x n–1 = (n–1)d x n+1 = (n+1)d x n = nd u n–1 u n u n+1 d d l θ v Wavelength λ x λ E = 0 t = 0 (2πt/ T) = 0 t = T (2πt/ T) = 2π t = T/2 (2πt/ T) = π t = 3T/8 (2πt/ T) = 3π/4 t = T/4 (2πt/ T) = π/2 t = T/8 (2πt/ T) = π/4 E = E max E = 0.7E max E = 0.7E max E = E max E = E max Period T t E π / π / π / π ◦
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xn–1 = (n–1)d xn+1 = (n+1)dxn = nd
un–1
un
un+1
d d
l
θ
v
Wavelength
λ
x
λ
E = 0
t = 0
(2πt/T) = 0
t = T
(2πt/T) = 2π
t = T/2
(2πt/T) = π
t = 3T/8
(2πt/T) = 3π/4
t = T/4
(2πt/T) = π/2
t = T/8
(2πt/T) = π/4
E = Emax
E = 0.7Emax
E = 0.7Emax
E = Emax
E = Emax
Period
T
t
E π / π /π / π ◦
λ
x
v
E
π /λπ /λ π ◦
λ λ
(π − π
λ
)
π / − π /λ
π / + π /λ +
= =π / − π /λ
φ π / − π /λ+φφ π ◦
=
= / / /π / + π
π
t
t
t = 0 T/2 T
E cos (2πt
T)
E cos (2πt
T )π
2+
π/
π / +φπ +φ = /ω +φ ω = π = π/
ωπ
ω +φ
ωω
× −
x or t
Amplitude
t
I
E cos(ωt)
[E cos(ωt)]2 ∝
Laser
Screen
Two slits
ω ωω = π = π/
/
E1
E2
E1 E2+
ω + ω = ω
◦ λ/
◦ π
E1
E2E1 E2+ = 0
◦
◦
λ λλ
λ/ λ/ λ/
ω ω
◦ π
◦ π
θ
Δ θ
d
Source 1 Source 2
Δl =
d s
in θA
B
C
Line AC ≈ line BC
θ
θ
90°
∠
Δ = θ
Δ = θ
Δθ Δ = θ Δ λ/ λ/ λ
θ Δ = θ
< λ θ = ◦
θ = ◦ θ λ < λ/
= λΔ = θ
Δ = θλ/ λ/ ◦
d = 4.5λ
θ
Max0Max1
Max2
Max3
Max4
Min1Min2
Min3
Min4
Min5
I
Δl = d sin
θ
θ = ◦
θ = ◦
◦ ◦
= λθ = ◦ θ =− ◦
◦ − ◦ λλ ◦
θ = θ
= λ θ
θ θ
θθ
Δ = θ λ λ λΔ = θ λ/ λ/ λ/
< λ/
Δ
==
=
Laser
Screen
Two slits
θ L
d
x
Intensity
θ = λ θ = λ/ θ ≈θ ≈ θ λ/ ≈ /
λ= =× − × −
= × − =
λ
× −
× −
~1 10–10 m+ λ λ λ
λ≈
Incoming x-rays
Outgoing x-rays
s s
1
2
2
1
θθ
θ
Incoming x-rays
Outgoing x-rays
d
θ θ
1 2
21
d s
in θ
d sin θ
θ
θθ
× −
λ× −
× −
θθ
θ
θ λλ
θ θ
θ
θ = λ
λ θ
θ= λ/ θ
End view
λθ θ = λ
θ = λ
× −× −
λ�
1
2
s1 s2
= Δ =
12
s1s2
=/Δ
θ = λ
≈ × −
λ ≈ × −
λ
Δ =
θ = λ
Thin film with N layers;
thickness λ/2
λ/4
λ/4
Layer (N/2 + 1)
Path difference
λ/2
/ + λ/× − / × −
/ + =
/ + = / ==
λ/λ/
λ/ λ/ λ/ λ/λ/ λ/ λ/ λ/
λ′ = λ/ λ
= λ×
Δ = /
= × − = × −
= × −
θ
θ=
Incident lightθ
λ θθ
θ = λλ ≈
θ = λ/
θ
θ = λ/ λθ
=
= × − ×× = ×
= × −θ = ◦
≈≈
Distant
light
bulb
CD
Eye
Distant
light
bulb
CD
Eye
Violet
Redθ = λ/
θ ◦
θ = λ
=λ
θ≈ × −
◦ = × − =
θ ≈ ◦
θ = λ
=λ
θ≈ × −
◦ = × − =
Lamp
Screen
λ/
λ/2
Molecule 1 Molecule 2
Incoming
light
Scattered
light
λ/ λ/=
λ/
λ
λ/
λ/
λ
= × × − ××
×
≈
λ/
√λ√
=
Δθ =λ
θ
θ = λ
θθ+Δθ Δθ
/Incident light
θ +Δθ
λ2λN
2+
λ+λ/
/ / λλ/ ( )
λ+ λ
/ +λ/
θ+Δθ
θ+Δθ = λ+ε ε λ
/
λ+ ε =
( )λ+ λ
ε
ε=λ
Δθ
Δθ
θ+Δθ = λ+ε= λ+λ
θ Δθ+ θ Δθ = λ+λ
Δθ
Δθ ≈ Δθ ≈Δθ
θ+ θΔθ ≈ λ+λ
θ = λ
θΔθ ≈ λΔθ ≈ λ
θ
=
θ ◦ θ Δθ
Δθ ≈ λ
Δθ≈ λ/ ≈ × − / × − = × −
θ
Incident
light
Δθ
λ
= d sin θ
d
Min
Max
Min
Max
θΔθ
λ /N
d cos θ
θ θ+Δθ
Δθ
Δθ ε= λ/θ
Δθ
Δθ ≈ λ/
θ=
λ
θ=
λ
θ
=θ
◦ θ
Δθ ≈ λ
λ+λ
Incident lightW
= N
d
W
λΔθ =
Δθ ≈ λ/
Slit
ϕ
= /
Δθ λ/
=λ × −
= =
λ/λ/
Δθ≈λ/
Δθλ Δθ
φ Δθ
λ λ
λ θ =
Δθ
∂
∂=
∂
∂
=∂ /∂
∂ /∂
≈ω =
√/
xn–1 = (n–1)d xn+1 = (n+1)dxn = nd
un–1 un un+1
d d
=
+
= − = + + + − + −= + −= − = + − − + − −= − −= −= + − − − −
u
x
(x0,u0)
(x4,u4)
(x6,u6)
(x5,u5)
(x3,u3)(x2,u2)
(x1,u1)
+ −∂ /∂
Δ
∂
∂=
Δ →Δ
Δ
≈ + −
∂
∂
(∂
∂
)=
Δ →Δ ∂ /∂
Δ
≈ + − / − − − /
∂
∂≈ + − − − −
≈ ∂
∂
=
=/ =
+ − +− − + + − − −
∂ /∂∂ /∂
≈ = =
= +
∂ +
∂=
∂
∂
/
∂
∂= =
∂
∂
∂
∂=
∂
∂
=
=±√
| |
+
+
= −
+=
−
= −x
u Δx = vΔt
v
Δ = ΔΔ +Δ +Δ
Δ = Δ
+Δ − +Δ = −
Δ − Δ = = Δ /Δ=
−=
= − = −
= +
+ = +
= −
= −∂
∂=
∂
∂= −
∂
∂=
∂
∂
(∂
∂
)=
∂
∂
(−
)=
(−
)∂
∂=
∂
∂=
∂
∂=
∂
∂=
∂
∂
(∂
∂
)=
∂
∂
( )=
( )∂
∂=
=−
∂
∂=
∂
∂
=√
/
x
u(x,t0)v
x1 x3x2
=
= = −
=
u(x1,t)
tt0 tt0 tt0
u(x2,t) u(x3,t)
(a) (c)(b)
= / ρρ= /
=
√=
√=
√=
√ρ
=√
/ρρ
=Δ
Δρ/ρ=
ρ ρ
= +
∂ +
∂=
∂ +
∂
∂
∂+
∂
∂=
∂
∂+
∂
∂
xn–1 = (n–1)d xn+1 = (n+1)dxn = nd
un–1
un
un+1
d d
l
θ
θ
/ = θ ≈ ◦◦
−= −
θ ≈ θ ≈ / ◦◦
= θ − θ −≈ − =
= θ − θ −
≈[
+ − − − −]
(∂
∂
)≈ + −
(∂
∂
)≈ + −
≈[(
∂
∂
)−(∂
∂
) ]
∂
∂=
∂
∂
(∂
∂
)≈
(∂
∂
)−(∂
∂
)=
( )
=∂
∂
∂
∂=
∂
∂
∂
∂=
∂
∂
=
=±√
μ = /
=
√=
√μ
√
λ
=
(π
λ− π
)
=
(π
λ
(− λ
))
=
(π
λ( − )
)
Wavelength
λ
x
v =λ
T
Period TAmplitude
− = λ/λ
λπ
◦
π/λ π/ω
= −ω
ω/ = π/ / π/λ = λ/ =
= /
ω = π/ ω = π
ω
ω
(∂
∂
)
λ= λ/ = λ
=
∫ λ (∂
∂
)
/ λ/ =
/= −ω
∂
∂=−ω −ω
=
∫ λ
ω −ω
=ω
∫ λ
−ω
θ+ θ =
θ− θ = θ
θ = − θ
∫ λ
− −ω =λ
=ω λ
/
= −ω
=∂
∂
= −ω
=−= ω
λ= λ/ = λ
=
∫ λ
/ λ/ == −ω
=
∫ λ
ω −ω
=ω
∫ λ
−ω
=ω λ
+ =ω λ
/ μ
+ = μω λ
=μω λ
= μω
ρ ×
μ μ= μ = ρ ×
ρ ω
ρω
/
ω
v v
tϕ
t + T/8ϕ + π/4
t + T/4ϕ + π/2
t + 3T/4ϕ + 3π/4
t + T/2ϕ + π
Nodes
λ
= λ
λ=
λ=
L
λ = 2L
λ = L
λ =λ =
λ = / λ = / λ = / λ = /
= λ / λ = /
3(λ/2) = L
λ = 2L/3
4(λ/2) = L
λ = 2L/4
5(λ/2) = L
λ = 2L/5
6(λ/2) = L
λ = 2L/6
D
λ = 2D
= /λ = / = / = / =/ = / = / = /
/
1
1
1
1
1
2
2
2
2
2
Sum of 1 and 2
= /
= =
=
/
/
= /λ
= /λ λ
Light Electron
Metal
I
K
EphotonW
K
−
Frequency f = c/λ f0
K
−
= =λ
= × −
=
λ= =× − · ×
× − = × − =
=
ElectronPhoton
Photon
Electron
θ
ϕ
p1, E1
p3, E3
p2, E2
= +=
Δλ
= +
+ = += − +
= + = −
• = − • −
= + − • = + − θ
= + − θ
− = + − θ = −− + −= + − θ
− = − θ(−
)= − θ(
λ − λ)= − θ =
λ
λ −λ = − θ
θ
=× − ·
× − × = × −
= /λ
× −◦
Photomultipliers
Laser
Two slits
θ L
d
θ = λ
θ= λλ θ
After 10 electrons
After 1000 electrons
After 100 electrons
After 50 electrons
θ = λλ
λ=
= × − ·=
λ= =
Δ
=
=√
= Δ
λ= = √Δ
λ = θ
Δ
θ
λθ
λ= /
θ = λ
θ = λ
Δ
/ /
=(
π)+
(π
)+
(π
)+ · · ·
∫ (π
)=
∫ [ (π
)+
(π
)+ · · ·
] (π
)
∫ (π
) (π
)= =
∫ (π
) (π
)= =/
∫ (π
)= + + · · ·+ + + + + · · ·
=
∫ (π
)
1
0
L/2 L/2
+ < < // < <
=
∫ (π
)=
∫ / (π
)+
=
[−π
(π
)] /
=π
[−
( π)]
/π=
= = / = = / = / = / = = /= /
S2S1
Laser
Section R
Here, E = Elaser + ES1 + ES2 + ER = 0
+ + + =
+
Laser
Now, E = Elaser + ER
Section R
+ + + =
+ =− +
+
S2S1
E = ES1 + ES2
= ∇• =ρ
ε
= ∇• =
= ∇× =−∂
∂
= ∇× = μ
(+ε
∂
∂
)
= 〈 〉 = 〈 〉
∇= 〈 ∂
∂
∂
∂
∂
∂〉
= 〈 ∂
∂〉= 〈 〉
= 〈 ∂
∂〉= 〈 〉
= 〈−∂
∂
∂
∂〉= 〈 −∂
∂〉
= 〈∂∂
−∂
∂〉= μ ε 〈 ∂
∂〉
+ −
∂
∂=−∂
∂
−∂
∂= μ ε
∂
∂
∂
∂
(∂
∂
)=− ∂
∂
(∂
∂
)=− ∂
∂
(∂
∂
)
∂ /∂
∂
∂= μ ε
∂
∂
∂
∂=
μ ε
∂
∂
= /√μ ε
= ×
λ= =λ
π − πλ+φ
= /ω = πλφ = == λ
/ /
Δ = θ λλ λ
Δ = θ λ/λ/ λ/
< λ/
Δ
θ = λ
θ
λ −λ = − θ
θ = λ
Δθ ≈ λ
θ ≈ λ/
∂
∂=
∂
∂
=
√=
√ρ
=
√=
√μ
= /λ / / /
λ=
= × −
=(
π)+
(π
)+
(π
)+ · · · =
∫ (π
)
= λ/< λ/
d = λ /3.5
I
=×
=[
/ − / +π ]
ωλ +
− ==
λ −
= λ
y
xz outCopper wires
in z direction
Unpolarized
electromagnetic
radiation heading
in the –x direction
θ
L =
2.5λ
Detector
θ
θ ◦θ
× −◦
× −× −
× −
× −
× −
× − × −
= ×
0.0
4 m
x
80 m/s
1.8 m
−
−
= +
× −
× −
= π =−π
t
t = 0 T
E cos (2πt
T )+ π
◦ ◦
◦
× −λ≈ × −
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