Saturday, November 29, 2014
Friday, November 28, 2014
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Thursday, November 27, 2014
23 Electromagnetic Waves
23 Electromagnetic Waves
v = λf
23.2 Speed of a Electromagnetic Wave
E = cB (traveling wave) (23.1)
c = 1 / ( ε0 μ0)½ (23.2)
c = 3.00 × 108 m/s
transverse wave
23.3 Electromagnetic Spectrum
700nm – 400nm
23.4 Sinusoidal Waves
plane wave
E = Emax sin 2π( t / T – x / λ) = Emax sin (ωt – kx)
B = Bmax sin 2π( t / T – x / λ) = Bmax sin (ωt – kx) (23.3)
Emax = cBmax. (23.4)
E = ─Emax sin 2π( t / T + x / λ) = ─Emax sin (ωt + kx)
B = Bmax sin 2π( t / T + x / λ) = Bmax sin (ωt + kx) (23.5)
23.5 Energy I Electromagnetic Waves
u = ε0E2 / 2 + B2 / (2μ0) (23.6)
B = E / c = ( ε0 μ0)½ E
u = ε0E2 / 2 + B2 / (2μ0) = ε0E2 / 2 + ( ( ε0 μ0)½ E)2 / (2μ0) = ε0E2 (23.7)
ΔU = u ΔV = (ε0E2) (AcΔt)
S = ( ΔU / Δt ) = ε0cE2
S = ( ΔU / Δt ) = ε0cE2 = ( ε0 / μ0)½ E2 = EB / μ0 = cu (23.8)
Sav = ε0cEmax2/2 = ( ε0 / μ0)½ Emax2/2 = EmaxBmax / (2μ0) = cu (sinusoidal wave) (23.9)
Sav = uav / ( ε0 μ0)½ = cuav (sinusoidal wave) (23.10)
I = Sav = ε0cEmax2/2 = ( ε0 / μ0)½ Emax2/2 = EmaxBmax / (2μ0) (sinusoidal wave) (23.11)
I intensity
Radiation Pressure
p / V = ε0E2 / c = EB / (μ0c2) = S / c2 (23.12)
( Δp / Δt ) / A = ε0E2/2 = (Sav / c2) c= Sav / c = I / c (sinusoidal wave) (23.13)
23.6 Nature of Light
wave front, rays, geometric optic, physical optics
23.7 Reflection and Refraction
n = c / v (23.14)
θr = θa (23.15)
sin θa / sin θb = nb / na na sin θa = nb sin θb (23.16)
λ = λ0 / n (23.17)
nb sin θb = na sin θa → sin θb = sin θa · na / nb
sin θcrit = sin 90º · na / nb = 1 · na / nb → sin θcrit = na / nb (23.18)
23.9 Dispersion c = c( f ), n = n( f )
23.10 Polarization
linarly polarized, polarizing filter, dichroizm, polarizing axis, polarizer,
I = Imax cos2 ϕ (23.19)
Polarization by Reflection
plane of incidence, polarizing angle θp, refracted beam, completely polarized
sin θp / cos θp = tan θp = nb / na Brewster's law (23.20)
Photoelasticity
23.11 Huygens's Principle
sin θa / sin θb = va / vb (23.21)
23.12 Scattering of light
Tuesday, November 25, 2014
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Thursday, November 20, 2014
Tuesday, November 18, 2014
Quiz Electromagnetic Induction is open
Quiz Electromagnetic Induction is open. Deadline is Friday, 11/21/2014 12:00.
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Thursday, November 13, 2014
Chapter 21 Electromagnetic Induction, Problems 1, 2, 4, 7, 9, 12, 13, 18, 23, 25
Chapter 21, problem 1. A circular area with a radius of x1 cm lives in the x-y plane. What is the magnitude of the
magnetic flux through this circle due to a uniform magnetic field B
= x2 T that point
(a)
in the +z
direction?
(b)
at an angle of x3ᵒ from the +z
direction?
(c)
in the +y
direction?
Chapter 21, problem
2. The magnetic field B in a certain region is x1 T, and its direction is
that of the +z axis in figure 21.47.
(a) What is the magnetic flux across the surface abcd in the figure?
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| Add caption |
(b) What is the magnetic flux across the surface befc?
(c) What is the magnetic flux across the surface aefd?
(d) What is the net magnetic flux through all five surfaces that enclose the shaded volume?
Chapter 21, Problem 4. A single loop of wire with an area of
x1 m2 is in a
uniform magnetic field that has an initial value of x2 T, is perpendicular to the plane of the loop, and is
decreased with the constant rate of x3
T/s.
(a) What emf is induced in this loop?
(a) What emf is induced in this loop?
(b) If the loop has a resistance of x4 Ω, find the current induced in the loop.
Chapter 21, problem 7.
A closely wound rectangular coil of N turns has dimensions of x1
cm by x2 cm. The plane of the coil is rotated from a position where
it makes an angle of x3ᵒ with a magnetic field of x4 T to
a position perpendicular to the field. The rotation takes x5 s.
What is average emf induced on the coil?
What is average emf induced on the coil?
Chapter 21, Problem 9
A x1 cm × x2
cm rectangular circuit containing a x3 Ω resistor is
perpendicular to a uniform magnetic field that start out at x4
T and steadily decreases at x5 T/s. While this field
is changing, what does the ammeter read? (What is electric current in the circuit?)
Chapter 21, Problem 12
(a) The current in winding A is from a to b and is increasing.
(b) The current in winding A is from b to a and is decreasing.
(c) The current in winding A is from b to a and is increasing.
Chapter 21, Problem 13
The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when
(a)
B is increasing;
(b)
B is decreasing;
(c)
B is constant with a value of B0.
Explain your reasoning.
Explain your reasoning.
Chapter 21, Problem 18
(a)
The loop is dropped.
(b)
The magnet is dropped.
(c)
Both the loop and magnet are dropped at the same
instant.
Chapter 21, Problem 23
You’re driving at x1 km/h in a direction x2° east of north, in a region where the earth’s
magnetic field of x3 × 10-5
T is horizontal and points due north. If your car measures x4 m from its underbody to its roof, calculate the
induced emf between roof and
underbody. (You can assume the sides of the car are straight and vertical.) Is
the roof of the car at a higher or lower potential than the underbody?
Chapter 21, Problem 25
(a) Find the magnitude of the emf induced in the rod when it is moving toward the right with a speed x2 m/s.
(b) In what direction does the current flow
in the rod?
(c) If the resistance of the circuit abdc is a constant x3 Ω, find the magnitude and direction of the force
required to keep the rod moving to the right with a constant speed of x4 m/s.
College Physics: Problems
Hugh D. Young, College Physics, 9th edition, Addison Wesley 2011
Chapter 21
Multiple-Choice Problems
1. A square loop of wire is pulled upward out of the space
between the poles of a magnet, as shown in Figure 21.35. As this is done, the
current induced in this loop, as viewed from the N pole of the magnet, will be
directed
A. clockwise.
B. counterclockwise.
C. zero.
2. The two solenoids in Figure 21.36 are coaxial and fairly
close to each other. While the resistance of the variable resistor in the
left-hand solenoid is increased at a constant rate, the induced current through
the resistor R will
A. flow from a to b. B. flow from b to a. C. be zero because the rate is
constant.
3. A metal ring is oriented with the plane of its area
perpendicular to a spatially uniform magnetic field that increases at a steady
rate. After the radius of the ring is doubled, while the rate of increase of
the field is cut in half, the emf induced in the ring
A. remains the same.
B. increases by a factor of 2.
C. increases by a factor of 4.
D. decreases by a factor of 2.
4. The slide wire of the variable resistor in Figure 21.37
is moved steadily to the right, increasing the resistance in the circuit. While
this is being done, the current induced in the small circuit A is directed
A. clockwise.
B. counterclockwise.
C. zero.
5. The slide wire on the variable resistor in Figure 21.38
is moved steadily to the left. While this is being done, the current induced in
the small circuit A is directed
A. clockwise.
B. counterclockwise.
C. zero.
6. A metal loop moves at constant velocity toward a long
wire carrying a steady current I, as shown in Figure 21.39. The current induced
in the loop is directed
A. clockwise.
B. counterclockwise.
C. zero.
7. The primary coil
of an ideal
transformer carries a current of 2.5 A, while the secondary coil carries
a current of 5.0 A. The ratio of number of turns of wire in the primary to that
in the secondary is
A. 1:1 B.
1:2 C. 2:1
8. A metal loop is held above the S pole of a bar magnet, as
shown in Figure 21.40, when the magnet is suddenly dropped from rest. Just
after the magnet is dropped, the induced current in the loop, as viewed from above it, is directed
A. clockwise.
B. counterclockwise. C. zero.
9. A steady current of 1.5 A flows through the solenoid
shown in Figure 21.41. The current induced in the loop, as viewed from the
right, is directed
A. clockwise.
B. counterclockwise.
C. zero.
10. A vertical bar moves horizontally at constant velocity
through a uniform magnetic field, as shown in Figure 21.42. We observe that
point b is at a higher potential than point a. We can therefore conclude that
the magnetic field must have a component that is directed
A. vertically downward. C. perpendicular to the
plane of the paper, outward.
B. vertically upward. D. perpendicular to
the plane of the paper, inward.
11. The vertical loops A and C in Figure 21.43 are parallel
to each other and are centered on the same horizontal line that is
perpendicular to both of them. Just after the switch S is closed, loop A will
A. not be affected by loop C. B. be attracted by loop C.
C. be repelled by loop C. D. move upward.
E. move downward.
12. After the switch S in Figure 21.43 has been closed for a
very long time, loop A will
A. not be affected by loop C. B. be attracted by loop C.
C. be repelled by loop C. D. move upward.
E. move downward.
13. After the switch S in the circuit in Figure 21.44 is
closed,
A. The current is zero 1.5 ms (one time constant) later.
B. The current is zero for a very long time afterward.
C. The largest current
is 5.0 A and
it occurs just
after S is closed.
D. The largest current is 5.0 A and it occurs a very long
time after S has been closed.
14. A square metal loop is pulled to the right at a constant
velocity perpendicular to a uniform
magnetic field, as shown in Figure 21.45. The current
induced in this loop is directed
A. clockwise.
B. counterclockwise.
C. zero.
15. A metal loop
is being pushed
at a constant velocity into a uniform magnetic
field, as shown in Figure 21.46, but is only partway into the field. As a result
of this motion,
A. End a of the resistor R is at a higher potential than end
b.
B. End b of the resistor R is at a higher potential than end
a.
C. Ends a and b are at the same potential.
Problems
1. A circular area with a radius of 6.50 cm lies in the x-y
plane. What is the magnitude of the magnetic flux through this circle due to a
uniform magnetic field B = 0.230 T that points (a) in the +z direction? (b) at
an angle of 53.1° from the +z direction?
(c) in the +y direction?
2. The magnetic field B in a certain region is 0.128 T, and
its direction is that of the +z axis in Figure 21.47. (a) What is the magnetic
flux across the surface abcd in the figure? (b) What is the magnetic flux
across the surface befc? (c) What is the magnetic flux across the surface aefd?
(d) What is the net flux through all five surfaces that enclose the shaded
volume?
3. An open plastic soda bottle with an opening diameter of
2.5 cm is placed on a table. A uniform 1.75 T magnetic field directed upward
and oriented 25° from vertical encompasses the bottle. What is the total
magnetic flux through the plastic of the soda bottle?
4. A single loop of wire with an area of 0.0900 m2 is in a
uniform magnetic field that has an initial value of 3.80 T, is perpendicular to
the plane of the loop, and is decreasing at a constant rate of 0.190 T/s (a)
What emf is induced in this loop? (b) If the loop has a resistance of 0.600 Ω
find the current induced in the loop.
5. A coil of wire with 200 circular turns of radius 3.00 cm
is in a uniform magnetic field along the axis of the coil. The coil has R =
40.0 Ω. At what rate, in teslas per second, must the magnetic field be changing
to induce a current of 0.150A in the coil?
6. In a physics laboratory experiment, a coil with 200 turns
enclosing an area of 12 cm2 is rotated from a position where its plane is
perpendicular to the earth’s magnetic field to one where its plane is parallel
to the field. The rotation takes 0.040 s. The earth’s magnetic field at the
location of the laboratory is 6.0 × 10-5 T (a) What is the total magnetic flux
through the coil before it is rotated? After it is rotated? (b) What is the
average emf induced in the coil?
7. A closely wound rectangular coil of 80 turns has
dimensions of 25.0 cm by 40.0 cm. The plane of the coil is rotated from a
position where it makes an angle of 37.0° with a magnetic field of 1.10 T to a
position perpendicular to the field. The rotation takes 0.0600 s. What is the
average emf induced in the coil?
8. A very long, straight solenoid with a cross-sectional
area of 6.00 m2 is wound with 40 turns of wire per centimeter, and the windings
carry a current of 0.250 A. A secondary winding of 2 turns encircles the
solenoid at its center. When the primary circuit is opened, the magnetic field
of the solenoid becomes zero in 0.0500 s. What is the average induced emf in
the secondary coil?
9. A 30.0 cm × 60.0 cm rectangular circuit containing a 15 Ω
resistor is perpendicular to a uniform magnetic field that starts out at 2.65 T
and steadily decreases at 0.25 T/s (See Figure 21.48.) While this field is
changing, what does the ammeter read?
10. A circular loop of wire with a radius of 12.0 cm is
lying flat on a tabletop. A magnetic
field of 1.5 T is directed vertically upward through the loop (Figure 21.49).
(a) If the loop is removed from the field region in a time interval of 2.0 ms,
find the average emf that will be induced in the wire loop during the
extraction process. (b) If the loop is viewed looking down on
it from above, is
the induced current in
the loop clockwise
or counterclockwise?
11. A flat, square coil with 15 turns has sides of length
0.120 m. The coil rotates in a magnetic field of 0.0250 T. (a) What is the
angular velocity of the coil if the maximum emf produced is 20.0 mV? (b) What
is the average emf at this angular velocity?
12. A cardboard tube is wrapped with two windings of
insulated wire, as shown in Figure 21.50. Is the induced current in the
resistor R directed from left to right or from right to left in the following
circumstances? The current in winding A is directed (a) from a
to b and
is increasing, (b) from b
to a and
is decreasing, (c) from b to a and is increasing, and (d) from b to a
and is constant.
13. A circular loop of wire is in a spatially uniform
magnetic field, as shown in Figure 21.51. The magnetic field is directed into the plane of the figure. Determine the
direction (clockwise or counterclockwise) of the induced current in the loop
when (a) B is increasing; (b) B is decreasing; (c) B is constant with a value
of B0. Explain your reasoning.
14. Using Lenz’s law, determine the direction of the current
in resistor ab of Figure 21.52 when (a)
switch S is opened after having been closed for several minutes; (b)
coil B is brought closer to coil A with the switch closed; (c) the resistance
of R is decreased while the switch remains closed.
15. A solenoid carrying a current I is moving toward a metal
ring, as shown in Figure 21.53. In what direction, clockwise or
counterclockwise (as seen from the solenoid) is a current induced in the
ring? In what direction will the induced
current be if the solenoid now stops moving toward the ring, but the current in
it begins to decrease?
16. A metal bar is pulled to the right perpendicular to a
uniform magnetic field. The bar rides on parallel metal rails connected through
a resistor, as shown in Figure 21.54, so the apparatus makes a complete
circuit. Find the direction of the current induced in the circuit in two ways:
(a) by looking at the magnetic force on the charges in the moving bar and (b)
using Lenz’s law.
17. Two closed loops A and C are close to a long wire
carrying a current I. (See Figure 21.55.) Find the direction (clockwise or
counterclockwise) of the current induced in each of these loops if I is
steadily increasing.
18. A bar magnet is held above a circular loop of wire as
shown in Figure 21.56. Find the direction (clockwise or counterclockwise, as
viewed from below the loop) of the current induced in this loop in each of the
following cases. (a) The loop is dropped. (b) The magnet is dropped. (c) Both
the loop and magnet are dropped at the same instant.
19. The current in Figure. 21.57 obeys the equation Ie =
I0e-2bt where b > 0. Find the direction (clockwise or counterclockwise) of
the current induced in the round coil for t > 0.
20. A bar magnet is close to a metal loop. When this magnet
is suddenly moved to the left away from the loop, as shown in Figure 21.58, a
counterclockwise current is induced in the coil, as viewed by an observer
looking through the coil toward the magnet. Identify the north and south poles
of the magnet.
21. A very thin 15.0 cm copper bar is aligned horizontally
along the east–west direction. If it moves horizontally from south to north at
11.5 m/s in a vertically upward magnetic field of 1.22 T, (a) what potential
difference is induced across its ends, and (b) which end (east or west) is at a
higher potential? (c) What would be the potential difference if the bar moved
from east to west instead?
22. When a thin 12.0 cm iron rod moves with a constant
velocity of 4.50 m/s perpendicular to the rod in the direction shown in Figure
21.59, the induced emf across its ends is measured to be 0.450V. (a) What is
the magnitude of the magnetic field? (b) Which point is at a higher potential,
a or b? (c) If the bar is rotated clockwise by 90° in the plane of the paper, but
keeps the same velocity, what is the potential difference induced across its
ends?
23. You’re driving at 95 km/h in a direction 35° east of
north, in a region where the earth’s magnetic field of 5.5 × 10-5 T is
horizontal and points due north. If your car measures 1.5 m from its underbody
to its roof, calculate the induced emf between roof and underbody. (You can
assume the sides of the car are straight and vertical.) Is the roof of the car
at a higher or lower potential than the underbody?
24. A 1.41 m bar moves through a uniform, 1.20 T magnetic
field with a speed of 2.50 m/s (Figure 21.60). In each case, find the emf
induced between the ends of this bar and identify which, if any, end (a or b)
is at the higher potential. The bar moves in the direction of (a) the +x-axis;
(b) the –y-axis;
(c) the +z-axis. (d) How should this bar move so that the
emf across its ends
has the greatest
possible value with b
at a higher potential than a, and what
is this maximum emf?
25. The conducting rod ab shown in Figure 21.61 makes
frictionless contact with metal rails ca and db. The apparatus is in a uniform
magnetic field of 0.800 T, perpendicular to the plane of the figure. (a) Find
the magnitude of the emf induced in the rod when it is moving toward the right
with a speed 7.50 m/s. (b) In what direction does the current flow in the rod?
(c) If the resistance of the circuit abdc is a constant 1.50 Ω, find the
magnitude and direction of the force required to keep the rod moving to the
right with a constant speed of 7.50 m/s.
26. Measuring blood flow. Blood contains positive and
negative ions and therefore is a conductor. A blood vessel, therefore, can be
viewed as an electrical wire. We can even picture the flowing blood as a series
of parallel conducting slabs whose thickness is the diameter d of the vessel
moving with speed v (a) If the blood vessel is placed in a magnetic field B
perpendicular to the vessel, as in the figure, show that the motional potential
difference induced across it is E = vBd. (b) If you expect that the blood will
be flowing at for a vessel 5.0 mm in diameter, what strength of magnetic field
will you need to produce a potential difference of 1.0 mV? (c) Show that the
volume rate of flow (R) of the blood is equal to R = πEd/(4B).
27. A toroidal solenoid has a mean radius of 10.0 cm and a
cross-sectional area of 4.00 cm2 and is wound uniformly with 100 turns. A
second coil with 500 turns is wound uniformly on top of the first. What is the
mutual inductance of these coils?
28. A 10.0-cm-long solenoid of diameter 0.400 cm is wound
uniformly with 800 turns. A second coil with 50 turns is wound around the
solenoid at its center. What is the mutual inductance of the combination of the
two coils?
29. Two coils are wound around the same cylindrical form,
like the coils in Example 21.8. When the current in the first coil is
decreasing at a rate of 0.242 A/s, the induced emf in the second coil has
magnitude 1.65 mV. (a) What is the mutual inductance of the pair of coils? (b)
If the second coil has 25 turns, what is the average magnetic flux through each
turn when the current in the first coil equals 1.20 A? (c) If the current in
the second coil increases at a rate of 0.360 A/s, what is the magnitude of the
induced emf in the first coil?
30. One solenoid is centered inside another. The outer one
has a length of 50.0 cm and contains 6750 coils, while the coaxial inner
solenoid is 3.0 cm long and 0.120 cm in diameter and contains 15 coils. The
current in the outer solenoid is changing at 37.5 A/s. (a) What is the mutual
inductance of these solenoids? (b) Find the emf induced in the inner solenoid.
31. Two toroidal solenoids are wound around the same form so
that the magnetic field of one passes through the turns of the other. Solenoid
1 has 700 turns, and solenoid 2 has 400 turns. When the current in solenoid 1
is 6.52 A, the average flux through each turn of solenoid 2 is 0.0320 Wb. (a)
What is the mutual inductance of the pair of solenoids? (b) When the current in
solenoid 2 is 2.54 A, what is the average flux through each turn of solenoid 1?
32. A 4.5 mH toroidal inductor has 125 identical equally
spaced coils. (a) If it carries an 11.5 A current, how much magnetic flux
passes through each of its coils? (b) If the potential difference across its
ends is 1.16 V, at what rate is he current in it changing?
33. At the instant when the current in an inductor is
increasing at a rate of 0.0640 A/s, the magnitude of the self-induced emf is
0.0160 V. What is the inductance of the inductor?
34. An inductor has inductance of 0.260 H and carries a
current that is decreasing at a uniform rate of 18.0 mA/s. Find the self-induced emf in this inductor.
35. A 2.50 mH toroidal solenoid has an average radius of
6.00 cm and a cross-sectional area of 2.00 cm2 (a) How many coils does it have?
(Make the same assumption as in Example 21.10.) (b) At what rate must the
current through it change so that a potential difference of 2.00 V is developed
across its ends?
36. Self-inductance of a solenoid. A long, straight solenoid
has N turns, a uniform cross-sectional area A, and length l. Use the definition
of self-inductance expressed by Equation 21.13 to show that the inductance of
this solenoid is given approximately by the equation L = μ0AN2/l. Assume that
the magnetic field is uniform inside the solenoid and zero outside.
37. When the current in a toroidal solenoid is changing at a
rate of 0.0260 A/s, the magnitude of the induced emf is 12.6 mV. When the
current equals 1.40 A, the average flux through each turn of the solenoid is
0.00285 Wb. How many turns does the solenoid have?
38. A transformer consists of 275 primary windings and 834
secondary windings. If the potential difference across the primary coil is 25.0
V, (a) what is the voltage across the secondary coil, and (b)
what is the effective load resistance of the secondary coil if it is
connected across a 125 Ω resistor?
39. Off to Europe! You plan to take your hair blower to
Europe, where the electrical outlets put out 240 V instead of the 120V seen in
the United States. The blower puts out 1600 W at 120 V. (a) What could you do
to operate your blower via the 240 V line in Europe? (b) What current will your
blower draw from a European outlet? (c) What resistance will your blower appear
to have when operated at 240 V?
40. You need a transformer that will draw 15 W of power from
a 220 V (rms) power line, stepping the voltage down to 6.0 V (rms). (a)
What will be the current in the secondary coil? (b) What should be the
resistance of the secondary circuit? (c) What will be the equivalent resistance
of the input circuit?
41. A step-up transformer. A transformer connected to a 120
V (rms) ac line is to supply 13,000 V (rms) for a neon sign. To reduce the
shock hazard, a fuse is to be inserted in the primary circuit and is to blow
when the rms current in the secondary circuit exceeds 8.50 mA. (a) What is the
ratio of secondary to primary turns of the transformer? (b) What power must be
supplied to the transformer when the rms secondary current is 8.50 mA? (c)What
current rating should the fuse in the primary circuit have?
42. An air-filled toroidal solenoid has a mean radius of
15.0 cm and a cross-sectional area of 5.00 cm2. When the current is 12.0 A, the
energy stored is 0.390 J. How many turns does the winding have?
43. Energy in a typical inductor. (a) How much energy is
stored in a 10.2 mH inductor carrying a 1.15 A current? (b) How much current
would such an inductor have to carry to store 1.0 J of energy? Is this a
reasonable amount of current for ordinary laboratory circuit elements?
44. (a) What would have to be the self-inductance of a
solenoid for it to store 10.0 J of energy when a 1.50 A current runs through it? (b) If this
solenoid’s cross-sectional diameter is 4.00 cm, and if you could wrap its coils
to a density of 10 coils/mm, how long would the solenoid be? (See problem 36.)
Is this a realistic length for ordinary laboratory use?
45. A solenoid 25.0 cm long and with a cross-sectional area
of 0.500 cm2 contains 400 turns of wire and carries a current of 80.0 A.
Calculate: (a) the magnetic field in the solenoid; (b) the energy density in
the magnetic field if the solenoid is filled
with air; (c)
the total energy
contained in the
coil’s magnetic field (assume the field is uniform); (d) the inductance
of the solenoid.
46. Large inductors have been proposed as energy-storage
devices. (a) How much electrical energy is converted to light and thermal
energy by a 200 W lightbulb in one day? (b) If the amount of energy calculated
in part (a) is stored in an inductor in which the current is 80.0 A, what is
the inductance?
47. When a certain inductor carries a current I, it stores
3.0 mJ of magnetic energy. How much current (in terms of I ) would it have to
carry to store 9.0 mJ of energy?
48. A 12.0 V dc battery having no appreciable internal
resistance, a 150.0 Ω resistor, an 11.0 mH inductor, and an open switch are all
connected in series. After the switch is closed, what are (a) the time constant
for this circuit, (b) the maximum current that flows through it, (c) the
current 73.3 μs after the switch is closed, and (d) the maximum energy stored
in the inductor?
49. An inductor with an inductance of 2.50 H and a resistor
with a resistance of 8.00 Ω are connected to the terminals of a battery with an
emf of 6.00 V and negligible internal resistance. Find (a) the initial rate of
increase of the current in the circuit, (b) the initial potential difference
across the inductor, (c) the current 0.313 s after the circuit is closed, and
(d) the maximum current.
50. In Figure 21.63, both switches S1 and S2 are initially
open. S1 is then closed and left closed until a constant current is
established. Then S2 is closed just as S1, is opened, taking the battery out of
the circuit. (a) What is the initial current in the resistor just after S2 is
closed and S1 is opened? (b) What is the time constant of the circuit? (c) What
is the current in the resistor after a large number of time constants have
elapsed?
51. In the circuit
shown in Figure 21.64, the battery and the inductor have no appreciable
internal resistance and there is no current in the circuit. After the switch is
closed, find the readings of the ammeter (A) and voltmeters (V1 and V2) (a) the
instant after the switch is closed; (b) after the switch has been closed for a
very long time. (c) Which answers in parts (a) and (b) would change if the
inductance were 24.0 mH instead?
52. A 35.0 V battery with negligible internal resistance, a
50.0 V resistor, and a 1.25 mH inductor with negligible resistance are all
connected in series with an open switch. The switch is suddenly closed. (a) How
long after closing the switch will the current through the inductor reach
one-half of its maximum value? (b) How long after closing the switch will the
energy stored in the inductor reach one-half of its maximum value?
53. A 1.50 mH inductor is connected in series with a dc
battery of negligible internal resistance, a 0.750 kΩ resistor, and an open
switch. How long after the switch is closed will it take for (a) the current in
the circuit to reach half of its maximum value, (b) the energy stored in the
inductor to reach half of its maximum value?
54. A 12.0 μF capacitor and a 5.25 mH inductor are connected
in series with an open switch. The capacitor is initially charged to 6.20 μC.
What is the angular frequency of the charge oscillations in the capacitor after
the switch is closed?
55. A 5.00 μF capacitor is initially charged to a potential
of 16.0 V. It is then connected in series with a 3.75 mH inductor. (a) What is
the total energy stored in this circuit? (b) What is the maximum current in the
inductor? What is the charge on the capacitor plates at the instant the current
in the inductor is maximal?
56. A 15.0 μF capacitor is charged to 175 μC and then
connected across the ends of a 5.00 mH inductor. (a) Find the maximum current
in the inductor. At the instant the current in the inductor is maximal, how
much charge is on the capacitor plates? (b) Find the maximum potential across
the capacitor. At this instant, what is the current in the inductor? (c) Find
the maximum energy stored in the inductor. At this instant, what is the current
in the circuit?
57. An inductor is connected to the terminals of a battery
that has an emf of 12.0 V and negligible internal resistance. The current is
4.86 mA at 0.725 ms after the connection is completed. After a long time the
current is 6.45 mA. What are (a) the resistance R of the inductor and (b) the
inductance L of the inductor?
58. A rectangular circuit is moved at a constant velocity of
3.0 m/s into, through, and then out of a uniform 1.25 T magnetic field, as
shown in Figure 21.65. The magnetic field region is considerably wider than
50.0 cm. Find the magnitude and direction (clockwise or counterclockwise) of
the current induced in the circuit as it is (a) going into the magnetic field,
(b) totally within the magnetic field, but still moving, and (c) moving out of
the field. (d) Sketch a graph of the current in this circuit as a function of
time, including the preceding three cases.
59. The rectangular loop in Figure 21.66, with area A and
resistance R, rotates at uniform angular velocity ω about the y axis. The loop
lies in a uniform magnetic field B in the direction of the x axis. Sketch
graphs of the following quantities, as functions of time, letting t = 0 when
the loop is in the position shown in the figure: (a) the magnetic flux through
the loop, (b) the rate of change of flux with respect to time, (c) the induced
emf in the loop, (d) the induced emf if the angular velocity is doubled.
60. A flexible circular loop 6.50 cm in diameter lies in a
magnetic field with magnitude 0.950 T, directed into the plane of the page as
shown in Figure 21.67. The loop is pulled at the points indicated by the
arrows, forming a loop of zero area in 0.250 s. (a) Find the average induced
emf in the circuit. (b) What is the direction of the current in R: from a to b
or from b to a? Explain your reasoning.
61. An electromagnetic car alarm. Your latest invention is a
car alarm that produces sound at a particularly annoying frequency of 3500 Hz.
To do this, the car-alarm circuitry must produce an alternating electric
current of the same frequency. That’s why your design includes an inductor and
a capacitor in series. The maximum voltage across the capacitor is to be 12.0 V
(the same voltage as the car battery). To produce a sufficiently loud sound,
the capacitor must store 0.0160 J of energy. What values of capacitance and
inductance should you choose for your car-alarm circuit?
62. In the circuit shown in Figure 21.68, S1 has been closed
for a long enough time so that the current reads a steady 3.50 A. Suddenly, S2
is closed and S1 is opened at the same instant. (a) What is the maximum charge
that the capacitor will receive? (b) What is the current in the inductor at
this time?
63. Consider the circuit in Figure 21.69. (a) Just after the
switch is closed, what is the current through each of the resistors? (b) After
the switch has been closed a long time, what is the current through each
resistor? (c) After S has been closed a
long time, it is opened again. Just after it is opened, what is the current
through the 20.0 Ω resistor?
Chapter 22
Multiple-Choice Problems
1. A piece of electrical equipment in an ac circuit draws a
root-mean-square current of 5.00 A. The average current over each cycle is
A. 5√2 = 7.07 A
B. 5.00 A. C. 5/√2 = 3.54 A D. 0.
2. A sinusoidal current is described by i = Icosωt, where ω
= 1.57 rad/s. At some time t', where 2s < t' < 4s, the current is +3.0 A.
Which phasor can represent the current at time t'?
3. A lightbulb is the resistance in a series R–L–C circuit
having an ac voltage source v = Vcosωt. As the frequency of the source is
adjusted closer and closer to the value 1/√(LC) what happens to the brightness
of the bulb?
A. It increases. B.
It decreases. C. It does not change.
4. A series R–L–C ac circuit with a sinusoidal voltage
source of angular frequency ω has a total reactance X. If this frequency is
doubled, the reactance becomes
A. 4X. B. 2X. C. X/2 D. X/4 E.
none of the above.
5. In a series R–L–C circuit powered by an ac sinusoidal
voltage source, which phasor diagram best illustrates the relationship between
the current i and the potential drop vR across the resistor?
6. In a series R–L–C circuit powered by an ac sinusoidal voltage
source, which phase diagram best illustrates the relationship between the
current i and the potential drop vc across the capacitor?
7. In a series R–L–C circuit powered by an ac sinusoidal
voltage source, which phase diagram best
illustrates the relationship between the current i and the potential
drop vL across the inductor?
8. A series circuit contains an inductor, a resistor, a
capacitor, and a sinusoidal voltage source of angular frequency ω. If we double
this frequency (there may be more than one correct choice),
A. the inductive reactance is doubled.
B. the capacitive reactance is doubled.
C. the total reactance is doubled.
D. the impedance is doubled.
9. In order to double the resonance frequency of a series
R–L–C ac circuit, you could
A. double both the inductance and capacitance.
B. double the resistance.
C. cut the resistance in half.
D. cut both the inductance and capacitance in half.
Problems
1. You have a special lightbulb with a very delicate wire
filament. The wire will break if the current in it ever exceeds 1.50 A, even
for an instant. What is the largest root-mean-square current you can run
through this bulb?
2. The plate on the back of a certain computer scanner says
that the unit draws 0.34 A of current from a 120 V, 60 Hz line. Find (a) the
root-mean-square current, (b) the current amplitude, (c) the average current,
and (d) the average square of the current.
3. A capacitance C and an inductance L are operated at the
same angular frequency. (a) At what angular frequency will they have the same
reactance? (b) If L = 5.00 mH and C = 3.50 μF, what is the numerical value of
the angular frequency in part (a), and what is the reactance of each element?
4. (a) Compute the reactance of a 0.450 H inductor at
frequencies of 60.0 Hz and 600 Hz. (b) Compute the reactance of a 2.50 μF
capacitor at the same frequencies. (c) At what frequency is the reactance of a
0.450 H inductor equal to that of a 2.50 μF capacitor?
5. A radio inductor. You want the current amplitude through
a 0.450-mH inductor (part of the circuitry for a radio receiver) to be 2.60 mA
when a sinusoidal voltage with amplitude 12.0 V is applied across the inductor.
What frequency is required?
6. A 2.20 μF capacitor is connected across an ac source
whose voltage amplitude is kept constant at 60.0 V, but whose frequency can be
varied. Find the current amplitude when the angular frequency is (a) 100 rad/s
(b) 1000 rad/s (c) 10000 rad/s.
7. The voltage amplitude of an ac source is 25.0 V, and its
angular frequency is 1000 rad/s. Find the current amplitude if the capacitance
of a capacitor connected across the source is (a) 0.0100 μF (b) 1.00 μF (c) 100
μF.
8. Find the current amplitude if the self-inductance of a
resistanceless inductor that is connected
across the source of the previous problem is (a) 0.0100 H, (b) 1.00 H,
(c) 100 H.
9. A sinusoidal ac voltage source in a circuit produces a
maximum voltage of 12.0 V and an rms current of 7.50 mA. Find (a) the voltage
and current amplitudes and (b) the rms voltage of this source.
10. A 65 Ω resistor, an 8.0 μF capacitor, and a 35 mH
inductor are connected in series in an ac circuit. Calculate the impedance for
a source frequency of (a) 300 Hz and (b) 30.0 Hz.
11. In an R–L–C series circuit, the rms voltage across the
resistor is 30.0 V, across the capacitor it is 90.0 V, and across the inductor
it is 50.0 V. What is the rms voltage of the source?
12. A 1500 Ω resistor is connected in series with a 350 mH
inductor and an ac power supply. At what frequency will this combination have
twice the impedance that it has at 120 Hz?
13. (a) Compute the impedance of a series R–L–C circuit at
angular frequencies of 1000, 750, and
500 rad/s. Take R = 200 Ω, L = 0.900 H, and C = 2.00 μF. (b) Describe
how the current amplitude varies as the angular frequency of the source is
slowly reduced from 1000 rad/s to 500 rad/s (c) What is the phase angle of the
source voltage with respect to the current when ω = 1000 rad/s? (d) Construct a
phasor diagram when ω = 1000 rad/s.
14. A 200 Ω resistor is in series with a 0.100 H inductor
and a 0.500 μF capacitor. Compute the impedance of the circuit and draw the
phasor diagram (a) at a frequency of 500 Hz, (b) at a frequency of 1000 Hz. In
each case, compute the phase angle of the source voltage with respect to the
current and state whether the source voltage lags or leads the current.
15. The power of a certain CD player operating at 120 Vrms
is 20.0 W. Assuming that the CD player behaves like a pure resistance, find (a) the
maximum instantaneous power, (b) the rms current, and (c) the resistance of
this player.
16. A series R–L–C circuit is connected to a 120 Hz ac
source that has Vrms = 80.0 V. The circuit has a resistance of 75.0 Ω and an
impedance of 105 Ω at this frequency. What average power is delivered to the
circuit by the source?
17. The circuit in Problem 13 carries an rms current of
0.250 A with a frequency of 100 Hz. (a) What is the average rate at which
electrical energy is converted to heat in the resistor? (b) What average power
is delivered by the source? (c) What is the average rate at which electrical
energy is dissipated (converted to other forms) in the capacitor? in the
inductor?
18. A series ac circuit contains a 250 Ω resistor, a 15 mH
inductor, a 3.5 μH capacitor, and an ac power source of voltage amplitude 45 V
operating at an angular frequency of 360 rad/s (a) What is the power factor of
this circuit? (b) Find the average power delivered to the entire circuit. (c)
What is the average power delivered to the resistor, to the capacitor, and to
the inductor?
19. An ac series R–L–C circuit contains a 120 Ω resistor, a
2.0 μF capacitor, and a 5.0 mH inductor. Find (a) the resonance angular
frequency and (b) the length of time that each cycle lasts at the resonance
angular frequency.
20. (a) At what angular frequency will a 5.00 μF capacitor
have the same reactance as a 10.0 mH inductor? (b) If the capacitor and
inductor in part (a) are connected in an L–C circuit, what will be the
resonance angular frequency of that circuit?
21. In an R-L-C series circuit, R = 150 Ω, L = 0.750 H, and
C = 0.0180 μF. The source has voltage amplitude and a frequency equal to the
resonance frequency of the circuit. (a) What is the power factor? (b) What is
the average power delivered by the source? (c) The capacitor is replaced by one
with C = 0.0360 μF and the source frequency is adjusted to the new resonance
value. Then what is the average power delivered by the source?
22. You need to make a series ac circuit having a resonance
angular frequency of using a resistor, a capacitor, and an inductor. (a) What
should be the inductance of the inductor, and (b) what is the impedance of this
circuit when you use it with an ac voltage source having an angular frequency
of 1525 Hz?
23. A series circuit consists of an ac source of variable
frequency, a 115 Ω resistor, a 1.25 μF capacitor, and a 4.50 mH inductor. Find
the impedance of this circuit when the angular frequency of the ac source is
adjusted to (a) the resonance angular frequency, (b) twice the resonance angular
frequency, and (c) half the resonance angular frequency.
24. In a series R–L–C circuit, R = 400 Ω, L = 0.350 H, and C
= 0.0120 μF (a) What is the resonance angular frequency of the circuit? (b) The
capacitor can withstand a peak voltage of 550 V. If the voltage source operates
at the resonance frequency, what maximum voltage amplitude can it have if the
maximum capacitor voltage is not exceeded?
25. In a series R–L–C circuit, L = 0.200 H, C = 80.0 μF, and
the voltage amplitude of the source is 240 V. (a) What is the resonance angular
frequency of the circuit? (b) When the source operates at the resonance angular
frequency, the current amplitude in the circuit is 0.600 A. What is the
resistance R of the resistor? (c) At the resonance frequency, what are the peak
voltages across the inductor, the capacitor, and the resistor?
26. In an R–L–C series circuit, R = 300 Ω, L = 0.400 H, and
C = 6.00 × 10-8 μF. When the ac source operates at the resonance frequency of
the circuit, the current amplitude is 0.500 A. (a) What is the voltage
amplitude of the source? (b) What is the amplitude of the voltage across the
resistor, across the inductor, and across the capacitor? (c) What is the
average power supplied by the source?
27. A 125 Ω resistor, an 8.50 μF capacitor, and an 1.2 mH
inductor are all connected in parallel across an ac voltage source of variable
frequency. (a) At what angular frequency will the impedance have its maximum
value, and (b) what is that value?
28. For the circuit in Figure 22.23, R = 300 Ω, L = 0.500 H
and C = 0.600 μF. The voltage amplitude of the source is 120 V. (a) What is the
resonance frequency of the circuit? (b) Sketch the phasor diagram at the
resonance frequency. (c) At the resonance frequency, what is the current
amplitude through the source? (d) At the resonance frequency, what is the
current amplitude through the resistor? Through the inductor? Through the
branch containing the capacitor?
29. For the circuit in Figure 22.23, R = 200 Ω, L = 0.800,
and C = 5.00 μF. When the source is operated at the resonance frequency, the
current amplitude in the inductor is 0.400 A. Determine the current amplitude
(a) in the branch containing the capacitor and (b) through the resistor.
30. (a) Use the phasor diagram for a parallel R–L–C circuit
(see Figure 22.21) to show that the current amplitude I for the current i
through the source is given by I = √(IR2 + (IC – IL)2) (b) Show that the result
of part (a) can be written as I = V/Z with 1/Z = I = √(1/R2 + (ωC – 1/ωL)2).
31. A coil has a resistance of 48.0 Ω. At a frequency of
80.0 Hz, the voltage across the coil leads the current in it by 52.3°.
Determine the inductance of the coil.
32. A large electromagnetic coil is connected to a 120 Hz ac
source. The coil has resistance 400 Ω and at this source frequency the coil has
inductive reactance 250 Ω. (a) What is the inductance of the coil? (b) What
must the rms voltage of the source be if the coil is to consume an average
electrical power of 800 W?
33. A parallel-plate capacitor having square plates 4.50 cm
on each side and 8.00 mm apart is placed in series with an ac source of angular
frequency 650 rad/s and voltage amplitude 22.5 V, a 75.0 Ω resistor, and an
ideal solenoid that is 9.00 cm long, has a circular cross section 0.500 cm in
diameter, and carries 125 coils per centimeter. What is the resonance angular
frequency of this circuit?
34. At a frequency ω1 the reactance of a certain capacitor
equals that of a certain inductor. (a) If the frequency is changed to ω2 = 2ω1,
what is the ratio of the reactance of the inductor to that of the capacitor?
Which reactance is larger? (b) If the frequency is changed to ω3 = ω1/3 what is
the ratio of the reactance of the inductor to that of the capacitor? Which
reactance is larger?
35. Five voltmeters, calibrated to read rms values, are
connected as shown in Figure 22.22. Let R = 200 Ω, L = 0.400 H, and C = 6.00
μF. The source voltage amplitude is V = 30.0 V. What is the reading of each
voltmeter if (a) ω = 200 rad/s (b) ω = 1000 rad/s?
36. Consider the circuit sketched in Figure 22.22. The
source has a voltage amplitude of 240 V, R = 150 Ω and the reactance of the
capacitor is 600 Ω. The voltage amplitude across the capacitor is 720 V. (a)
What is the current amplitude in the circuit? (b) What is the impedance? (c)
What two values can the reactance of the inductor have?
37. In a series R–L–C circuit, the components have the
following values: L = 20.0 mH, C = 140 nF, and R = 350 Ω. The generator has an
rms voltage of 120 V and a frequency of 1.25 kHz. Determine (a) the power
supplied by the generator; and (b) the power dissipated in the resistor.
38. (a) Show that for an R–L–C series circuit the power
factor is equal to R/Z. (b) Show that for any ac circuit, not just one
containing pure resistance only, the average power delivered by the voltage
source is given by Pav = I2rmsR.
39. In an R–L–C series circuit the magnitude of the phase angle is 54.0°, with the source voltage
lagging the current. The reactance of the capacitor is 350 Ω and the resistor
resistance is 180 Ω. The average power delivered by the source is 140 W. Find
(a) the reactance of the inductor; (b) the rms current; (c) the rms voltage of
the source.
40. In a series R–L–C circuit, R = 300 Ω, XC = 300 Ω, and
XL= 500 Ω. The average power consumed in the resistor is 60.0 W. (a) What is
the power factor of the circuit? (b) What is the rms voltage of the source?
41. In a series R–L–C circuit, the phase angle is 40.0°,
with the source voltage leading the current. The reactance of the capacitor is
400 Ω and the resistance of the resistor is 200 Ω. The average power delivered
by the source is 150 W. Find (a) the reactance of the inductor, (b) the rms
current, (c) the rms voltage of the source.
42. A 100 Ω resistor, a 0.100 μF capacitor, and a 300.0 mH
inductor are connected in series to a voltage source with amplitude 240 V. (a)
What is the resonance angular frequency? (b) What is the maximum current in the
resistor at resonance? (c) What is the maximum voltage across the capacitor at
resonance? (d) What is the maximum voltage across the inductor at resonance?
(e) What is the maximum energy stored in the capacitor at resonance? in the
inductor?
43. Consider the same circuit as in the previous problem,
with the source operated at an angular frequency of 400 rad/s (a) What is the
maximum current in the resistor? (b) What is the maximum voltage across the
capacitor? (c) What is the maximum voltage across the inductor? (d) What is the
maximum energy stored in the capacitor? in the inductor?
44. What is the dc impedance of the electrode, assuming that
it behaves as an ideal capacitor?
A. 0 B. Infinite C. √2 × 104 Ω D. √2 × 106 Ω
45. If the electrode is oscillated between two points 20 μ
apart with a frequency of (5000/π) Hz, what is the impedance of the electrode?
A. 0
B. Infinite C. √2 × 104
Ω D. √2 × 106 Ω
46. The signal from the oscillating electrode is fed into an
amplifier, which reports the measured voltage as an rms value, Vrms. However,
the number of interest for analyzing currents driven by the cell is the
peak-to-peak voltage difference (Vpp), that is, the voltage difference between
the two extremes of the electrode’s excursion. What is the value of Vpp in
terms of Vrms?
A. Vrms/√2
B. Vrms/2√2 C.
√2Vrms D.
2√2Vrms
Chapter 23
Multiple-Choice Problems
1. Light having a certain frequency, wavelength, and speed
is traveling through empty space. If the frequency of this light were doubled,
then
A. its wavelength would remain the same, but its speed would
double.
B. its wavelength would remain the same, but its speed would
be halved.
C. its wavelength would be halved, but its speed would
double.
D. its wavelength would be halved, but its speed would
remain the same.
E. both its speed and its wavelength would be doubled.
2. Unpolarized light with an original intensity passes
through two ideal polarizers having their polarizing axes turned at 120° to
each other. After passing through both polarizers, the intensity of the light
is
A. (√3/2)Io B.
(1/2)Io C. (√3/4)Io D. (1/4)Io E. (1/8)Io
3. Light travels from water (with index of refraction 1.33)
into air (index of refraction 1.00). Which of the following statements about
this light is true?
A. The light has the same frequency in the air as it does in
the water.
B. The light travels faster in the air than in the water.
C. The light has the same wavelength in the air as it does
in the water.
D. The light has the same speed in the air as in the water.
E. The wavelength of the light in the air is greater than
the wavelength in the water.
4. If a sinusoidal electromagnetic wave with intensity has
an electric field of amplitude E, then a wave of the same wavelength will have
an electric field of amplitude
A. 4E B.
2√2E C. 2E D. √2E
5. A plane electromagnetic wave is traveling vertically
downward with its magnetic field pointing northward. Its electric field must be pointing
A. toward the south.
B. toward the east.
C. toward the west.
D. vertically upward.
E. vertically downward.
6. Suppose that a reflective solar sail (see Example 23.5)
is deployed not perpendicular to the sun’s rays but at some other angle. In
what direction will the sail accelerate?
A. In the direction the sun’s rays are moving.
B. Perpendicular to the surface of the sail.
C. At an angle somewhere between that of the sun’s rays and
the perpendicular to the surface of the sail.
D. The sail will not accelerate unless it is perpendicular
to the sun’s rays.
7. The index of refraction, n, has which of the following
range of values?
A. n ≥ 1 B. 0
≤ n ≤ 1 C. n ≥ 0
8. A ray of light going from one material into another
follows the path shown in Figure 23.47.
What can you conclude about the relative indexes of refraction of these
two materials?
A. na ≥ nb B.
na > nb C. na < nb D. na ≤ nb
9. Which of the following statements about radio waves,
infrared radiation, and x rays are correct?
A. They all have the same wavelength in vacuum.
B. They all have the same frequency in vacuum.
C. They all have exactly the same speed as visible light in
vacuum.
D. The short-wavelength x rays travel faster through vacuum
than the long-wavelength radio waves.
10. Two lasers each produce 2 mW beams. The beam of laser B
is wider, having twice the cross-sectional area as the beam of laser A. Which
of the following statements about these two laser beams are correct?
A. Both of the beams have the same average power.
B. Beam A has twice the intensity of beam B.
C. Beam B has twice the intensity of beam A.
D. Both beams have the same intensity.
11. A ray of light follows the path shown in Figure 23.48 as
it reaches the boundary between two transparent materials. What can you
conclude about the relative indexes of refraction of these two materials?
A. n1 ≥ n2 B.
n1 > n2 C. n1 < n2 D. n1 ≤ n2
12. A light beam has a wavelength of 300 nm in a material of
refractive index 1.5. In a material of refractive index 3.0, its wavelength
will be
A. 450 nm B. 300
nm C. 200 nm D. 150 nm E. 100 nm
13. A light beam
has a frequency
of 300 MHz
in a material
of refractive index 1.5. In a material of refractive index 3.0, its
frequency will be
A. 450 MHz.
B. 300 MHz. C. 200
MHz. D. 150 MHz. E. 100 MHz.
14. You are sunbathing in the late afternoon when the sun is
relatively low in the western sky. You are lying flat on your back, looking
straight up through Polaroid sunglasses. To minimize the amount of light
reaching your eyes, you should lie with your feet pointing in what direction?
A. north. B.
east. C. south. D. west. E. some other direction.
15. A beam of light takes time t to travel a distance L in a
certain liquid. If we now add water to the liquid to reduce its index of
refraction by half, the time for the beam to travel the same distance will be
A. 2t. B.
√2t C. t/√2 D. t/2
Problems
1. When a solar flare erupts on the surface of the sun, how
many minutes after it occurs does its light show up in an astronomer’s
telescope on earth?
2. TV ghosting. In a TV picture, faint, slightly offset
ghost images are formed when the signal from the transmitter travels to the
receiver both directly and indirectly after reflection from a building or some
other large metallic mass. In a 25 inch set, the ghost is about 1.0 cm to the
right of the principal image if the reflected signal arrives 0.60 μs after the
principal signal. In this case, what is the difference in the distance traveled
by the two signals?
3. a) How much time does it take light to travel from the
moon to the earth, a distance of 384,000 km? (b) Light from the star Sirius
takes 8.61 years to reach the earth. What is the distance to Sirius in
kilometers?
4. A geostationary communications satellite orbits the earth
directly above the equator at an altitude of 35,800 km. Calculate the time it
takes for a signal to travel from a point on the equator to the satellite and
back to the ground at another point on the equator exactly halfway around the
earth.
5. Consider electromagnetic waves propagating in air. (a)
Determine the frequency of a wave with a wavelength of (i) 5.0 km, (ii) 5.0 μm
(iii) 5.0 nm. (b) What is the wavelength (in meters and nanometers) of (i) gamma rays of frequency
6.50 × 1021 Hz (ii) an AM station radio wave of
frequency 590 kHz?
6. Most people perceive light having a wavelength between
630 nm and 700 nm as red and light with
a wavelength between 400 nm and 440 nm as violet. Calculate the
approximate frequency ranges for (a) violet light and (b) red light.
7. The electric field of a sinusoidal electromagnetic wave
obeys the equation E = -(375 V/m)sin[(5.97 × 1015 rad/s)t + (1.99 × 107
rad/m)x] (a) What are the amplitudes of the electric and magnetic fields
of this wave? (b) What are the
frequency, wavelength, and period of the wave? Is this light visible to humans?
(c) What is the speed of the wave?
8. A sinusoidal electromagnetic wave having a magnetic field
of amplitude 1.25 μT and a wavelength of 432 nm is traveling in the +x
direction through empty space. (a) What is the frequency of this wave? (b) What
is the amplitude of the associated electric field? (c) Write the equations for
the electric and magnetic fields as functions of x and t in the form of
Equations (23.3).
9. Visible light. The wavelength of visible light ranges
from 400 nm to 700 nm. Find the corresponding ranges of this light’s (a)
frequency, (b) angular frequency, (c) wave number.
10. Ultraviolet radiation. There are two categories of
ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 nm to
400 nm. It is not so harmful to the skin and is necessary for the production of
vitamin D. UVB, with a wavelength between 280 nm and 320 nm, is much more
dangerous, because it causes skin cancer. (a) Find the frequency ranges of UVA
and UVB. (b) What are the ranges of the wave numbers for UVA and UVB?
11. Medical x rays. Medical x rays are taken with
electromagnetic waves having a wavelength around 0.10 nm. What are the
frequency, period, and wave number of such waves?
12. Radio station WCCO in Minneapolis broadcasts at a
frequency of 830 kHz. At a point some distance from the transmitter, the
magnetic-field amplitude of the electromagnetic wave from WCCO is 4.82 × 10-11
T. Calculate (a) the wavelength, (b) the
wave number, (c) the angular frequency, and (d) the electric-field amplitude.
13. A sinusoidal electromagnetic wave of frequency travels
in vacuum in the +x-direction. The magnetic field is parallel to the y-axis and
has amplitude (a) Find the magnitude and direction of the electric field. (b)
Write the wave functions for the electric and magnetic fields in the form of
Equations (23.3).
14. Consider each of the electric- and magnetic-field
orientations given next. In each case, what is the direction of propagation of
the wave? (a) E in the +x direction, B in the +y direction. (b) E in the –y
direction, B in the +x direction. (c) E in the +z direction, B in the –x
direction. (d) E in the +y direction, B in the -y direction.
15. An electromagnetic wave has a magnetic field given by B
= (8.25 × 10-9 T) sin[(ωt + 1.38 × 104 rad/s)x], with the magnetic field in the
+y direction. (a) In which direction is the wave traveling? (b) What is the
frequency f of the wave? (c) Write the wave function for the electric field.
16. Laboratory lasers. He–Ne lasers are often used in
physics demonstrations. They produce light of wavelength 633 nm and a power of
0.500 mW spread over a cylindrical beam 1.00 mm in diameter (although these
quantities can vary). (a) What is the intensity of this laser beam? (b) What
are the maximum values of the electric and magnetic fields? (c) What is the
average energy density in the laser beam?
17. Fields from a lightbulb. We can reasonably model a 75 W
incandescent lightbulb as a sphere 6.0 cm in diameter. Typically, only about 5%
of the energy goes to visible light; the rest goes largely to nonvisible
infrared radiation. (a) What is the visible light intensity (in W/m2) at the
surface of the bulb? (b) What are the amplitudes of the electric and magnetic
fields at this surface, for a sinusoidal wave with this intensity?
18. Threshold of vision. Under controlled darkened
conditions in the laboratory, a light receptor cell on the retina of a person’s
eye can detect a single photon (more on photons in Chapter 28) of light of
wavelength 505 nm and having an energy of 3.94 × 10-19 J. We shall assume that
this energy is absorbed by a single cell during one period of the wave. Cells
of this kind are called rods and have a diameter of approximately 0.0020 mm.
What is the intensity (in W/m2) delivered to a rod?
19. High-energy cancer treatment. Scientists are working on
a new
technique to kill
cancer cells by
zapping them with ultrahigh-energy (in the range of pulses of light that last for an extremely
short time (a few nanoseconds). These short pulses scramble the interior of a
cell without causing it to explode, as long pulses would do. We can model a
typical such cell as a disk in diameter, with the pulse lasting for 4.0 ns with
an average power of 2.0 × 1012 W. We shall assume that the energy is spread
uniformly over the faces of 100 cells for each pulse. (a) How much energy is
given to the cell during this pulse? (b) What is the intensity (in W/m2)
delivered to the cell? (c) What are the maximum values of the electric and
magnetic fields in the pulse?
20. At the floor of a room, the intensity of light from
bright overhead lights is 8.00 W/m2. Find the radiation pressure on a totally
absorbing section of the floor.
21. The intensity at a certain distance from a bright light
source is 6.00W/m2. Find the radiation pressure (in pascals and in atmospheres)
on (a) a totally absorbing surface and (b) a totally reflecting surface.
22. A sinusoidal electromagnetic wave from a radio station
passes perpendicularly through an open window that has area 0.500 m2. At the
window, the electric field of the wave has rms value 0.0200 V/m. How much
energy does this wave carry through the window during a 30.0 s commercial?
23. Two sources of sinusoidal electromagnetic waves have
average powers of 75 W and 150 W and emit uniformly in all directions. At the
same distance from each source, what is the ratio of the maximum electric field
for the 150 W source to that of the 75 W source?
24. Radiation falling on a perfectly reflecting surface
produces an average pressure p. If radiation of the same intensity falls on a
perfectly absorbing surface and is spread over twice the area, what is the
pressure at that surface in terms of p?
25. A sinusoidal electromagnetic wave emitted by a cellular
phone has a wavelength of 35.4 cm and an electric field amplitude of 5.40 ×
10-2 V/m at a distance of 250 m from the antenna. Calculate: (a) the frequency
of the wave; (b) the magnetic-field amplitude; (c) the intensity of the wave.
26. Two plane mirrors intersect at right angles. A laser
beam strikes the first of them at a point 11.5 cm from their point of
intersection, as shown in Figure 23.49. For what angle of incidence at the
first mirror will this ray strike the midpoint of the second mirror (which is
28.0 cm long) after reflecting from the first mirror?
27. Three plane mirrors intersect at right angles. A beam of
laser light strikes the first of them at an angle θ with respect to the normal.
(a) Show that when this ray is reflected off of the other two mirrors and
crosses the original ray, the angle α between these two rays will be α = 180° -
2θ (b) For what angle θ will the two
rays be perpendicular when they cross?
28. Two plane mirrors A and B intersect at a 45° angle.
Three rays of light leave point P (see Figure 23.51) and strike one of the
mirrors. What is the subsequent path of each of the following rays until they
no longer strike either of the mirrors? (a) Ray 1, which strikes A at 45° with
respect to the normal. (b) Ray 2, which
strikes B traveling perpendicular to mirror A. (c) Ray 3, which strikes B
perpendicular to its surface.
29. Prove that when a ray of light travels at any angle into
the corner formed by two mirrors placed at right angles to each other, the
reflected ray emerges parallel to the original ray (see Figure 23.52).
30. A light beam travels at 1.94 × 108 m/s in quartz. The
wavelength of the light in quartz is 355 nm. (a) What is the index of
refraction of quartz at this wavelength? (b)
If this same light travels through air, what is its wavelength there?
31. Using a fast-pulsed laser and electronic timing
circuitry, you find that light travels 2.50 m within a plastic rod in 1.5 ns.
What is the refractive index of the plastic?
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