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ECE 107: Electromagnetism Set 7: Dynamic fields

Instructor: Prof. Vitaliy Lomakin Department of Electrical and Computer Engineering

University of California, San Diego, CA 92093

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Maxwell’s equations • Maxwell’s equations

•  Goals of this chapter –  To “close the loop” between dynamics and statics –  To explain effects introduced by time variations –  To extend some results from static to dynamics

t∂∇× = −∂BE

t∂∇× = +∂DH J

ρ∇⋅ =D

0∇⋅ =B

12 2 1

12 2 1

12 2 1

12 2 1

ˆ ( ) 0ˆ ( )ˆ ( )ˆ ( ) 0

s

s

ρ× − =⋅ − =× − =⋅ − =

n E En D Dn H H Jn B B

εµ

==

D EB H

E ⋅d l∫ = − ddt

B ⋅ds∫∫H ⋅d l∫ = d

dtD ⋅ds∫∫ + I

D ⋅ds∫∫ = Q

B ⋅ds∫∫ = 0

εµ

==

D EB H

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Faraday’s law •  Faraday’s law: • Magnetic flux: •  Electromotive force voltage: •  induces currents in a wire

loop. The currents produce the flux that opposes the inducing flux (Lenz’s law)

•  allows for transformers, generators, motors, etc.! Its existence is a foundation of Electromagnetics

E ⋅d l

C∫ = − ddt

B ⋅dsS∫∫

SdΦ = ⋅∫∫ B s

emf S

d dV ddt dtΦ= − = − ⋅∫∫ B s

emfV

emfV

•  Transformers: –  Current à mutual flux –  Flux à voltage

–  Energy conservation à

Faraday’s law (2)

I1V1 = I2V2

⇒ I1I2

= N2

N1

V1 = −N1dΦdt, V2 = −N2

dΦdt

⇒ V1V2

= N1N2

•  Electric generator: –  Rotating loop in a magnetic field à EMF

•  Electric motor: –  Opposite to the generator –  An AC current is sent to the coil in

a constant magnetic field à coil spins

Faraday’s law (2)

•  Eddy currents: –  Time varying magnetic field à Electric field à (Eddy) current

•  Examples: –  Eddy current waist separator –  Eddy current break –  Transcranial magnetic stimulation

Faraday’s law (2)

∇×E = − ∂B

∂t⇒ J =σE

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Ampere’s law •  Ampere’s law: •  Displacement current: •  Displacement current density: •  Displacement current is required to establish

consistent equations of Electromagnetics. Its introduction resulted in Maxwell’s equations

•  To explain its importance, consider a capacitor under an ac voltage.

H ⋅d l

C∫ = I + ddt

D ⋅dsS∫∫

d dS S

dI d ddt

= ⋅ = ⋅∫∫ ∫∫J s D s

dddt

= DJ

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Charge-current continuity relation •  Start with the Ampere’s law •  Apply divergence

•  Continuity relation:

•  The amount of charge change equals the current needed to compensate the change

•  For statics –  The current and charge are independent –  Kirchhoff’s current law

t∂∇× = +∂DH J

∇⋅(∇× H) =0 (vector identity)

= ∇⋅ ∂D∂t

+∇⋅J ⇒∇⋅J = − ∂∂t

∇⋅D =ρvGauss law

⇒∇⋅Jv = − ∂∂t

ρv

⇒ Jv ⋅dsS∫∫ = − ∂

∂tρvV∫∫∫ ⇒ I = − ∂

∂tQ

Jv ⋅ds

S∫∫ = 0⇒ 0ii I =∑

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Complex permittivity (1) • Modified Ampere’s law

–  Ampere’s law –  Constitutive relation –  Current

–  Modified Ampere’s law

•  Complex permittivity •  Loss tangent

jω∇× = +H D Jε=D E

J = Jcconductivity

Jc=σE

+ Ji impressedgiven source

⇒∇× H = jωεE+ Jc + Ji⇒∇× H = jωεE+σE+ Ji

⇒∇× H = jω ε − jσω

⎛⎝⎜

⎞⎠⎟

εc

E+ Ji

| | jc c

j j e δσε ε ε ε εω

−′ ′′= − = − =

tan ( )δ ε ε σ ωε′′ ′= =

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Complex permittivity (2) • Maxwell’s equations in conducting media

–  Differential equations

–  Boundary conditions

–  Continuity relation

∇× E = − jωµ H∇× H = jωεc

E+ Ji∇⋅ D = ρi∇⋅ B = 0

n12 × ( E2 − E1) = 0n12 ⋅(εc2

E2 − εc11E1) = ρis

n12 × ( H2 − H1) = J is

n12 ⋅(µ2H2 − µ1

H1) = 0

i isjωρ∇⋅ = −J

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Electromagnetic potentials (1) •  Fields radiated in free space by a charge

distribution and current distribution (recall that and are related )

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Vt

µ= ∇×

∂= −∇ −∂

H A

AE

V (R,t)

retartedpotential

= 14πε

ρv ( t − ′R vp

retardation due to causality

)

′R′V

∫∫∫ d ′v

A(R,t) = µ4π

Jv (t − ′R vp )

′R′V

∫∫∫ d ′v

( , )v tJ r( , )v tρ r

v v tρ∇⋅ = −∂ ∂Jvρ vJ

83 10pv c m s= = ×

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Electromagnetic potentials (2) •  Frequency domain fields radiated by a charge

distribution and current distribution (recall that and are related )

⇒H = 1

µ∇× A

E = −∇ V − jω A

V (R) = 14πε

ρve− jk ′R

′R′V

∫∫∫ d ′v

A(R) = µ4π

Jve− jk ′R

′R′V

∫∫∫ d ′v

Jv

ρv (r)

2 wavenumberp

kvω π

λ= = −

ρv ∇⋅ Jv = − jω ρv

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Radiation from a short dipole (1) •  Consider an electric dipole •  Its time derivative results in

current

• When then is uniform •  Physically, the dipole is

represented by two wires •  Vector potential

ˆql=p z

ddt

p = Il z ⇒ jω p = Il z

l λ= I

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•  Expressions for the radiated fields

• When the expressions reduce to the expression due to a static dipole with

• Magnetic field is present only in the dynamic case •  No electric field is in the direction, hence the

field is TM to z •  The short dipole is one of the simplest antennas

⇒

0ω →( )p Il jω=

ϕ

η0 =µ0

ε0

= 120π

characteristic impedance of free space

Radiation from a short dipole (2)

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•  Far field approximation: or –  The terms can be neglected –  Far field approximation fields

–  Characteristics of far field fields of ANY antenna §  Relation between E and H is simply through the impedance

of free space similar to the case of transmission lines §  The phase behavior is like for TLs with z replaced by R §  The field decays but only as (compare to decay

for a static charge and decay for an electric dipole)

R λ kR12 3~1 ( ) ,1 ( )kR kR

1 R 21 R31 R

Radiation from a short dipole (3)