72. (30.2) Interaction between two parallel current
carrying wires.
Two parallel wires carrying currents exert forces on
each other. Each current produces a magnetic field in
which the other current is placed.
B1
F21
I1
I2
l
a
1
2
The magnitude of the magnetic force exerted on segment l
of a wire by the other wire (infinite) is
µ l
F21 = 0 ⋅ I1I2
2πa
Parallel "currents" attract and antiparallel repel.
Proof.
Wire 1 produces a magnetic field which has the
following value at the other wire:
r
r
µ
0
B1 =
⋅ I × a$
2 πa
Therefore the magnetic force exerted on the second wire is:
r
r
r
µ l r r
F21 = I2l × B1 = 0 ⋅ I2 × (I1 × aˆ ) =
2πa
r r r
µ l r
µl r r
= 0 ⋅ ((I2 ⋅ aˆ )I1 − (I1 ⋅ I 2 )⋅ aˆ ) = − 0 (I1 ⋅ I 2 )⋅ aˆ
2πa
2 πa
101
73. (30.4, 30.5) The magnetic field of a solenoid - a long
tightly wound helical coil.
I
N
S
L
I
From the symmetry of the currents we can
approximate the outside field to be zero.
Bout = 0
and the magnetic field inside is uniform (approximately).
Its direction is parallel to the axis, and the magnitude of the
fields is:
Bin = µ 0nI
where n is the number of turns per unit length of the
solenoid, and I is the current in the solenoid.
102
Proof. Imagine an amperian loop along a line of the
magnetic field. The contour integral of the magnetic field
along this loop is:
∫ B ⋅ds = ∫ B in ds + ∫ B out ds =Bin L
in
out
The current through the surface enclosed by this loop is
IA = N⋅I
where N is the number of turns in the solenoid.
From Amper-Maxwell's equation we obtain:
Bin L = µ 0NI
therefore
Bin = µ 0
N
I = µ 0nI
L
103
74. (30.9) Magnetic properties of matter.
When
a substance is placed in a (external) magnetic
r
field (B 0 ), its molecules acquire a magnetic moment related
to the rexternal field. This creates an additional magnetic
field (B m ) (internal). The magnetic staterof a substance is
described by the magnetization vector, M, defined by the
(internal) magnetic field of the substance.
r
r
Bm = µ0M
The net magnetic field is different when the substance
is present than when it is not. Except for ferromagnetic, the
magnetic susceptibility χ of the substance relates the two
fields.
r r
r
r
B = B 0 + B m = (1 + χ )B 0
Depending on the reaction of a material to an external
magnetic field, the substance can be a
a) ferromagnetic - a substance that produces a strong
magnetic field aligned with the external field (magnetic
susceptibility χ>>0), which is sustained even after the
external field is removed - ferromagnetic substances exhibit
hysteresis of magnetization.
b) paramagnetic - a substance that produces a magnetic
field aligned with the external field (χ>0)
c) diamagnetic - a substance that produces a magnetic
field opposed to the external field (χ<0).
104
75. (31.intro, 31.1, 31.2) Magnetic Induction
a) An induced electromotive
force appears in a conducting
loop when the magnetic flux
through the surface enclosed
by the loop is changing. This
effect is called electromagnetic
induction.
ε
N
ΦB
b) The electromotive force induced in a coil with N loops
has value
dΦ
ε = −N B
dt
where ΦB is the magnetic flux through the cross-sectional
surface of the coil.
76. (31.3) Lenz rule
The polarity of an induced electromotive force is such
that it tends to produce an induced current creating a
magnetic field that opposes the change in flux causing the
emf.
105
77. (32.4, 33.9) Mutual induction
dΦ
Φ
dI1
+
ε2
I2
If two coils are arranged in such a way that the magnetic
fluxes though cross-sectional areas of each coil are related,
a changing current in one coil induces an emf in the second
coil. The effect is called mutual induction.
The emf induced in the second coil is proportional to
the rate of change in the current in the first coil. The
proportionality coefficient M is called the mutual inductance.
dI
ε2 = −M 1
dt
(The SI unit of mutual inductance is H=Vs/A)
106
78. (32.1) Self-induction
A change in current dI in a coil induces an
electromotive force εL the same coil.
The emf is proportional to the rate of change in
current.
dΦ m
d
d N 
µ 0 N 2 A dI
εL = − N ⋅
= − N ⋅ (A ⋅ B) = − NA ⋅  µ0 ⋅ I  = −
⋅
dt
dt
dt  l 
l
dt
A coil is an inductor with an inductance
µ 0N 2A
I
∆I
L=
l
+
εL
+
−
−
Voltage across an inductor is due
to the induced emf only.
ε
79. Application of electric induction (common examples)
a)
b)
c)
d)
e)
f)
g)
h)
a microphone
the playback head of a tape deck or VCR
an electric guitar pickup
an AC generator
an inductor
a transformer
the back emf of an electric motor
ground fault interupter
107
80. (32.3) Magnetic energy
a) The energy stored in an inductor depends on the
inductance of the inductor and the current through the
inductor:
1
U m = LI2
2
Proof.
The rate at which energy is delivered to the solenoid
depends on the voltage across the solenoid and the current
through the solenoid:
dI
Pel = Iε = I ⋅ L
dt
Therefore the energy at time t stored in the magnetic field
of the solenoid is
I (t )
t
t
dI
1
U m (t ) = ∫ Pel dt ' = ∫ LI(t ') dt ' = ∫ LIdI = LI 2
dt '
2
0
0
0
b) Magnetic energy density is defined as the energy
stored in a magnetic field per unit volume of the field
dU m
um =
dV
c) If, at a certain location, the magnitude of the magnetic
field vector is B, the magnetic energy density at this
location is:
1
um =
⋅ B2
2µ 0
108
Proof.
For a solenoid
B = µ0
N
I
L
Therefore
2
um
2
U m 12 LI
1 1 µ0 N 2A  B ⋅ l 
1
 =
=
=
= ⋅
⋅
⋅ 
⋅ B2
A ⋅l A ⋅l 2 A ⋅l
l
2µ 0
 µ0 N 
81. (30.9) Magnetic moment of an electron.
In classical physics, the magnetic moment of a particle
is always associated with the angular momentum (in a
circular motion) of the particle. There is evidence that an
electron has a magnetic moment which is not associated
with an orbital motion (Stern-Gerlach experiment). The
corresponding "angular momentum", called the spin, has a
magnitude (very small) of
S=
3
h = 9.2 × 10−35 J ⋅ s
2
The electron magnetic moment vector associated with the
spin vector has the value:
r
er
µB = − s
m
Spin is an intrinsic property of an electron (like mass
and charge) and should not be confused with a spinning of
the electron!
109
82. Radiation (20.7)
a) The process of emitting energy in
the form of waves or particles;
b) Energy radiated in the form of
waves or particles;
83. (34.1) Electromagnetic radiation
Electromagnetic radiation is associated with
electromagnetic waves (oscillations of the electric and
magnetic field) or beams of particles called photons.
All electromagnetic waves travel in a vacuum with a
speed of
c = 1 ≈ 3 ⋅ 108 m / s
µ0ε0
in all inertial frames.
In other media, the speed of light is less and depends
on the index of refraction n. The index of refraction of a
medium is defined as the ratio of the speed of light in a
vacuum c to the speed of light in the medium v.
c
n≡
v
Any electromagnetic radiation can be considered as a
superposition of sinusoidal (harmonic) electromagnetic
waves.
110
84. (34.2) Electromagnetic wave
z
k
y
x
a) Electromagnetic waves always
move in a direction perpendicular to
their electric and magnetic fields
(transverse oscillations).
For a monochromatic, plane
wave, vector k, called the
propagation vector, indicates the
direction of the wave propagation.
Ei (r , t ) = Em, i cos(k ⋅ r − ωt)
Bi (r , t) = Bm , i cos(k ⋅ r − ωt )
In an electromagnetic wave, the electric and magnetic fields
are perpendicular to each other and to the propagation
vector.
b) If the oscillations of the fields are harmonic, the wave
is monochromatic (definition).
Monochromatic light
produces the sensation of a defined color; but a certain
color can be caused by non-monochromatic light as well.
c) If the oscillations of the fields follow a certain simple
pattern we say that the electromagnetic wave is polarized.
When the fields oscillate along certain straight lines, the
wave is linearly polarized. When the fields rotate without
changing their magnitude the wave has circular polarization
(an attribute of the wave).
111
85. (34.2) Wave equation.
a) Each of the fields satisfies the wave equation. In an
empty space
∂2E = µ ε ∂2E
0 0
∂x2
∂t 2
∂2B = µ ε ∂2B
0 0
∂x2
∂t 2
and
proof.
y
Applying Faraday's law
∂Φ
∫ E ⋅ ds = − ∂tB
to the amperian loop marked in
the figure, we obtain an
equation relating the magnetic
and electric fields.
E
dy
dx
x
z
B
∂E
∂E
∫ E ⋅ ds = E(x + dx)dy + 0 − E( x)dy + 0 = E(x) + ∂x dx − E(x)dy = ∂x dxdy
− ∂Φ B = − ∂ (Bdxdy) = − ∂B dxdy
∂t
∂t
∂t
Therefore
∂E = − ∂B
∂x
∂t
Similarly, from the Ampère-Maxwell law we can show
∂B
∂E
= − µ0 ε 0
∂x
∂t
112
Combining the last two equations leads to the wave
equations
∂2E = − ∂ ∂B = − ∂ ∂B = − ∂  −µ ε ∂E = µ ε ∂2E
0 0 2
∂x ∂t
∂t ∂x
∂t  0 0 ∂t 
∂x2
∂t
∂2B = ∂  −µ ε ∂E  = −µ ε ∂ ∂E = −µ ε ∂  − ∂B = µ ε ∂2B
0 0 ∂t ∂x
0 0 ∂t  ∂t 
0 0 2
∂x2 ∂x  0 0 ∂t 
∂t
b) The simplest solution to the wave equation is a plane
wave:
E(x, t) = Em cos (kx − ωt)
B(x, t) = Bm cos (kx − ωt)
with
E Em ω
1
=
= =
=c
B Bm k
µ0ε 0
proof
Recall that the wavelength and frequency of the above
electromagnetic wave are λ = 2π and f = ω , respectively.
k
2π
The wave travels with a phase speed of
c = λf = 2π ⋅ ω = 1 .
k 2π
µ0ε0
113
Direct substitution in the wave equations
∂2E
∂2E
k E(x , t ) = − 2 = −µ0 ε0 2 = µ 0ε0 ⋅ ω2 E(x , t )
∂x
∂t
2
leads to
ω
=
k
1
µ0ε0
Through a direct substitution is a differential equation
relating magnetic and electric field
− kB m sin (kx − ωt ) =
∂B
∂E
= −µ 0ε0
= −µ 0ε 0ωE m sin (kx − ωt )
∂x
∂t
we obtain
E Em
1 k
1
=
=
⋅ =
B Bm µ 0 ε0 ω
µ 0 ε0
114
86. (34.3) The energy carried by electromagnetic waves
a) The (total) energy density u of an electromagnetic
wave is defined as the energy dU of both the electric and
magnetic fields per unit volume of the wave.
dU
u≡
dV
3
(SI unit: J/m )
The average energy density comprised by a sinusoidal
wave depends on the amplitudes of the magnetic and
electric fields
1
u av = ε0 E 2m = ε0 E 2rms
2
1 2
1
u av =
Bm = B 2rms
2µ 0
µ0
where Erms and Brms are the root-mean-square values of the
electric and magnetic fields respectively.
proof. The (instantaneous) total energy density at a
certain location is equal to the sum of the energy density
associated with each field separately
ε 0 E 2 B2
ε0 E2
E2
2
u( t ) =
+
=
+
=
ε
E
0
2
2µ 0
2
2µ 0 c 2
115
From the definition of average value of a function we find
t
∫ u(t')dt'
uav = lim 0
t
∫ ε0E2m cos 2(kx − ωt)dt'
t
∫ 2 + 2 cos 2(kx − ωt')dt'
= lim 0
= ε 0E2m lim 0
t−0
t →∞
t − 0t
t →∞
 t'+ 1 sin 2(kx − ωt')
 2ω
 0 ε 0E 2m
ε E2
= 0 m lim
=
2 t →∞
t−0
2
t →∞
1
1
t−0
=
b) The rate of energy flow across a unit area in
electromagnetic wave is described by the Poynting vector
defined by the following expression
1
S≡
E×B
µ0
proof.
Consider, a plane electromagnetic wave. If we imagine a
surface (with area dA) perpendicular to the path of the
wave, the amount of energy passing this surface in a
differential time dt is
dU = u ⋅ (dA ⋅ cdt )
where u(x,t) is the instantaneous energy density at the
surface. The rate at which the energy is transferred across
the surface is therefore
dU
= c ⋅ u( x , t ) ⋅ dA
dt
116
Considering the magnitude of the Poynting vector at the
location of the surface we find
µ 0ε 0 E2
EB E2
ε 0 E2
1 dU
S( x, t ) = S =
=
=
=
= c ⋅ u( x , t ) =
⋅
µ 0 cµ 0
µ0
µ 0ε 0
dA dt
Note that the direction of the Poynting vector coincides
with the direction of the wave propagation while its
magnitude is equal to the rate of energy transfer per unit
area.
c) Wave intensity I is the ratio of the average rate at
which an electromagnetic wave carries energy through an
imaginary surface, which is perpendicular to the
propagation direction, to the area of the surface.
(SI unit W/m2)
I ≡ Sav
d) Intensity I and energy density u of a wave are
proportional with the speed of light being the
proportionality constant.
I = Sav = ( c ⋅ u ( t )) av = cu av
For a certain electromagnetic radiation (a
superposition of sinusoidal waves), the function relating the
intensity of each sinusoidal wave with the wavelength is
called the spectrum of that radiation.
117
87. (34.7) Electromagnetic radiation that stimulates the
human eye is called visible light. This includes waves with
wavelengths between 400nm and 700nm. The maximum
sensitivity of the human eye is to light with a wavelength of
about 555nm, which corresponds to the maximum intensity
of the sun's spectrum.
Electromagnetic radiation also includes radio waves,
infrared and ultraviolet (black) light, X-rays and γ-rays.
118
LIGHT AND OPTICS
88. (35.3, 35.4, 35.5) Reflection and refraction
At the boundary of two media, light undergoes both
reflection and refraction.
angle of
incidence
incident ray
angle of
reflection
θ1 θ’1
θ2
reflected ray
refracted ray
angle of
refraction
a) The law of reflection
The reflected ray lies in the plane of incidence and the
angle of reflection θ1' is equal to the angle of incidence θ1
θ'1 = θ1
b)
The law of refraction (Snell's law)
The refracted ray lies in the plane of incidence and the
angle of refraction θ2 is related to the angle of incidence by
n1 sin θ1 = n2 sin θ2
In general, index of refraction of a medium is a function of
the radiation wavelength.
119
89. (35.7) Total internal reflection
Total internal reflection occurs when light travels from
a medium of high index of refraction to one of lower index
of refraction. The smallest angle θc at which total internal
reflection occurs is called the critical angle.
θc = sin− 1 n2
n1
( n1 > n2 )
because
n1 sin θc = n2 sin 90°
90. In order to be seen, an object
must send light from each of its points
in many directions. The eye collects
some of the light emitted from a point
allowing the brain to interpret the
location of the point.
In some situations diverging rays are interpreted as
originating from a single point creating an image of a point.
120