College Physics
10th Edition
ISBN: 9781285737027
Author: Raymond A. Serway, Chris Vuille
Publisher: Cengage Learning
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Textbook Question
Chapter 15, Problem 33P
Three identical charges (q = −5.0 μC.) lie along a circle of radius 2.0 m at angles of 30°, 150°, and 270°, as shown in Figure P15.33 (page 524). What is the resultant electric field at the center of the circle?
Figure P15.33
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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.
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Chapter 15 Solutions
College Physics
Ch. 15.1 - A suspended object A is attracted to a neutral...Ch. 15.2 - Object A has a charge of +2 C, and object B has a...Ch. 15.3 - A test charge of + 3 C is at a point P where the...Ch. 15.3 - A circular ring of charge of radius b has a total...Ch. 15.3 - A free electron and a free proton are placed in an...Ch. 15.4 - Rank the magnitudes of the electric field at...Ch. 15.8 - Calculate the magnitude of the flux of a constant...Ch. 15.8 - Suppose the electric field of Quick Quiz 15.7 is...Ch. 15.8 - Find the electric flux through the surface in...Ch. 15.8 - For a closed surface through which the net flux is...
Ch. 15 - A glass object receives a positive charge of +3 nC...Ch. 15 - The fundamental charge is e = 1.60 1019 C....Ch. 15 - Each of the following statements is related to...Ch. 15 - Two uncharged, conducting spheres are separated by...Ch. 15 - Four concentric spheres S1, S2, S3, and S4 are...Ch. 15 - IF a suspended object A is attracted to a charged...Ch. 15 - Positive charge Q is located at the center of a...Ch. 15 - Consider point A in Figure CQ15.8 located an...Ch. 15 - A student stands on a thick piece of insulating...Ch. 15 - In fair weather, there is an electric field at the...Ch. 15 - A charged comb often attracts small bits of dry...Ch. 15 - Why should a ground wire be connected to the metal...Ch. 15 - There are great similarities between electric and...Ch. 15 - A spherical surface surrounds a point charge q....Ch. 15 - If more electric field lines leave a Gaussian...Ch. 15 - A student who grew up in a tropical country and is...Ch. 15 - What happens when a charged insulator is placed...Ch. 15 - A 7.50-nC charge is located 1.80 m from a 4.20-nC...Ch. 15 - A charged particle A exerts a force of 2.62 N to...Ch. 15 - Rocket observations show that dust particles in...Ch. 15 - A small sphere of mass m = 7.50 g and charge q1 =...Ch. 15 - The nucleus of 8Be, which consists of 4 protons...Ch. 15 - A molecule of DNA (deoxyribonucleic acid) is 2.17...Ch. 15 - Two uncharged spheres are separated by 2.00 in. If...Ch. 15 - Four point charges are at the corners of a square...Ch. 15 - Two small identical conducting spheres are placed...Ch. 15 - Calculate the magnitude and direction of the...Ch. 15 - Three charges are arranged as shown in Figure...Ch. 15 - A positive charge q1 = 2.70 C on a frictionless...Ch. 15 - Three point charges are located at the corners of...Ch. 15 - Two identical metal blocks resting on a...Ch. 15 - Two small metallic spheres, each of mass m = 0.20...Ch. 15 - Panicle A of charge 3.00 104 C is at the origin,...Ch. 15 - A small object of mass 3.80 g and charge 18.0 C is...Ch. 15 - (a) Determine the electric field strength at a...Ch. 15 - An electric field of magnitude 5.25 105 N/C...Ch. 15 - An electron is accelerated by a constant electric...Ch. 15 - Charge q1 = 1.00 nC is at x1 = 0 and charge q2 =...Ch. 15 - A small sphere of charge q = +68 C and mass m =...Ch. 15 - A proton accelerates from rest in a uniform...Ch. 15 - (a) Find the magnitude and direction of the...Ch. 15 - Four point charges are located at the corners of a...Ch. 15 - A helium nucleus of mass m = 6.64 1027 kg and...Ch. 15 - A charged dust particle at rest in a vacuum is...Ch. 15 - A particle of mass 1.00 109 kg and charge 3.00 pC...Ch. 15 - Two equal positive charges are at opposite corners...Ch. 15 - Three point charges are located on a circular are...Ch. 15 - In Figure P15.31, determine the point (other than...Ch. 15 - Three charges are at the corners of an equilateral...Ch. 15 - Three identical charges (q = 5.0 C.) lie along a...Ch. 15 - Figure P15.31 shows the electric held lines for...Ch. 15 - (a) Sketch the electric field lines around an...Ch. 15 - (a) Sketch the electric field pattern around two...Ch. 15 - Two point charges are a small distance apart. (a)...Ch. 15 - Three equal positive charges are at the corners of...Ch. 15 - Refer 10 Figure 15.20. The charge lowered into the...Ch. 15 - The dome of a Van de Graaff generator receives a...Ch. 15 - If the electric field strength in air exceeds 3.0 ...Ch. 15 - In the Millikan oil-drop experiment illustrated in...Ch. 15 - A Van de Graaff generator is charged so that a...Ch. 15 - A uniform electric field of magnitude E = 435 N/C...Ch. 15 - An electric field of intensity 3.50 kN/C is...Ch. 15 - The electric field everywhere on the surface of a...Ch. 15 - Four closed surfaces, S1 through S4, together with...Ch. 15 - A charge q = +5.80 C is located at the center of a...Ch. 15 - Figure P15.49 shows a closed cylinder with...Ch. 15 - A charge of q = 2.00 109 G is spread evenly on a...Ch. 15 - A point charge q is located at the center of a...Ch. 15 - A charge of 1.70 102 C is at the center of a cube...Ch. 15 - Suppose the conducting spherical shell of Figure...Ch. 15 - A very large nonconducting plate lying in the...Ch. 15 - In deep spare, two spheres each of radius 5.00 m...Ch. 15 - A nonconducting, thin plane sheet of charge...Ch. 15 - Three point charges are aligned along the x-axis...Ch. 15 - A small plastic ball of mass m = 2.00 g is...Ch. 15 - A proton moving at v0 = 1.50 106 m/s enters the...Ch. 15 - The electrons in a particle beam each have a...Ch. 15 - A point charge +2Q is at the origin and a point...Ch. 15 - A 1.00-g cork ball having a positive charge of...Ch. 15 - Two 2.0-g spheres are suspended by 10.0-cm-long...Ch. 15 - a point charge of magnitude 5.00 C is at the...Ch. 15 - Two hard rubber spheres, each of mass m = 15.0 g,...Ch. 15 - Prob. 66APCh. 15 - A solid conducting sphere of radius 2.00 cm has a...Ch. 15 - Three identical point charges, each of mass m =...Ch. 15 - Each of the electrons in a particle beam has a...Ch. 15 - Protons are projected with an initial speed v0 = 9...
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