Reminders on waves
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Traveling waves on a string obey the wave equation:
∂ 2 y(x,t) 1 ∂ 2 y(x,t)
= 2
2
∂x
v
∂t 2
y=wave function
λ
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A mechanical wave is a disturbance created by a vibrating object that travels through a medium
from one location to another
General solution: y(x,t) = f1(x-vt) + f2(x+vt)
The general solution can be expressed as a superposition of harmonic
wave functions: y(x,t) = A sin(kx-ωt) y(x,t) = sin(kx+ ωt) A = amplitude
k = 2π/λ = wave number λ = wavelength
f = frequency T = 1/f = period
ω = 2πf=2π/T angular frequency
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Reminders: waves
Classification of waves:

Transverse wave: medium particles move in direction perpendicular to
direction of wave
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Longitudinal wave: medium particles move in direction parallel to direction
of wave
Eg: EM waves
sound waves
Velocity of waves: general form of wave on the right
y(x,0) = A sin(ax)
y(x=0,t = 0) = A sin(0) = 0 and y(λ/2,0)=Asin(aλ/2) =0 ⇒aλ/2=π
a = 2π/λ ⇒ y(x,0)=Asin(2π/λ x)=Asin(kx)
If the wave moves of a distance vt, at a later time t it is:
y(x,t) = A sin[k(x-vt)]
The wave travels a distance λ in one period T (by definition):
0=Asin(0)=Asin[k(λ-vT)] ⇒ λ=vT ⇒ v = λ/T=λf
λ/2
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Maxwell’s equations in empty space
∫ E ⋅ dA = 0
S
∫
L
E ⋅ ds = −
(Gauss' Law)
∫ B ⋅ dA = 0
S
dΦB
(Faraday - Henry)
dt
∫ B ⋅ ds = µ0ε0
L
dΦE
(Ampere - Maxwell law)
dt
From these equations we get EM wave equations!
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λ
∂2y 1 ∂2y
= 2 2
2
∂x
v ∂t
€
Wave composed of E and B fields!
Equations for a plane transverse wave
propagating in empty space with
velocity
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Hertz’s Experiment (1887)

R
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T
First generation and detection of EM waves in Lab
Transmitter T: 2 spherical electrodes separated by
narrow gap charged until air gap ionized
The discharge between the electrodes exhibits
oscillatory behavior of frequency f ~ 4 x 107 Hz
This ‘oscillating dipole’ emits E and B plane waves
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When resonance frequency of T and R match sparks
also in R = receiver
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Hertz’s hypothesis: the energy transmitted from T to
R is carried by waves

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He measured λ by having waves from T reflected
on a metal sheet so he obtained standing waves
with nodes at a distance λ/2
By knowing f of T he measured the speed of the
radiation close to c

Radiation has wave properties: interference,
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diffraction, reflection, refraction and polarization
Solutions of Maxwell’s equations

λ
The simplest solution to partial differential
equations is sinusoidal wave propagating along x:
 E = E
max sin (kx – ωt) i

B = Bmax sin (kx – ωt) k

Angular wave number: k = 2π/λ λ = wavelength

Angular frequency is ω = 2πƒ ƒ= wave frequency
E·B=0

The speed of the electromagnetic wave is
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E and B are orthogonal

An easy way to understand this:
3
2
1
θ
E
I
ΦB
1.
2.
3.
= B A cosθ
Max flux θ=0
Less flux
Null flux θ=90°
B parallel to area and E perpendicular
to circuit so E ⊥ B
E orthogonal to B!
Increasing B-field
dΦB
∫ E ⋅ ds = - dt
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∂E
∂B
=−
∂x
∂t
Let’s demonstrate:

E = Emax sin (kx – ωt)
B = Bmax sin (kx – ωt)
A wave at instant t in x and x+dx:
the E field varies from E to E+dE
∂E
E(x + dx,t) ≈ E(x,t) +
dx ⇒
∂x
∫ E ⋅ ds =[ E(x + dx,t) − E(x,t)]l =
€
∂E
dxl
∂x
Magnetic flux through rectangle:
dΦB = Bldx
dΦB
∂B
= ldx
⇒
dt
∂
t
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∫ E ⋅ ds =
∂E
dΦ
∂B
dxl = − B = −dxl
∂x
dt
∂t
Relation between E and B

E = Emax sin (kx – ωt)

B = Bmax sin (kx – ωt)

From:
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First derivatives:
∂E
= kE max cos(kx − ωt)
∂x
∂B
= −ωBmax cos(kx − ωt)
∂t
∂E
∂B
=−
∂x
∂t
This relation comes from
Maxwell’s equations!
EM Waves from an Antenna
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Accelerated charged particles are sources of EM waves:
EM waves are radiated by any circuit carrying alternating current
Two rods connected to an AC source. (a) Ends of rods charged and Efield parallel to rods
As oscillations continue, the rods become less charged, the field near
the charges decreases and the field at t = T/4 is zero (b)
The charges and field reverse (c)
The oscillations continue (d)
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Energy carried by EM waves
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Rate at which energy flows through a unit area perpendicular to
direction of wave propagation
Let’s consider a cylinder with axis along x of area A and length L and the
time for the wave to travel L is Δt=L/c
The average power is
U av uav AL
Pav =
Δt
=
Δt
= uav Ac
In a given volume, the energy is
shared equally by the two fields
And the intensity (average P/area)
Pav
1
B2
E2
2
uE = ε0 E = uB =
= 2
Iav =
= uav c €
2
2µ0 2c µ0
A
Total instantaneous energy density (E=cB)
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ExB
B 2 EB
u = uE + uB = ε0 E =
=
µ0 µ0c
2
Poynting Vector
Wave intensity I = time average over one or more cycle
<sin2(kx - ωt)> = 1/2 then <E2> = Emax2/2 and <B2> = Bmax2/2

Iav = uav c =
E max Bmax
2µ 0
Define vector with magnitude= power per unit area (J/s.m2 =
W/m2)
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
Its direction is the direction of propagation of the EM wave
 Its magnitude varies in time
 Its magnitude reaches a maximum at the same instant as E and B