University Physics Volume 2
18th Edition
ISBN: 9781938168161
Author: OpenStax
Publisher: OpenStax
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Textbook Question
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Chapter 16, Problem 4CQ
If the electric field of an
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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 16 Solutions
University Physics Volume 2
Ch. 16 - Check Your Understanding When the emf across a...Ch. 16 - Check Your Understanding Could a purely electric...Ch. 16 - Check Your Understanding The wave equation was...Ch. 16 - Check Your Understanding What conclusions did our...Ch. 16 - Check Your Understanding How would the speed and...Ch. 16 - Check Your Understanding How do the...Ch. 16 - Explain how the displacement current maintains the...Ch. 16 - Describe the field lines of the induced magnetic...Ch. 16 - Why is it much easier to demonstrate in a student...Ch. 16 - If the electric field of an electromagnetic wave...
Ch. 16 - In which situation shown below will the...Ch. 16 - In which situation shown below will the...Ch. 16 - Under what conditions might wires in a circuit...Ch. 16 - Shown below is the interference pattern of two...Ch. 16 - When you stand outdoors in the sunlight, y can you...Ch. 16 - How does the intensity of an electromagnetic wave...Ch. 16 - What is the physical significance of the Poynting...Ch. 16 - A 2.0-mW helium-neon laser transmits a continuous...Ch. 16 - Why is t1 radiation pressure of an electromagnetic...Ch. 16 - Why did the early Hubble Telescope photos of...Ch. 16 - (a) If the electric field and magnetic field in a...Ch. 16 - Compare the speed, wavelength, and frequency of...Ch. 16 - Accelerating electric charge emits electromagnetic...Ch. 16 - Compare and contrast the meaning of the prefix...Ch. 16 - Part of the light passing through the air is...Ch. 16 - When a bowl of soup is removed from a microwave...Ch. 16 - Certain orientations of a broadcast television...Ch. 16 - What property of light corresponds to loudness in...Ch. 16 - Is the visible region a major portion of the...Ch. 16 - Can the human body detect electromagnetic...Ch. 16 - Radio waves normally have their E and B fields in...Ch. 16 - Give an example of resonance in the reception of...Ch. 16 - Illustrate that the size of details of an object...Ch. 16 - In which pan of the electromagnetic spectrum are...Ch. 16 - In what range of electromagnetic radiation are the...Ch. 16 - If a microwave oven could be modified to merely...Ch. 16 - A leaky microwave oven in a home can sometimes...Ch. 16 - When a television news anchor in a studio speaks...Ch. 16 - Show that the magnetic field at a distance r from...Ch. 16 - Express the displacement current in a capacitor in...Ch. 16 - A potential difference V(t) = V0sin tis maintained...Ch. 16 - Suppose the parallel-plate capacitor shown below...Ch. 16 - The potential difference V(t) between parallel...Ch. 16 - A parallel-plate capacitor has a plate area of...Ch. 16 - The voltage across a parallel-plate capacitor with...Ch. 16 - The voltage across a parallel-plate capacitor with...Ch. 16 - If the Sun suddenly turned off, we would not know...Ch. 16 - What is the maximum electric field strength in an...Ch. 16 - An electromagnetic wave has a frequency of 12 MHz....Ch. 16 - If electric and magnetic field strengths vary...Ch. 16 - The electric field of an electromagnetic wave...Ch. 16 - A plane electromagnetic wave of frequency 20 GHz...Ch. 16 - The following represents an electromagnetic wave...Ch. 16 - While outdoors on a sunny day, a student holds a...Ch. 16 - A plane electromagnetic wave travels northward. At...Ch. 16 - The electric field of an electromagnetic wave is...Ch. 16 - A radio station broadcasts at a frequency of 760...Ch. 16 - The filament in a clear incandescent light bulb...Ch. 16 - At what distance does a 100-W lightbulb produce...Ch. 16 - An incandescent light bulb emits only 2.6 W of its...Ch. 16 - A 150-W lightbulb emits 5% of its energy as...Ch. 16 - A small helium-neon laser has a power output of...Ch. 16 - At the top of Earth’s atmosphere, the...Ch. 16 - The magnetic field of a plane electromagnetic wave...Ch. 16 - What is the intensity of an electromagnetic wave...Ch. 16 - Assume the helium-neon lasers commonly used in...Ch. 16 - An AM radio transmitter broadcasts 50.0 kW of...Ch. 16 - Suppose the maximum safe intensity of microwaves...Ch. 16 - A 2.50-rn-diameter university communications...Ch. 16 - Lasers can be constructed that produce an...Ch. 16 - A 1-W lightbulb emits 5% of its energy as...Ch. 16 - What pressure does light emitted uniformly in all...Ch. 16 - A microscopic spherical dust particle of radius 2m...Ch. 16 - A Styrofoam spherical ball of radius 2 mm and mass...Ch. 16 - Suppose that S avg for sunlight at a point on the...Ch. 16 - reaches the ground with an intensity of about...Ch. 16 - Suppose a spherical particle of mass m and radius...Ch. 16 - How many helium atoms, each with a radius of about...Ch. 16 - If you wish to detect details of the size of atoms...Ch. 16 - Find the frequency range of visible light, given...Ch. 16 - (a) Calculate the wavelength range for AM radio...Ch. 16 - Radio station WWVB, operated by the National...Ch. 16 - An outdoor WIFi unit for a picnic area has a...Ch. 16 - The prefix “mega” (M) and “kilo” (k), when...Ch. 16 - A computer user finds that his wireless router...Ch. 16 - (a) The ideal size (most efficient) for a...Ch. 16 - What are the wavelengths of (a) X-rays of...Ch. 16 - For red light of =660nm , what are f,, and k?Ch. 16 - A radio transmitter broadcasts plane...Ch. 16 - (a) Two microwave frequencies authorized for use...Ch. 16 - During normal beating, the heart creates a maximum...Ch. 16 - Distances in space are often quoted in units of...Ch. 16 - A certain 60.0-Hz ac power line radiates an...Ch. 16 - (a) What is the frequency of the 193-nm...Ch. 16 - In a region of space, the electric field is...Ch. 16 - A microwave oven uses electromagnetic waves of...Ch. 16 - Galileo proposed measuring the speed of light by...Ch. 16 - Show that the wave equation in one dimension...Ch. 16 - On its highest power setting, a microwave oven...Ch. 16 - A certain microwave oven projects 1.00 kW of...Ch. 16 - E1ecmagnedc radiation from a 5.00-mW laser is...Ch. 16 - A 200-turn flat coil of wire 30.0 cm in diameter...Ch. 16 - Suppose a source of electromagnetic waves radiates...Ch. 16 - A radio station broadcasts its radio waves with a...Ch. 16 - The Poynting vector describes a flow of energy...Ch. 16 - The Sun’s energy strikes Earth at an intensity of...Ch. 16 - If a Lightsail spacecraft were sent on a Mars...Ch. 16 - Lunar astronauts placed a reflector on the Moon’s...Ch. 16 - Radar is used to determine distances to various...Ch. 16 - Calculate the ratio of the highest to lowest...Ch. 16 - How does the wavelength of radio waves for an AM...Ch. 16 - A parallel-plate capacitor with plate separation d...Ch. 16 - A particle of cosmic dust has a density =2.0g/cm3...
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