Maxwell's Equations & Displacement Current - Lecture 19

1. 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

2. 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

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

4. Maxwell’s Equations & Displacement Current Usual approximations

5. 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 Fields and Waves I

6. Maxwell’s Equations – static models For Electrostatics For Magnetostatics 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. 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 First introduced by Maxwell in 1873 Fields and Waves I

9. Maxwell’s Equations & Displacement Current Displacement current

10. 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) 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 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 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: 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= 0 no conduction between the plates The electric field between the capacitors d :spacing between the plates 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 Fields and Waves I

16. 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 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. Fields and Waves I

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

22. 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. Fields and Waves I

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

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

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

33. Example – Displacement Current Fields and Waves I

34. Example – Displacement Current Fields and Waves I

35. 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. 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. 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 Fields and Waves I

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