Electromagnetism. Electromagnetic Waves : Topics. University of Twente Department Applied Physics. First-year course on

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1 Universit of Twente Department Applied Phsics First-ear course on lectromagnetism lectromagnetic Waves : Topics F.F.M. de Mul 1

2 Presentations: lectromagnetism: Histor lectromagnetism: lectr. topics lectromagnetism: Magn. topics lectromagnetism: Waves topics Capacitor filling (complete) Capacitor filling (partial) Divergence Theorem -field of a thin long charged wire -field of a charged disk -field of a dipole -field of a line of dipoles -field of a charged sphere -field of a polaried object -field: field energ lectromagnetism: integrations lectromagnetism: integration elements Gauss Law for a clindrical charge Gauss Law for a charged plane Laplace s and Poisson s Law B-field of a thin long wire carring a current B-field of a conducting charged sphere B-field of a homogeneousl charged sphere

3 lectromagnetic Waves Contents: 1. Gauss and Farada s Laws for (D)-field. Gauss and Mawell s Laws for B (H)-field 3. Mawell s quations and The Wave quation 4. Harmonic olution of the Wave quation 5. Plane waves 1) orientation of field vectors ) comple wave vector 3) B - correspondence 6. The Ponting Vector and Intensit 3

4 1. Gauss and Farada s Laws for Gauss Law : V dv 1 1 d dv div. dv div V Div = micro-flu per unit of volume V d B c V ind Farada s Law : d rot Bd db c n dl rot d Rot = micro-circulation per unit of area 4

5 . Gauss and Mawell s Laws for B B V dv Gauss Law : Bd div B V divb. dv Mawell s Fi for Ampere s Law : j L L rot rot Bdl jd B j H j f dd d d d 5

6 3. Mawell s quations and Wave quation In vacuum ( = and j = ): db B db d... ( H) d ( B)... d D ( ) d } H j f dd d This is a 3-Dimensional Wave quation General form: A 1 A... v = velocit: v t A v 1 v = m/s = light velocit 6

7 4. Harmonic olution of thewave quation d ma have 3 components: Choose X-ais // = = ( Polariation direction = X-ais). Question: Does a plane-wave epression for satisf the wave equation? = ep{i (t-k)} : Propagation = amplitude + polariation vector; +Z-ais = direction of propagation NB. has a similar form for all -values!! (like rolling sea waves) 7

8 4. Harmonic olution of the Wave quation: Plane Waves d ma have 3 components: Choose X-ais // = = (Polariation direction = X-ais). Does a plane-wave epression for satisf the wave equation? = ep{i (t-k)} : = amplitude + polariation vector; +Z-ais = direction of propagation Propagation Insertion into wave equation: t k = =( /c ) k = / c = / k = wave number ; = wavelength 8 (in 3D-case: k = wave vector )

9 5. Plane waves (1): orientation of fields uppose: // -ais;.. propagation // +-ais: k // e.. = ep i (t-k).. = = Question: what is direction of B? H Propagation Remember: div A A A A A In div and rot: onl /, due to: ep i (ωt-k)! (1) div = () div B = -ik e. = -ik e.h =,D e H,B e 9

10 5. Plane waves (1): orientation of fields uppose: // -ais;.. propagation // +-ais: k // e.. = ep i (t-k).. = = div =,D e div H = H,B e What is mutual direction of,d and H,B? In div and rot: onl /, due to: ep i (ωt-k)! Remember: rot X X H e X Propagation e X e X (3) rot = - db/ -ik e = -ih,d H,B (4) rot H = j f + dd/ j f = -ik e H = +i,d H,B 1

11 5. Plane waves (1): orientation of fields uppose: // -ais;.. propagation // +-ais: k // e.. = ep i (t-k) Propagation H (1) div = () div B = (3) rot = - db/ (4) rot H = j f + dd/ j f = Consequences: (1)+(): and H e (3)+(4): H If chosen // -ais, then H,B // -ais H,B also harmonic: B = B ep i (t-k) 11

12 5. Plane waves (): comple wave vector uppose: // -ais;.. propagation // +-ais: k // e.. = ep i (t-k) Propagation (3) -ik e = -ih (4) -ik e H = +i H ik e H ik e ( ke ) i ( i ). and with: e e = - ik =(+i). k comple: k = k Re + ik Im ik ( i ). ep (-ik) = ep (-ik Re ). ep (k Im ) } harmonic } k Im < : absorption > : amplification ( laser ) 1

13 5. Plane waves (3): B- correspondence uppose: // -ais; B // -ais;.. propagation // +-ais: k // e.. = ep i (t-k).. B = B ep i (t-k) Farada: rot = - db/ e e e t. e B. e. e Y - components are B onl Propagation t B k B Result : B k : cb Mawell (for j=): rot H = dd/: similar result Mawell equations are (partl) redundant. 13

14 { { 6. The Ponting vector and Intensit Definition (for free space) : = H Direction of : // k k H = (H) = H() - (H) = 1 B = H(-dB/) - (j f +dd/) j f t t Integrate over wave volume V (with surface A) and appl Divergence Theorem A da v ( ) Outflu of energ [J/s] = [W] dv 1 d v B { Loss of lectromagnetic field energ [J/s] dv v ( j f ) dv Joule heating losses [J/s] = energ outflu per m = Intensit [W/m ] 14

15 Universit of Twente Department Applied Phsics First-ear course on lectromagnetism The end lectromagnetic Waves : Topics F.F.M. de Mul 15