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Electromagnetic
Field Theory
Ahmed Farghal, Ph.D.Electrical Engineering, Sohag University
Lecture 7
The Uniform Plane Wave
Nov. 13, 2019
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2
Taking the partial derivative of any field quantity w. r. t. time is
equivalent to multiplying the corresponding phasor by 𝑗𝜔.
We can express Eq. (8)𝜕𝐻𝑦
𝜕𝑧= −𝜖0
𝜕𝐸𝑥
𝜕𝑡(using sinusoidal fields) as
where, in a manner consistent with (19):
On substituting the fields in (21) into (20), the latter equation
simplifies to
Phasor Form
(20)
(22)
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Phasor Form
3
For sinusoidal or cosinusoidal time variation,
𝐄 = 𝐸𝑥𝐚𝑥 𝐸𝑥 = 𝐸 𝑥, 𝑦, 𝑧 cos 𝜔𝑡 + 𝜓
Using Euler’s Identity 𝑒𝑗𝜔𝑡 = cos𝜔𝑡 + 𝑗 sin𝜔𝑡
The field is 𝐸𝑥 = Re 𝐸 𝑥, 𝑦, 𝑧 𝑒𝑗𝜓𝑒𝑗𝜔𝑡
The field in the phasor form is 𝐸𝑥𝑠 = 𝐸 𝑥, 𝑦, 𝑧 𝑒𝑗𝜓 𝐄𝑠 = 𝐸𝑥𝑠𝐚𝑥
For any sinusoidal time variation for field𝜕
𝜕𝑡𝐸 = 𝑗𝜔𝐸 and
𝜕
𝜕𝑡𝐻
= 𝑗𝜔𝐻
The Maxwell’s equation in phasor form
Eqs. (23) through (26) may be used to obtain the sinusoidal
steady-state vector form of the wave equation in free space.
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Wave Equation
4
Using Maxwell’s Equation the steady state wave equation can be
written as:
Since
𝑘𝑜, the free space wavenumber, 𝑘𝑜 = 𝜔 𝜇𝑜𝜀𝑜
The wave equation for the 𝑥-components is
Using the definition of Del operator
The field components does not change with x or y so the wave
equation is lead to ordinary differential equation
the solution of which
Vector Helmholtz Wave Equation
s
2
ss
2
ss EH∇E∇-)E∇(∇)E∇(∇ ooo-j
0E∇s s
2
s
2 EE∇ ok
xs
2
xs
2 EE∇ ok
xsoxsxsxs Ek
z
E
y
E
x
E 2
2
2
2
2
2
2
∂
∂
∂
∂
xsoxs Ek
dz
Ed 2
2
2
(31)
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5
Given E𝑠, 𝐇𝑠 is most easily obtained from (24):
which is greatly simplified for a single 𝐸𝑥𝑠 component varying only
with 𝑧,
Using (31) for𝐸𝑥𝑠 = 𝐸𝑥0𝑒−𝑗𝑘0𝑧 + 𝐸𝑥0
′ 𝑒𝑗𝑘0𝑧, we have
In real instantaneous form, this becomes
where 𝐸𝑥0 and 𝐸𝑥0′ are assumed real.
WAVE PROPAGATION IN FREE SPACE
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In general, we find from (32) that the electric and magnetic field
amplitudes of the forward-propagating wave in free space are
related through
We also find the backward-propagating wave amplitudes are related
through
where the intrinsic impedance of free space is defined as
The dimension of 𝜂0 in ohms is immediately evident from its
definition as the ratio of E (in units of V/m) to H (in units of A/m).
WAVE PROPAGATION IN FREE SPACE
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The fields
7
The uniform plane cannot exists because it extends to infinity in
two dimensions at least and represents an infinite a mount of
energy.
The far field of the antenna can be represent by a uniform plane
wave
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8
Thank you