with charge 52.0 nC, are...Ch. 22 - Review. An electric dipole in a uniform horizontal...
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