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

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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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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