Lecture 6: Maxwell s Equations, Boundary Conditions.

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1 Whites, EE 382 Lecture 6 Page 1 of 7 Lecture 6: Maxwell s Equations, Boundar Conditions. In the last four lectures, we have been investigating the behavior of dnamic (i.e., time varing) electric and magnetic fields. In the previous lecture, we discussed Maxwell s law (i.e., Ampère s law with the added displacement current term). For a capacitor, we found that displacement current completes the path of the current where conduction ends. Notice in the definition of capacitor displacement current dv Id = C that a time varing electric field in space is producing a conduction current, which subsequentl produces a time varing magnetic field. Amazing! Conversel, in Farada s law dψ m emf = a magnetic field effect produces an emf (a source voltage). This is a beautiful dualit between these two effects: 212 Keith W. Whites

2 Whites, EE 382 Lecture 6 Page 2 of 7 Maxwell s law: Farada s law: E t B t B t E t Since a time varing electric field produces a magnetic force and vice versa, we now speak of an electro-magnetic field, rather than electric and magnetic fields separatel. Because of this dualit, we will see shortl that electromagnetic signals can propagate as waves! It is because of this fantastic circumstance that there exists light, radio communications, satellite remote sensing, RADAR, fiber optic networks, CAT scans, etc. Maxwell s Equations The laws of classical electromagnetics can be neatl summarized into a concise collection called Maxwell s equations. In point form, Maxwell s equations read:

3 Whites, EE 382 Lecture 6 Page 3 of 7 B t E t = Farada s law Dt H t = + J t Ampère s law Dt = ρv t Gauss law, I B t = Gauss law, II In integral form, Maxwell s equations read d E t dl = B t ds cs Farada s law sc d H t dl = D t ds J t ds cs + Ampère s law sc sc Dt ds= ρv tdv Gauss law, I sv vs B t ds = Gauss law, II sv In addition, the continuit equation (conservation of charge) reads in point form: ρv J t = t and in integral form d J t ds = ρv t dv sv vs

4 Whites, EE 382 Lecture 6 Page 4 of 7 These laws describe all of classical (i.e., non-quantum mechanical) electromagnetism. Maxwell s equations are an amazingl short and concise set of equations. However, these equations are usuall difficult to solve for real-world problems. Interdependent Equations As it turns out, not all of these equations are independent for dnamic fields. For example, if we take the divergence of Ampère s law: D ( H) = = J + we find that it reduces to ρ J = v which is the continuit equation. There are other examples of interdependencies among Maxwell s equations for dnamic fields. Constitutive Equations For dnamic electromagnetism, the constitutive equations are still applicable:

5 Whites, EE 382 Lecture 6 Page 5 of 7 Dt = ε Et B t = μh t J t = σ E t However, for sinusoidal stead state problems, the material parameters are often a function of frequenc. That is ε ε ω μ μ ω σ = σ ω = = Boundar Conditions The boundar conditions for dnamic EM fields remain the same as were derived earlier in EE 381 for static fields: â 21 Tangential components aˆ 21 E2 t E1 t = aˆ 21 H2 t H1 t = Js = K Normal components aˆ 21 D2 t D1 t = ρs t aˆ 21 B2 t B1 t =

6 Whites, EE 382 Lecture 6 Page 6 of 7 Example N6.1: The electric field E( x, t) = aˆ zeocos( ωt+ x) V/m exists in free space. Determine H ( t ) and consistent with this electric field and all of Maxwell s equations. B From Farada s law: E =. For this example aˆ ˆ ˆ x a az Ez B ( = ) ( = ) = aˆ = x z x E Therefore, B Ez = = Eo cos( ωt + x) x x = Eo 1sinωt+ x ωt+ x x B or = Eo sin( ωt+ x) So, B = Eocos( ωt+ x) + C ω z The constant C cannot be a function of time. It is often taken as zero for dnamical problems if there are no sources present for constant magnetic fields. Therefore, H t = aˆ Eocos( ωt+ x) [A/m] ωμ

7 Whites, EE 382 Lecture 6 Page 7 of 7 D From Ampere s law: H = + J. Then aˆ ˆ ˆ x a az H E ( = ) ( = ) = aˆ z = ε x z x H Therefore, Eocos( ωt+ x) = ε Eocos( ωt+ x) x ωμ = Ez or ωt+ x ωt+ x Eo( 1sin ) ( ωt+ x) = εeo( 1sin ) ( ωt+ x) ωμ x = = ω So that 2 ε ω =± ω μ ε [rad/m] ωμ = or D ( ε E) E ε z = = =. consistent with Maxwell s z equations. B B = =. consistent with Maxwell s equations.

EELE 3332 Electromagnetic II Chapter 9. Maxwell s Equations. Islamic University of Gaza Electrical Engineering Department Dr.

EELE 3332 Electromagnetic II Chapter 9 Maxwell s Equations Islamic University of Gaza Electrical Engineering Department Dr. Talal Skaik 2013 1 Review Electrostatics and Magnetostatics Electrostatic Fields