Integral Expression of EM Fields Summary

Transcription

1 Integal Expeion of EM Field ummay 5 Integal Expeion of EM Field.doc 08/07/0 5- In tem of tangential and nomal component of E and on a ρ E= jωφ φ φ d + + { jωφ ( nˆ ( nˆ E φ φ ( nˆ E } d ρ = jω φ φ φ d { jωφ ( nˆ E ( nˆ φ φ ( nˆ } d Altenatiely, in tem of tangential component only a E = jωφ φ + φ ρ d + jωφ( n ( n E φ + [( n ] φ d jω = jωφ + φ + φ ρ d + jωφ( n E ( n φ + [( n E ] φ d jω 5-2 We can conclude that field can be uniquely detemined by the ouce inide V and on only., N E, n d, d, V nρ = n n n E ρ =n E It eem that one ha to calculate 6 tem by integation independently. Pactical calculation, on the contaty, i athe taightfowad. Once two type of potential, that i fo electic line cuent and fo magnetic line cuent, ae obtained, ix tem can be deied ytematically, efeing to the fomula uing ecto potential unde Loentz condition.

2 5 Integal Expeion of EM Field.doc 08/07/0 5-2 ow to calculate the field fom engineeing point of iew?? Two type of ouce. (n : nomal into inide V Electic cuent in V and equialent uface electic cuent = n on. 2 Magnetic cuent (non-phyical in V and equialent uface magnetic cuent = E n P n n V E on. E 3 Vecto potential A = φd φd π ( +, = + V π ( V Volume and uface integation. ae calculated. 4 With efeence to the ecto field deied in tem of potential (on the next page, duality i utilized and ix tem ae calculated by two potential A and a well a thei imple deiatie. ρ E = jωφ φ φ d + ρ = jωφ φ φ d + + = ni ρ = ni A = φd = E niρ = ni E = φd A A E = jω A +, = + jω jω jω

3 5 Integal Expeion of EM Field.doc 08/07/ If the obee i in the fa field of the ouce and the uface, futhe implification applie which dipene with deiatie. Fom electic cuent E ( ˆ( ˆ A =jω A A L, 2 ( ˆ A = EA = 20π L 4 Fom magnetic cuent =jω ( ˆ( ˆ E ( ˆ = = 20π L L 5, 6 3 Finally, we get E=jω( Aˆ( A ˆ ( ˆ ( jω L, 2, 3 = ( ˆ ( jωa jω( ˆ( ˆ L 4, 5, 6 Explicitly, field ae gien by E = jω( A ˆ ˆ ( ˆ θ + Aφ φ jω θ φ + φ θˆ = jω( θˆ φˆ ( φˆ θ + φ jω Aθ Aφ θˆ

4 Refeence : 5 Integal Expeion of EM Field.doc 08/07/0 5-4 = jωe + E = jω n E Vecto potential A P n V Vecto elmholtz Equation E 2 2 A + k A = Fee pace Geen' Function jk e A = d Field ae expeed in tem of ecto potential A a: = A ( A E = jω A+ jω φ When only the fa field ae conideed, we hae a TEM wae. E =jω Aˆ( A ˆ ( TEM = ( ˆ E = 20 π Duality in Maxwell' Equation E =jω = jω E+ Fee pace Geen' Function jk e = d Field ae expeed in tem of ecto potential A a : E= E= jω + jω ( =jω( ˆ( ˆ E =( ˆ E E ρ ρ When only the fa field ae conideed, we hae a TEM wae. ρ ρ

5 5 Integal Expeion of EM Field.doc 08/07/0 5-5 Application to pactical a well a appoximate calculation of field. Field on the uface ae appoximated and the field in the olume will be calculated in tem of ecto potential fo them. catteing effect ae not exactly but appoximately taken into account.. Phyical optic : Reflected wae ae appoximated by geometical optic auming the fequency i high enough and uface i eplaced with infinite plain one. = 2n, m = 0 2. Apetue antenna o catteing though apetue: Petubation of field due to catteing fom finite ize of window ae neglected and field on ae aumed to be i i identical with the incident one. = n, m = E n Poblem. uppoe that a plane wae i incident fom z axi. Deie the equialent uface electic and magnetic cuent on the x-y plane at z = 0 and calculate the field fom thee uface cuent in the egion z > 0 and z < 0. Poe and explain the field equialence pinciple. 2. uppoe that a plane wae i incident fom z axi. At z = 0 and in the x-y plane, a ectangula quae plate of electical conducto (2a x 2b i eflecting it. Deie the appoximate uface electic and magnetic cuent on plate and calculate the field fom thee uface cuent in the fa field egion fo z > 0 and z < 0. Pedict qualitatiely the geometical optic behaio of the total field. (Reflection, tanding wae, hadow etc.

6 Maxwell equation = jω E+ E=jω M E = = ρ ρ 5 Integal Expeion of EM Field.doc 08/07/0 5-6 duality E E M M ρ ρ ρ ρ e A = A jk d ( A E= jωa+ jω e M jk d E= = jω + jω ( jωa jω ˆ A E= jωa+ jω A = jω+ + jω jω ˆ jωa = A = M e jk d