Schedule. Time Varying electromagnetic fields (1 Week) 6.1 Overview 6.2 Faraday s law (6.2.1 only) 6.3 Maxwell s equations

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1 chedule Time Varying electrmagnetic fields (1 Week) 6.1 Overview 6.2 Faraday s law (6.2.1 nly) 6.3 Maxwell s equatins Wave quatin (3 Week) 6.5 Time-Harmnic fields 7.1 Overview 7.2 Plane Waves in Lssless Media 7.3 Plane Waves in Lssy Media 7.5 Flw f lectrmagnetic Pwer and the Pynting Vectr 7.6 Nrmal Incidence f Plane Waves at Plane undaries Intr.1

2 Intr.2 VI. TIM-VARYING LCTROMAGNTIC FILD 6.1 quatins fr static electric and magnetic fields J H D ρ v with the fllwing relatin between D,,,H: H H D µ µ µ ε ε ε r r

3 6.2 Faraday s Law: The first mdificatin in case f time-varying electrmagnetic fields is due t Faraday s Law, namely, if there is time varying change in the magnetic flux linking a clsed circuit, an e.m.f. will be induced in the circuit which is prprtinal t the rate f change f flux. C (t) C. dl. d t where is a surface bunded by the clsed line C. Intr.3

4 Nte that the negative sign means the induced e.m.f. ppses the directin f change f flux. In differential frm, t ddy currents: The time-varying magnetic field in a magnetic cre induces an emf, s that an hmic eddy current is induced arund the axis f the cre. J Intr.4

5 6.3 Displacement current: In time-varying fields, we must cnsider the current cntinuity equatin invlving the free current density J. Charge cnservatin requires that the net current leaving any clsed surface is equal t the rate f decrease f ttal charge enclsed within the surface, i.e. J. d d dt V ρ dv v V ρ v t dv In differential frm:.j ρ t v J Q Intr.5

6 We nw shw that Ampere s law in the static frm must be mdified in rder t satisfy the current cntinuity equatin. Taking the divergence f bth sides f the static Ampere s law gives:.( H).J ince fr any vectr A ( A) 0 it fllws that the static Ampere s law leads t which vilates the current cntinuity equatin.. J 0 Intr.6

7 T reslve the cnflict, Maxwell prpsed the fllwing Ampere s law in time-varying fields: H J D t D The term is called the displacement current. t We can shw that this frm is cnsistent with the current cntinuity equatin. Taking the divergence f bth sides f the equatin: (. D).( H ). J + t (. D) ρ v. J t t + Intr.7

8 Intr.8 In summary, Maxwell s equatin fr time-varying fields: In pint frm: D D J H v t t ρ In integral frm: + C C d Q d d t I d t d d 0.. D.. D l H. l.

9 xample: uppse the -field in a surce free (i.e. ρ v 0) regin is given by a wave travelling in the z-directin sin( ω t β z )a x Find the value f the H-field present. What must be the value f β s that bth fields satisfy Maxwell s equatins? lutin: ubstituting in Faraday s law, t z sin( ωt βz) a y β cs( ωt βz)a y Intr.9

10 Integrating with time gives: β cs( ω t βx )a y dt β sin( ωt βz) a + C ω y a y In time-varying fields, we can ignre the DC term, s that β H sin( ωt βz) µ ωµ This shws that an assciated time-varying H-field must cexist. a y Intr.10

11 T find the value f β, let s substitute the abve expressin f H int Ampere s law: ε t H z 2 β ωµ β ωµ cs( ωt Integrating with time, we find t be 2 sin( ωt βz) βz)a x β sin( ωt βz)a 2 x ω µ ε Cmparing with the given expressin f β ω µ ε a x Intr.11

12 ND Intr.12

13 The frmula fr vectr prduct in rectangular crdinates: A a A x x x a A y y y a A z z z Intr.13

14 Take the simple example f a parallel plate capacitr with ac current flwing. An ac current flws in the external circuit; but since n current can flw in the dielectric in the capacitr, it appears that the current is nt cntinuus. y including the cncept f a displacement current in the dielectric, the current is nw cntinuus, with the cnductin current J c in the external circuit equal t the displacement current J d in the dielectric. J c J d J c Intr.14

15 Curl in rectangular c-rdinates a x x x a y y y a z z z Intr.15

16 We can write Ampere s Law fr a clsed path in a magnetic material as:. dl I µ where the ttal current enclsed I T is the sum f free current I and magnetizatin current I m. It can be shwn that Therefre I I m M. dl I T I m M. dl µ T H. dl ( H M ) µ + Intr.16

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