Download 2012_spring online homework 12 solution

Online Homework 12 Solution
Electric and Magnetic Field Vectors Conceptual Question
Part A The electric and magnetic field vectors at a specific point in space
and time are illustrated. Based on this information,
in what direction does the electromagnetic wave
propagate?
Sol:)
由右手定則可判斷電磁波傳播方向為 + ẑ
Part B The electric and magnetic field vectors at a specific point in space and
r
r
time are illustrated. ( E and B are in the xy
plane. Both vectors make 45° angles with the +y
axis.) Based on this information, in what
direction does the electromagnetic wave
propagate?
Sol:)
由右手定則可判斷電磁波傳播方向為 − ẑ
Part C The magnetic field vector and the direction of propagation of an
electromagnetic wave are illustrated. Based on this
information, in what direction does the electric
field vector point?
Sol:)
已知磁場與電磁波傳播方向，由右手定則可判斷
電場方向為 + ŷ
Part D The electric field vector and the direction of propagation of an
r
electromagnetic wave are illustrated. ( E is in xz
plane and makes a 45° angle with the +x axis.) Based
on this information, in what direction does the
magnetic field vector point?
Sol:)
已知電場與電磁波傳播方向，由右手定則可判斷
磁場方向為 at a -45° angle in the xy plane
1
Traveling Electromagnetic Wave
Light, radiant heat (infrared radiation), X rays, and radio waves are all
examples of traveling electromagnetic waves. Electromagnetic waves
comprise combinations of electric and magnetic fields that are mutually
compatible in the sense that the changes in one generate the other.
The simplest form of a traveling electromagnetic wave is a plane wave. For
a wave traveling in the x direction whose electric field is in the y direction,
the electric and magnetic fields are given by
r
E = E0 sin (kx − ωt ) yˆ ,
r
B = B0 sin (kx − ωt )zˆ .
This wave is linearly polarized in the y direction.
Part A
In these formulas, it is useful to understand which variables are
parameters that specify the nature of the wave. The variables E 0 and B0 are
the __________ of the electric and magnetic fields. Choose the best answer to
fill in the blank.
Sol:)
amplitudes
Part B The variable ω is called the __________ of the wave. Choose the best
answer to fill in the blank.
Sol:)
angular frequency
Part C The variable k is called the __________ of the wave. Choose the best
answer to fill in the blank.
Sol:)
wave number
2
Part D
What is the mathematical expression for the electric field at the point
x = 0, y = 0, z at time t?
Sol:)
r
E = E 0 sin (− ωt ) yˆ
Part E For a given wave, what are the physical variables to which the wave
responds?
Sol:)
x and t
Part F What is the wavelength λ of the wave described in the problem
introduction? Express the wavelength in terms of the other given variables and
constants like π.
Sol:)
sin (kx ) = sin (kx + kλ )
sin (θ ) = sin (θ + 2π )
2π
⇒λ =
k
Part G What is the period T of the wave described in the problem
introduction? Express the period of this wave in terms of ω and any constants.
Sol:)
2π
T=
ω
Part H What is the velocity v of the wave described in the problem
introduction? Express the velocity in terms of quantities given in the
introduction (such as ω and k) and any useful constants.
Sol:)
2π
ω
v= = k =
T 2π
k
λ
ω
3
Poynting Flux and Power Dissipation in a Resistor
Consider a cylindrical resistor of radius r0, length l, and resistance R with
a steady current i flowing along the axis of the cylinder.
Part A Which of the following is the most accurate qualitative description of
r
the magnetic field vector B inside the cylindrical resistor?
Sol:).
The magnetic field vector is everywhere tangential to circles centered on the
axis of the cylinder.
r
Part B
Find the magnitude of the magnetic field B(r ) inside the cylindrical
resistor, where r is the distance from the axis of the cylinder, in terms of i, r, r0,
l, and other given variables. You will also need π and µ 0 . Ignore fringing
effects at the ends of the cylinder.
Sol:).
r r
B
∫ ⋅ dl = µ 0 iencl
B ⋅ (2πr ) = µ 0
iπr 2
πr02
r
ir
B(r ) = µ 0
2πr02
r
Part C What can you say about the electric field vector E inside the
resistor?
Sol:).
The electric field vector points along the axis of the resistor in the direction of
the current.
4
r
Part D
What is the magnitude of the electric field vector E ?
Sol:).
V = iR
r V iR
E = =
l
l
r
Part E
Sol:).
In what direction does the Poynting vector S point?
r 1 r r
S=
E×B
µ0
In cylindrical coordinate rˆ × θˆ = kˆ
kˆ × θˆ = −rˆ
So the answer is ” − r̂ ”.
r
Part F
Calculate S , the magnitude of the Poynting vector at the surface of
the resistor (not at the circular ends of the cylinder). To answer this you need to
take r = r0 .
Sol:).
r
r
From part B and part D we know E and B (r )
r 1 r r i2R
S=
E×B =
µ0
2πlr0
5
Triangle Electromagnetic Wave
In this problem we will investigate a triangle wave traveling in the x
direction whose electric field is in the y direction. This wave is linearly
polarized along the y axis; in other words, the electric field is always
directed along the y axis. Its electric and magnetic fields are given by the
following expressions:
E y ( x, t ) = E 0 ( x − vt ) / a
and BZ (x, t ) = B0 (x − vt ) / a ,
where E0 , B0 , and a are constants. The
constant a, which has dimensions of length,
is introduced so that the constants E0 and
B0 have dimensions of electric and
magnetic field respectively. This wave is
pictured in the figure at time t = 0 . Note that we have only drawn the field
vectors along the x axis. In fact, this idealized wave fills all space, but the
field vectors only vary in the x direction.
We expect this wave to satisfy Maxwell's equations. For it to do so, we will
find that the following must be true:
1. The amplitude of the electric field must be directly proportional to
the amplitude of the magnetic field.
2. The wave must travel at a particular velocity (namely, the speed of
light).
r
Part A What is the propagation velocity v of the electromagnetic wave
whose electric and magnetic fields are given by the expressions in the
r
introduction? Express v in terms of v and the unit vectors x̂ , ŷ , and ẑ . The
answer will not involve c; we have not yet shown that this wave travels at the
speed of light.
Sol:)
r
v = vxˆ
6
In the next few parts, we will use Faraday's law of induction to find a
relationship between E0 and B0 .
Faraday's law relates the line integral of the electric field around a closed
loop to the rate of change in magnetic flux through this loop:
r
r
r
∫ E (r , t ) ⋅ dl
C
=−
r
d r r
B (r , t ) ⋅ dA .
∫
dt S
Part B To use Faraday's law for this problem, you will need to constuct a
suitable loop, around which you will integrate the electric field. In which plane
should the loop lie to get a nonzero electric field line integral and a nonzero
magnetic flux?
Sol:)
The xy plane
Part C Consider the loop C1 shown in the figure. It is a square loop with
sides of length L, with one corner at the origin and the opposite corner at the
coordinates x = L , y = L . Recall that
r r
E (r , t ) = [E0 ( x − vt ) / a ]yˆ . What is the value of the
line integral of the electric field around loop C1
at arbitrary time t? Express the line integral in
terms of E0 , L, a, v, and/or t.
Sol:)
Since Ex=0, the line integral alone segment 1 and 3 are zero.
r
r
r
Consider segment 2, dl = dyyˆ , E ⋅ dl = E y dy
E 0 (L − vt )
E (L − vt ) L
E (L − vt )L
dy = 0
dy = 0
∫
0
0
0
a
a
a
r
r r
Consider segment 3, dl = −dyyˆ , E ⋅ dl = − E y dy
∫
L
E y ( x = L, t )dy = ∫
∫
L
− E y ( x = 0, t )dy = − ∫
0
0+
L
L
0
E 0 (− vt )
E vt L
E vtL
dy = 0 ∫ dy = 0
a
a 0
a
E 0 (L − vt )L
E vtL E 0 L2
+0+ 0
=
a
a
a
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Part D
r r
Recall that B(r , t ) = [B0 (x − vt ) / a ]zˆ . Find the value of the magnetic
flux through the surface S1 in the xy plane that is bounded by the loop C1 , at
arbitrary time t. Express the magnetic flux in terms of B0 , L, a, v, and/or t.
Sol:)
r
r
r r
dA = dxdyzˆ , B (r , t ) ⋅ dA = B z ( x, t )dxdy
r
r r
∫ B(r , t ) ⋅ dA = ∫ B (x, t )dxdy
z
S1
S1
L
x − vt
dx ∫ dy
0
0
a
 L2

L

 − vtL 
B0 L2  − vt 
2
L=
2

= B0 
a
a
= B0 ∫
L
Part E
Now use Faraday's law to establish a relationship between E0 and
B0 .
Sol:)
By using Faraday’s law, we know that
r
r
r
∫ E (r , t )⋅ dl
C1
=−
d
dt
r
r
r
∫ B(r , t )⋅ dA
S1
r E 0 L2
r r
等號左邊可由 part C 得知 ∫ E (r , t ) ⋅ dl =
C1
a
r
r r
d
d
等號右邊由 part D 可得 − ∫ B(r , t ) ⋅ dA = −
dt S1
dt
L

B0 L2  − vt 
2
2
 = B0 L v
a
a
B0 L2 v E0 L2
=
⇒ E0 = B0 v
a
a
If the electric and magnetic fields given in the introduction are to be
self-consistent, they must obey all of Maxwell's equations, including the
Ampère-Maxwell law. In these last few parts (again, most of which are
hidden) we will use the Ampère-Maxwell law to show that self-consistency
requires the electromagnetic wave described in the introduction to
propagate at the speed of light.
8
The Ampère-Maxwell law relates the line integral of the magnetic field
around a closed loop to the rate of change in electric flux through this
loop:
r
r
r
∫ B(r , t ) ⋅ dl
C
= µ 0 I + µ 0ε 0
r
d r r
E (r , t ) ⋅ dA .
∫
dt S
In this problem, the current I is zero. (For I to be nonzero, we would need
charged particles moving around. In this problem, there are no charged
particles present. We assume that the electromagnetic wave is propagating
through a vacuum.)
Part F To use the Ampère-Maxwell law you will once again need to
construct a suitable loop, but this time you will integrate the magnetic field
around the loop. In which plane should the loop lie to get a nonzero magnetic
field line integral and hence nonzero electric flux?
Sol:)
The zx plane
Part G
Use the Ampère-Maxwell law to find a new relationship between E 0
and B0 . Express E 0 in terms of B0 , µ 0 , ε 0 ,
and other quantities given
in the introduction.
Sol:)
r
r
r
∫ B(r , t ) ⋅ dl
C2
= µ 0ε 0
r
r r
d
E (r , t ) ⋅ dA
∫
dt S 2
如同 part C
r r
r
可得 ∫ B(r , t ) ⋅ dl =
C2
B0 L2
a
如同 part D
可得 ∫
S2
r
r r
E (r , t ) ⋅ dA =
L

− E0 L2  − vt 
2

a

L

− E0 L2  − vt  
2

r
r r
d
d
2
  = µ 0ε 0 E0 L v
µ 0ε 0 ∫ E (r , t ) ⋅ dA = µ 0ε 0 
dt S2
dt 
a
a





2
2
B0 L
B0
µε E Lv
= 0 0 0
⇒ E0 =
a
a
µ 0ε 0 v
9
Part H Finally we are ready to show that the electric and magnetic fields
given in the introduction describe an electromagnetic wave propagating at the
speed of light. If the electric and magnetic fields are to be self-consistent, they
must obey all of Maxwell's equations. Using one of Maxwell's equations,
Faraday's law, we found a certain relationship between E 0 and B0 . You
derived this in Part E. Using another of Maxwell's equations, the
Ampère-Maxwell law, we found what appears to be a different relationship
between E 0 and B0 . You derived this in Part I. If the results of Parts E and I
are to agree, what does this imply that the speed of propagation
Express v in terms of only µ 0 and ε 0 .
Sol:)
Part G E0 =
B0
µ 0ε 0 v
Part E E 0 = B0 v
v=
1
µ 0ε 0
10
must be?
Electromagnetic Waves Ranking Task
Part A Rank these electromagnetic waves on the basis of their speed (in
vacuum).
Rank from fastest to slowest. To rank items as equivalent, overlap them.
Sol:)
The relationship among wave speed, frequency, and wavelength is c = fλ
v AM = v FM = vinf rared = v yellow = v green = v x − ray
Part B Rank these electromagnetic waves on the basis of their wavelength.
Rank from longest to shortest. To rank
items as equivalent, overlap them.
Sol:)
λ AM > λ FM > λinf rared > λ yellow > λ green > λ x −ray
Part C Rank these electromagnetic waves on the basis of their frequency.
Rank from largest to smallest. To rank items as equivalent, overlap them.
Sol:)
f AM < f FM < f inf rared < f yellow < f green < f x − ray
11