Time Varying Electromagnetic Fields Short notes | Electromagnetics - Electronics and Communication Engineering (ECE) PDF Download

 Page 1


Time V arying Electromagnetic Fields Notes
Introduction
Time-varying electromagnetic fields describe electric and magnetic fields that
change with time, leading to dynamic inter actions. This area of electromagnetics
is governed b y Maxwell’ s equations and is fundamental to understanding wave
propagation, antennas, and electromagnetic devices.
K ey Concepts
• Time-V arying Fields : Electric ( E ) and magnetic ( B ) fields that vary with
time, inducin g each other .
• Maxwell’ s Equations : Describe the behavior of time-varying electromag-
netic fiel ds.
• Applications : Wireless communication, r adar , electromagnetic compati-
bility , and micr owave engineering.
Maxwell’ s Equations (Time-V arying Fields)
In differential form, for a medium with permittivity ? , permeability µ , and con-
ductivity s :
• Gauss’ s Law for Electric Fields :
?· E =
?
?
where ? is the charge density .
• Gauss’ s L aw for Magnetic Fields :
?· B = 0
• F ar ada y’ s L aw of Induction :
?× E =-
? B
?t
• Ampere’ s Law with Maxwell’ s Correction :
?× H = J+
? D
?t
where H =
B
µ
, D = ? E , and J = s E (conduction current density) .
1
Page 2


Time V arying Electromagnetic Fields Notes
Introduction
Time-varying electromagnetic fields describe electric and magnetic fields that
change with time, leading to dynamic inter actions. This area of electromagnetics
is governed b y Maxwell’ s equations and is fundamental to understanding wave
propagation, antennas, and electromagnetic devices.
K ey Concepts
• Time-V arying Fields : Electric ( E ) and magnetic ( B ) fields that vary with
time, inducin g each other .
• Maxwell’ s Equations : Describe the behavior of time-varying electromag-
netic fiel ds.
• Applications : Wireless communication, r adar , electromagnetic compati-
bility , and micr owave engineering.
Maxwell’ s Equations (Time-V arying Fields)
In differential form, for a medium with permittivity ? , permeability µ , and con-
ductivity s :
• Gauss’ s Law for Electric Fields :
?· E =
?
?
where ? is the charge density .
• Gauss’ s L aw for Magnetic Fields :
?· B = 0
• F ar ada y’ s L aw of Induction :
?× E =-
? B
?t
• Ampere’ s Law with Maxwell’ s Correction :
?× H = J+
? D
?t
where H =
B
µ
, D = ? E , and J = s E (conduction current density) .
1
K ey Relationships
• Electric and Magnetic Field Coupling : Time-varying E induces B , and vice
versa, via F ar ada y’ s an d Ampere’ s laws.
• W ave Equation : Derived from Maxwell’ s equations in a source-free, loss-
less medium :
?
2
E-µ?
?
2
E
?t
2
= 0, ?
2
B-µ?
?
2
B
?t
2
= 0
• W ave Pr opagation Speed : In a medium:
v =
1
v
µ?
In free space, v = c =
1
v
µ
0
?
0
˜ 3× 10
8
m/s, where µ
0
= 4p× 10
-7
H/m, ?
0
=
8.854×10
-12
F/m.
Electromagnetic W aves
• Plane W aves : Uniform waves with constant amplitude in a plane perpen-
dicular t o propagation direction. F or a wave in the z -direction:
E(z,t) = E
0
cos(?t-kz)ˆ x, B(z,t) =
E
0
c
cos(?t-kz)ˆ y
where k =
?
v
is the wave number , and ? = 2pf is the angular fr equency .
• W avelength : ? =
v
f
=
2p
k
.
• Po ynting V ector : Represents power flow direction and magnitude:
S = E× H
A ver age power dens ity: S
avg
=
1
2
Re{ E× H
*
} .
Boundary Conditions
• T angential Ele ctric Field : Continuous across a boundary: E
1t
= E
2t
.
• T angential Magnetic Field : Discontinuous if surface current exists: H
1t
-
H
2t
= J
s
× ˆ n .
• Normal E lectric Flux : D
1n
- D
2n
= ?
s
, where ?
s
is sur face charge density .
• Normal Magnetic Flux : Continuous: B
1n
= B
2n
.
Pr actical Consider ations
• Lossy Media : Conductivity s introduces attenuation, modifying the wave
equation w ith a damping term.
2
Page 3


Time V arying Electromagnetic Fields Notes
Introduction
Time-varying electromagnetic fields describe electric and magnetic fields that
change with time, leading to dynamic inter actions. This area of electromagnetics
is governed b y Maxwell’ s equations and is fundamental to understanding wave
propagation, antennas, and electromagnetic devices.
K ey Concepts
• Time-V arying Fields : Electric ( E ) and magnetic ( B ) fields that vary with
time, inducin g each other .
• Maxwell’ s Equations : Describe the behavior of time-varying electromag-
netic fiel ds.
• Applications : Wireless communication, r adar , electromagnetic compati-
bility , and micr owave engineering.
Maxwell’ s Equations (Time-V arying Fields)
In differential form, for a medium with permittivity ? , permeability µ , and con-
ductivity s :
• Gauss’ s Law for Electric Fields :
?· E =
?
?
where ? is the charge density .
• Gauss’ s L aw for Magnetic Fields :
?· B = 0
• F ar ada y’ s L aw of Induction :
?× E =-
? B
?t
• Ampere’ s Law with Maxwell’ s Correction :
?× H = J+
? D
?t
where H =
B
µ
, D = ? E , and J = s E (conduction current density) .
1
K ey Relationships
• Electric and Magnetic Field Coupling : Time-varying E induces B , and vice
versa, via F ar ada y’ s an d Ampere’ s laws.
• W ave Equation : Derived from Maxwell’ s equations in a source-free, loss-
less medium :
?
2
E-µ?
?
2
E
?t
2
= 0, ?
2
B-µ?
?
2
B
?t
2
= 0
• W ave Pr opagation Speed : In a medium:
v =
1
v
µ?
In free space, v = c =
1
v
µ
0
?
0
˜ 3× 10
8
m/s, where µ
0
= 4p× 10
-7
H/m, ?
0
=
8.854×10
-12
F/m.
Electromagnetic W aves
• Plane W aves : Uniform waves with constant amplitude in a plane perpen-
dicular t o propagation direction. F or a wave in the z -direction:
E(z,t) = E
0
cos(?t-kz)ˆ x, B(z,t) =
E
0
c
cos(?t-kz)ˆ y
where k =
?
v
is the wave number , and ? = 2pf is the angular fr equency .
• W avelength : ? =
v
f
=
2p
k
.
• Po ynting V ector : Represents power flow direction and magnitude:
S = E× H
A ver age power dens ity: S
avg
=
1
2
Re{ E× H
*
} .
Boundary Conditions
• T angential Ele ctric Field : Continuous across a boundary: E
1t
= E
2t
.
• T angential Magnetic Field : Discontinuous if surface current exists: H
1t
-
H
2t
= J
s
× ˆ n .
• Normal E lectric Flux : D
1n
- D
2n
= ?
s
, where ?
s
is sur face charge density .
• Normal Magnetic Flux : Continuous: B
1n
= B
2n
.
Pr actical Consider ations
• Lossy Media : Conductivity s introduces attenuation, modifying the wave
equation w ith a damping term.
2
• Reflection and Tr ansmission : At interfaces, waves reflect and tr ansmit
based on impedan ce mismatch: ? =
v
µ
?
.
• Applications : Antennas, waveguides, and RF circuits rely on time-varying
field princi ples.
• Numerical T ools : Software lik e Ansys HFSS or COMSOL simulates complex
time-varying fie lds.
3