1. Fields and Waves I Lecture 19 K. A. Connor Y. Maréchal
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
Welcome to Fields and Waves I
Before I start, can those of you with pagers and cell phones please turn them off? Thanks.

2. J. Darryl Michael – GE Global Research Center, Niskayuna, NY
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.
28 November 2018
Fields and Waves I

3. Overview Usual approximations of Maxwell’s equations
Displacement Current
Continuity Equation and boundary conditions
Quasi-Statics approximation
Conductors vs. Dielectrics
28 November 2018
Fields and Waves I

4. Maxwell’s Equations & Displacement Current
Usual approximations

5. Usual models in physics
Maxwell’s 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
Maxwell’s equations
Can be simplified for each model
28 November 2018
Fields and Waves I

6. Maxwell’s Equations – static models
For Electrostatics
For Magnetostatics
28 November 2018
Fields and Waves I

7. 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.
28 November 2018
Fields and Waves I

8. Full Maxwell’s Equations
Added term in curl H equation for time varying electric field that gives a magnetic field.
For Electromagnetism
For high frequencies
First introduced by Maxwell in 1873
28 November 2018
Fields and Waves I

9. Maxwell’s Equations & Displacement Current

10. Ampere’s Law – Curl H Equation
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)
28 November 2018
Fields and Waves I

11. Displacement Current Total current Conduction current density
Displacement current density
Connection between electric and magnetic fields under time varying conditions
28 November 2018
Fields and Waves I

12. 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
28 November 2018
Fields and Waves I

13. Example: Parallel Plate Capacitor
The wire is considered as a perfect conductor
I1d = 0 +
From circuit theory: -
Total current in the wire:
28 November 2018
Fields and Waves I

14. Example: Parallel Plate Capacitor
The dielectric is considered as perfect (zero conductivity)
Electrical charges can’t move physically through a perfect dielectric medium
I2c= no conduction between the plates
The electric field between the capacitors
d :spacing between the plates
28 November 2018
Fields and Waves I

15. 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
28 November 2018
Fields and Waves I

16. Order of magnitude Consider a conducting wire
Conductivity = 2.107S/m
Relative permittivity = 1
Current = sin(wt) A
w = 109 rad/s
Find the value of the displacement current
Phase quadrature
9 order of magnitude
Negligible in conductors
28 November 2018
Fields and Waves I

17. Maxwell’s Equations & Displacement Current
Maxwell’s equations, boundary conditions

18. 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.
28 November 2018
Fields and Waves I

19. Fully connected fields
Sources
Material property
Material property
Maxwell’s equations are fully coupled.
28 November 2018
Fields and Waves I

20. 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
28 November 2018
Fields and Waves I

21. 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.
28 November 2018
Fields and Waves I

22. Continuity Equation
Differential and integral forms of the Continuity Equation (Equation for Charge and Current Conservation) I2 I3
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.
28 November 2018
Fields and Waves I

23. Maxwell’s equations are fully coupled.
Summary
Maxwell’s equations are fully coupled.
28 November 2018
Fields and Waves I

24. 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
28 November 2018
Fields and Waves I

25. Boundary Conditions Case 1: REGIONS 1 & 2 are DIELECTRICS (Js = 0)
Material 1
dielectric
Material 2
dielectric
28 November 2018
Fields and Waves I

26. Boundary Conditions Case 2: REGIONS 1 is a DIELECTRIC
REGION 2 is a CONDUCTOR, D2 = E2 =0
Material 1
Material 2
conductor
28 November 2018
Fields and Waves I

27. Maxwell’s Equations & Displacement Current
Quasi static

28. 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.
28 November 2018
Fields and Waves I

29. 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 ?
28 November 2018
Fields and Waves I

30. 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.
28 November 2018
Fields and Waves I

31. A quasi-static approach
The total displacement current between the capacitor plates
Using phasor notation for the voltage and current
28 November 2018
Fields and Waves I

32. 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.
28 November 2018
Fields and Waves I

33. Example – Displacement Current
28 November 2018
Fields and Waves I

34. Example – Displacement Current
28 November 2018
Fields and Waves I

35. A quasi-static approach2 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.
28 November 2018
Fields and Waves I

36. 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.
28 November 2018
Fields and Waves I

37. 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
28 November 2018
Fields and Waves I

38. Conductors vs. Dielectrics
The material will behave mostly like a conductor when
Loss tangent of the material.
28 November 2018
Fields and Waves I