Fields and Waves I Lecture 19 Maxwells Equations

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 Fields and Waves I

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. 11/25/2020 Fields and Waves I 2

Overview § Usual approximations of Maxwell’s equations § Displacement Current § Continuity Equation and boundary conditions § Quasi-Statics approximation § Conductors vs. Dielectrics 11/25/2020 Fields and Waves I 3

Maxwell’s Equations & Displacement Current Usual approximations Fields and Waves I

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 11/25/2020 Fields and Waves I 5

Maxwell’s Equations – static models cs i t ta s r. E o F ro t c le to ne g a cs i t sta r. M o F 11/25/2020 Fields and Waves I 6

Maxwell’s Equations – quasi static models cs i t ta -s i s ua q eto n g Fo a M r Added term in curl E equation for time varying current or moving path that gives an electric field from a time-varying magnetic field. 11/25/2020 Fields and Waves I 7

Full Maxwell’s Equations sm i t e gn a m r Fo E Added term in curl H equation for time varying electric field that gives a magnetic field. ro t c le First introduced by Maxwell in 1873 11/25/2020 Fields and Waves I 8

Maxwell’s Equations & Displacement Current Displacement current Fields and Waves I

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) 11/25/2020 Fields and Waves I 10

Displacement Current Total current Conduction current density Displacement current density Connection between electric and magnetic fields under time varying conditions 11/25/2020 Fields and Waves I 11

Example: Parallel Plate Capacitor What are the meanings of these currents ? + - Imaginary surface S 1 ++++++++++++++ Imaginary surface S 2 S 1=cross section of the wire -------------- E-Field S 2=cross section of the capacitor I 1 c, I 1 d : conduction and displacement currents in the wire I 2 c, I 2 d : conduction and displacement currents through the capacitor 11/25/2020 Fields and Waves I 12

Example: Parallel Plate Capacitor The wire is considered as a perfect conductor I 1 d = 0 + From circuit theory: - Total current in the wire: 11/25/2020 Fields and Waves I 13

Example: Parallel Plate Capacitor The dielectric is considered as perfect (zero conductivity) Electrical charges can’t move physically through a perfect dielectric medium I 2 c= 0 no conduction between the plates The electric field between the capacitors d : spacing between the plates 11/25/2020 Fields and Waves I 14

Example : Parallel Plate Capacitor The displacement current I 2 d 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 11/25/2020 Fields and Waves I 15

Order of magnitude § Consider a conducting wire • • Conductivity = 2. 107 S/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 11/25/2020 Fields and Waves I 16

Maxwell’s Equations & Displacement Current Maxwell’s equations, boundary conditions Fields and Waves I

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. 11/25/2020 Fields and Waves I 18

Fully connected fields Sources Material property Maxwell’s equations are fully coupled. 11/25/2020 Fields and Waves I 19

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 11/25/2020 Fields and Waves I 20

Continuity Equation : integral form Now, integrate this equation over a volume. From the divergence theorem, the left hand side is Ulaby For a fixed volume, we can move the derivative outside the integral on the right to obtain the final form of this equation. 11/25/2020 Fields and Waves I 21

Continuity Equation Differential and integral forms of the Continuity Equation (Equation for Charge and Current Conservation) I 3 I 2 For statics, the current leaving some volume must sum to zero I 1 I 4 I 5 11/25/2020 If the charge is time varying, sum of currents is equal to this variation. A general form of the Kirchoff Current Law. Fields and Waves I 22

Summary Maxwell’s equations are fully coupled. 11/25/2020 Fields and Waves I 23

Boundary conditions derived for electrostatics and magnetostatics remain valid for time-varying fields: - For instance, tangential Components of E w Material 1 h h << w Material 2 Note: If region 2 is a conductor E 1 t = 0 Outside conductor E and D are normal to the surface 11/25/2020 Fields and Waves I 24

Boundary Conditions Case 1: REGIONS 1 & 2 are DIELECTRICS (Js = 0) Material 1 dielectric Material 2 dielectric 11/25/2020 Fields and Waves I 25

Boundary Conditions Case 2: REGIONS 1 is a DIELECTRIC REGION 2 is a CONDUCTOR, D 2 = E 2 =0 Material 1 Material 2 conductor 11/25/2020 Fields and Waves I 26

Maxwell’s Equations & Displacement Current Quasi static Fields and Waves I

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. 11/25/2020 Fields and Waves I 28

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 a. Find D between the plates. b. 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 d. V/dt. d. What is H between the plates? e. What is the induced emf ? 11/25/2020 Fields and Waves I 29

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 11/25/2020 Fields and Waves I 30

A quasi-static approach The total displacement current between the capacitor plates Using phasor notation for the voltage and current 11/25/2020 Fields and Waves I 31

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. 11/25/2020 Fields and Waves I 32

Example – Displacement Current 11/25/2020 Fields and Waves I 33

Example – Displacement Current 11/25/2020 Fields and Waves I 34

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. 11/25/2020 Fields and Waves I 35

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. 11/25/2020 Fields and Waves I 36

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 11/25/2020 Fields and Waves I 37

Conductors vs. Dielectrics The material will behave mostly like a conductor when Loss tangent of the material. 11/25/2020 Fields and Waves I 38