Fields and Waves I

1
Fields and Waves I

• Lecture 19
• Maxwells Equations Displacement Current
• K. A. Connor
• Electrical, Computer, and Systems Engineering
  Department
• Rensselaer Polytechnic Institute, Troy, NY
• Y. Maréchal
• Power Engineering Department
• Institut National Polytechnique de Grenoble,
  France

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These Slides Were Prepared by Prof. Kenneth A.
Connor Using Original Materials Written Mostly by
the Following

• Kenneth A. Connor ECSE Department, Rensselaer
  Polytechnic Institute, Troy, NY
• J. Darryl Michael GE Global Research Center,
  Niskayuna, NY
• Thomas P. Crowley National Institute of
  Standards and Technology, Boulder, CO
• Sheppard J. Salon ECSE Department, Rensselaer
  Polytechnic Institute, Troy, NY
• Lale Ergene ITU Informatics Institute,
  Istanbul, Turkey
• Jeffrey Braunstein Chung-Ang University, Seoul,
  Korea

Materials from other sources are referenced where
they are used. Those listed as Ulaby are figures
from Ulabys textbook.
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Overview

• Usual approximations of Maxwells equations
• Displacement Current
• Continuity Equation and boundary conditions
• Quasi-Statics approximation
• Conductors vs. Dielectrics

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Maxwells Equations Displacement Current

• Usual approximations

5
Usual models in physics
Maxwells equations Models all electromagnetism

• Models can vary according to

• Time
• Steady state
• Phasor
• Transient
• Frequency
• No
• Low
• High

• Material
• Linear / non linear
• Isotropic / anisotropic
• Hysteretic
• Scale
• Microscopic
• Usual

Maxwells equations

• Can be simplified for each model

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Maxwells Equations static models
For Electrostatics
For Magnetostatics
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Maxwells Equations quasi static models
For Magneto quasi-statics
Added term in curl E equation for time varying
current or moving path that gives an electric
field from a time-varying magnetic field.
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Full Maxwells Equations
Added term in curl H equation for time varying
electric field that gives a magnetic field.
For Electromagnetism
First introduced by Maxwell in 1873
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Maxwells Equations Displacement Current

• Displacement current

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Displacement Current
Amperes Law Curl H Equation
(quasi) Static field
Time varying field
Displacement current density
Integral Form of Amperes Law for time varying
fields
Displacement current
IC Conduction Current A linked to a
conductivity property
Electric Flux Density (Electric Displacement)
in C/unit area
Conduction Current Density (in A/unit area)
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Displacement Current
Total current
Conduction current density
Displacement current density
Connection between electric and magnetic fields
under time varying conditions
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Example Parallel Plate Capacitor
What are the meanings of these currents ?
Imaginary surface S1


Imaginary surface S2
E-Field
- - - - - - - - - - - - - - - - - - - - - - - - -
- -
-
S1cross section of the wire
S2cross section of the capacitor
I1c, I1d conduction and displacement currents
in the wire I2c, I2d conduction and
displacement currents through the capacitor
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Example Parallel Plate Capacitor
The wire is considered as a perfect conductor

I1d 0

From circuit theory
-
Total current in the wire
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Example Parallel Plate Capacitor
The dielectric is considered as perfect (zero
conductivity)
Electrical charges cant move physically through
a perfect dielectric medium
I2c 0 no conduction between the plates
The electric field between the capacitors
d spacing between the plates
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Example Parallel Plate Capacitor
The displacement current I2d
Displacement current doesnt carry real charge,
but behaves like a real current If wire has a
finite conductivity s then both wire and
dielectric have conduction AND displacement
currents
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Order of magnitude

• Consider a conducting wire
• Conductivity 2.107S/m
• Relative permittivity 1
• Current 2 . 10-3 sin(wt) A
• w 109 rad/s
• Find the value of the displacement current

Phase quadrature 9 order of magnitude Negligible
in conductors
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Maxwells Equations Displacement Current

• Maxwells equations, boundary conditions

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Maxwells Equations
Note that the time-varying terms couple electric
and magnetic fields in both directions. Thus, in
general, we cannot have one without the other.
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Fully connected fields
Sources
Material property
Material property
Maxwells equations are fully coupled.
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Continuity Equation
Begin by taking the divergence of Amperes Law
where we have used the vector identity that the
divergence of the curl of any vector is always
equal to zero. Now from Gauss Law,
or
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Continuity Equation integral form
Now, integrate this equation over a volume.
Ulaby
From the divergence theorem, the left hand side
is
For a fixed volume, we can move the derivative
outside the integral on the right to obtain the
final form of this equation.
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Continuity Equation
Differential and integral forms of the Continuity
Equation (Equation for Charge and Current
Conservation)
I3
I2
For statics, the current leaving some volume must
sum to zero If the charge is time varying, sum
of currents is equal to this variation.
I1
I4
I5
A general form of the Kirchoff Current Law.
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Summary
Maxwells equations are fully coupled.
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Boundary conditions
Boundary conditions derived for electrostatics
and magnetostatics remain valid for
time-varying fields
- For instance, tangential Components of E
w
Material 1
h ltlt w
h
Material 2
Note
If region 2 is a conductor E1t 0
Outside conductor E and D are normal to the
surface
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Boundary Conditions
Case 1
REGIONS 1 2 are DIELECTRICS (Js 0)
Material 1 dielectric
Material 2 dielectric
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Boundary Conditions
REGIONS 1 is a DIELECTRIC REGION 2 is a
CONDUCTOR, D2 E2 0
Case 2
Material 1
Material 2 conductor
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Maxwells Equations Displacement Current

• Quasi static

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A quasi-static approach
Because all four equations are coupled, in
general, we must solve them simultaneously. We
will see a general way to do this in the next
lecture, which will lead us to electromagnetic
waves. However, we will first look at the
coupled equations as a perturbation of what we
have done so far in electrostatics and
magnetostatics.
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Example
A parallel plate capacitor with circular plates
and an air dielectric has a plate radius of 5 mm
and a plate separation of d10 mm. The voltage
across the plates is where

• Find D between the plates.
• Determine the displacement current density,
  ?D/?t.
• c. Compute the total displacement current, ?
  ?D/?t ? ds , and compare it with the capacitor
  current, I C dV/dt.
• d. What is H between the plates?
• e. What is the induced emf ?

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A quasi-static approach
The electric field for a parallel plate capacitor
driven by a time-varying source is
The time-varying electric field now produces a
source for a magnetic field through the
displacement current . We can solve for the
magnetic field in the usual manner.
0
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A quasi-static approach
The total displacement current between the
capacitor plates
Using phasor notation for the voltage and current
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A quasi-static approach
Applying Amperes Law to a circular contour with
radius r lt a, the fraction of the displacement
current enclosed is
Amperes Law then gives us
Thus, we now have both electric and magnetic
fields between the plates.
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Example Displacement Current
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Example Displacement Current
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A quasi-static approach
2
3
1
?
In general, we should now use this magnetic field
to find a correction to the electric field by
plugging it into Faradays Law. However, under
what we call quasi-static conditions, we only
need to find this first term.
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Validity domain of quasi-static approach
Maxwells Equations.
Need a simultaneous solution for the electric and
magnetic fields
Lead to a wave equation identical in form to the
wave equation found for transmission lines
Quasi static approach
Valid if the system dimensions are small compared
to a wavelength. real meaning of low
frequencies. There is a reasonably complete
derivation of this condition in Unit 9 of the
class notes.
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Conductors vs. Dielectrics
The analysis of the capacitor under time-varying
conditions assumed that the insulator had no
conductivity. If we generalize our results to
include both and we will have both a
conduction and a displacement current.
The material will behave mostly like a dielectric
when
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Conductors vs. Dielectrics
The material will behave mostly like a conductor
when
Loss tangent of the material.