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Fields and Waves I. Lecture 19 Maxwell’s 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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fields and waves i

Fields and Waves I

Lecture 19

Maxwell’s 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

slide2
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 Ulaby’s textbook.

Fields and Waves I

overview
Overview
  • Usual approximations of Maxwell’s equations
  • Displacement Current
  • Continuity Equation and boundary conditions
  • Quasi-Statics approximation
  • Conductors vs. Dielectrics

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usual models in physics
Usual models in physics

Maxwell’s equations Models all electromagnetism

  • Time
    • Steady state
    • Phasor
    • Transient
  • Frequency
    • No
    • Low
    • High
  • Models can vary according to
  • Material
    • Linear / non linear
    • Isotropic / anisotropic
    • Hysteretic
  • Scale
    • Microscopic
    • Usual

Maxwell’s equations

  • Can be simplified for each model

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maxwell s equations static models
Maxwell’s Equations – static models

For Electrostatics

For Magnetostatics

Fields and Waves I

slide7
Maxwell’s 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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slide8
Full Maxwell’s 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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displacement current
Displacement Current

Ampere’s Law – Curl H Equation

(quasi) Static field

Time varying field

Displacement current density

Integral Form of Ampere’s 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 current1
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
Example: Parallel Plate Capacitor

What are the meanings of these currents ?

Imaginary surface S1

++++++++++++++++++++++++++++

+

Imaginary surface S2

E-Field

- - - - - - - - - - - - - - - - - - - - - - - - - - -

-

S1=cross section of the wire

S2=cross 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 capacitor1
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 capacitor2
Example: Parallel Plate Capacitor

The dielectric is considered as perfect (zero conductivity)

Electrical charges can’t 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 capacitor3
Example : Parallel Plate Capacitor

The displacement current I2d

Displacement current doesn’t carry real charge, but behaves like a real current

If wire has a finite conductivity σ then both wire and dielectric have conduction AND displacement currents

Fields and Waves I

order of magnitude
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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maxwell s equations displacement current2

Maxwell’s Equations & Displacement Current

Maxwell’s equations, boundary conditions

maxwell s equations
Maxwell’s 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
Fully connected fields

Sources

Material property

Material property

Maxwell’s equations are fully coupled.

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continuity equation
Continuity Equation

Begin by taking the divergence of Ampere’s 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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slide21
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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slide22
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
Summary

Maxwell’s equations are fully coupled.

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boundary conditions
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 << 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 conditions1
Boundary Conditions

Case 1:

REGIONS 1 & 2 are DIELECTRICS (Js = 0)

Material 1

dielectric

Material 2

dielectric

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boundary conditions2
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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a quasi static approach
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
Example

A parallel plate capacitor with circular plates and an air dielectric has a plate radius of 5 mm and a plate separation of d=10 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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slide30
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 approach1
A quasi-static approach

The total displacement current between the capacitor plates

Using phasor notation for the voltage and current

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slide32
A quasi-static approach

Applying Ampere’s Law to a circular contour with radius r < a, the fraction of the displacement current enclosed is

Ampere’s Law then gives us

Thus, we now have both electric and magnetic fields between the plates.

Fields and Waves I

a quasi static approach2
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 Faraday’s 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
Validity domain of quasi-static approach

Maxwell’s 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
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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slide38
Conductors vs. Dielectrics

The material will behave mostly like a conductor when

Loss tangent of the material.

Fields and Waves I

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