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Electrodynamics – Maxwell’s Equations in
Vacuum
Faraday’s law and electric field excited by magnetic field
Displacement current and magnetic field excited electric field
Maxell’s equations in vacuum
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Electron Acceleration and Time-Varying Current
• Accelerated electron creates time-varying current, timevarying current creates time-varying magnetic field, timevarying magnetic field will create electric field vortex
• Why?
• Explanation – isolated natural system tend to eliminate
any disturbance, with time evolution towards its eigen
state: once a time-varying magnetic field is established, a
electric field vortex will be induced, to establish yet
another magnetic field with an opposite change towards a
cancellation of the total field, hence Faraday’s law holds

t
 
 
 B  ds   E  dl  0
s
l


B
 E  
t
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Electric-Magnetic Field Coupling Caused by
Time-Varying Source
• Accelerated charge causes a time-varying charge distribution,
leading to a time-varying current; from Ampere’s law, time-varying
magnetic field exists; from Faraday’s law, electric vortex exists
• Therefore, not only electric field can be generated in its divergence
form by the static charge distribution, it can also be generated in its
curl form by the “temporary” charge distribution which
• Indicating that the electric field is brought in by both its divergence
and curl, following the Helmholtz theorem, the electric field is
complete
• Hence, time-varied magnetic field excites electric field, electric and
magnetic fields become coupled; however, such coupling is unidirectional
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Illustration of Relation between Electric and
Magnetic Fields
Static
charge
Div.
Time-varying charge
distribution (time-varying
current, temporary charge)
Moving charge, static charge
distribution (constant current)
Div.
Curl
Curl
Static Electric
Field
Time-varying
electric field
Static Magnetic
Field
Time-varying
magnetic field
Curl


J  
t
Charge conservation law
Gauss’s law
 
E 
0

B  0
Ampere’s law
  B  0 J
Faraday’s law


B
 E  
t
(Derived from Coulomb’s law)
(Derived from Biot-Savert’s law)
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Maxwell’s Equations
• Contradiction between Ampere’s law and the charge
conservation law


    B  0  J

0


 0

t
• Maxwell mended Ampere’s law, solved the problem


 


E
    B   0 (  J 
)
  B   0 J   0 0
t
t


 
Gaussian law is called   E 
0

0
0
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Introduction of Displacement Current
• Significance – likes a current, the time-varying electric field can
generate magnetic field
• Hence the time-varying rate of the electric displacement vector is
equivalent to a current, named as the displacement current; the
conventional current caused by the moving charge is then called
the conduction current to make a difference
• Not only time-varying magnetic field, equivalent to temporary
charges, can excite electric field, time-varying electric field,
equivalent to a “temporary” current (i.e., the displacement current),
can also excite magnetic field
• Finally, electric and magnetic fields are fully coupled through mutual
excitation
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Significance of Maxwell’s Equations
• Therefore
– 1. accelerated charge creates time-varying magnetic field in its
neighborhood (through the time-varying conduction current and
following Ampere’s law);
– 2. time-varying magnetic field excites electric field in its
neighborhood (through Faraday’s law)
– 3. time-varying electric field excites magnetic field in its
neighborhood (through Ampere’s law)
• Step 2 and 3 form a sustainable loop, and make electricmagnetic fields propagation! Such an electric-magnetic
non-local oscillation is named as the electromagnetic
wave
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Maxwell’s Equations in Vacuum


B
 E  
t



E
  B   0 J   0 0
t

  E   / 0

B 0
Faraday’s law
Maxwell modified Ampere’s law,
derived from Biot-Savert’s law
Gaussian law, derived from Coulomb’s law
Gaussian law, derived from Biot-Savert’s law
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Home Work 2
1. Give a set of consistent electric and magnetic field
expressions for an electron, the observer can either stay still
or move at a constant speed.
2. Derive the gravitational wave equation.
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