Download Lecture 31 Chapter 34 Electromagnetic Waves

Lecture 31
Chapter 34
Electromagnetic Waves
Review
• For an RLC circuit
– Voltages add up to emf
E = vR + vC + vL
– Maximum current given by
Em
I=
Z
– Impedance defined as
Z = R + ( X L − XC )
2
2
– Phase constant defined as
X L = ωd L
1
XC =
ωd C
XL − XC
tan φ =
R
Review
• For RLC circuit, resonance and the max
current I occurs when ωd = ω
• For an ac circuit, define rms
values
I
V
I rms =
V rms =
2
2
1
ωd = ω =
LC
E rms
• Average power dissipated to thermal
energy
Pavg = I
2
rms
R
E
=
2
Pavg = Erms I rms cosφ
Review
• Transformer –
– 2 coils (primary and secondary)
wound around same iron core
– Transformation voltage and
current are related to ratio of
the number of turns in the coils
NS
VS = VP
NP
NP
IS = IP
NS
– Equivalent resistance seen by
2
generator
R eq
 NP 
 R
= 
 NS 
EM Waves (1)
• Electromagnetic waves –
– Beam of light is a traveling wave of E and B fields
– All waves travel through free space with same speed
EM Waves (2)
• Generate electromagnetic (EM) waves –
– Sinusoidal current in RLC causes charge
ω=
and current to oscillate along rods of
antenna with angular frequency ω
– Changing E and B fields form EM wave that
travels away from antenna at speed of light, c
1
LC
EM Waves (3)
• E and B fields change with
time and have features:
– E and B fields ⊥ to direction
of wave’s travel – transverse
wave
– E field is ⊥ B field
– Direction of wave’s travel is
given by cross product r r
E×B
– E and B fields vary
• Sinusodially
• With same frequency and in
phase
EM Waves (4)
• Write E and B fields as
sinusoidal functions of
position x (along path of
wave) and time t
E = E m sin( kx − ω t )
B = Bm sin( kx − ω t )
• Angular frequency ω and
angular wave number k
• E and B components
cannot exist independently
ω = 2πf
k=
2π
λ
EM Waves (5)
• Speed of wave is v = ω
k
• Using definition of ω
and k, velocity is
2π
k=
ω = 2π f
λ
ω
2πf
v= =
= fλ
k 2π / λ
• In vacuum EM waves
move at speed of light
v = c = fλ
c = 3 × 10 m / s
8
EM Waves (6)
• Use Faraday’s and Maxwell’s laws of
induction
r
r
r
r
dΦ B
dΦ E
∫ B • ds = µ 0ε 0
∫ E • ds = −
dt
dt
• Can prove that speed of light c is given by
(proof done inEsection 34-3) 1
c=
m
Bm
c =
µ 0ε 0
c = 3 × 10 m / s
8
• Light travels at same speed regardless of
what reference frame its measured in
EM Waves (7)
• EM waves can transport energy and deliver it
to an object it falls on
• Rate of energy transported per unit area is
given by Poynting vector, S, and defined as
r 1 r r
S=
E×B
µ0
• SI unit is W/m2
• Direction of S gives wave’s direction of travel
EM Waves (8)
• Magnitude of S is given by
S=
1
µ0
EB
• Found relation c = Em
Bm
• Rewrite S in terms of E since
most instruments measure E
component rather than B
E
S =
E
µ0
c
• Instantaneous energy flow rate is
1
S =
1
cµ 0
E
2
EM Waves (9)
• Usually want time-averaged value of S also
called intensity I I = S =  energy / time  =  power 
avg
I =
[
E ]
µ c
1
2
avg
0
=

[
E
µ c
1
area
2
m
 ave
 area  ave
sin 2 ( kx − ω t )
]
avg
0
• Average value over full cycle of sin 2 θ = 1 / 2
Em
• Use the rms value
Erms =
2
• Rewrite average S or intensity as I = 1 E 2
µ 0c
rms
EM Waves (10)
• Find intensity, I , of point source
which emits light isotropically –
equal in all directions
 power 
 energy / time 
I = S avg = 
=



area
 ave  area  ave

• Find I at distance r from source
• Imagine sphere of radius r and area
Power
PS
I =
=
2
Area
4π r
• I decreases with square of distance
A = 4π r
2
EM Waves (11)
• Checkpoint #2 – Have an E field shown in
picture. A wave is transporting energy in the
negative z direction. What is the direction of
the B field of the wave?
• Poynting vector gives
r 1 r r
S=
E×B
µ0
• Use right-hand rule to find B field
Positive x direction