ASSIGN_6d - Assignment#6 PHYSICS 354/4-01 ELECTRICITY&...

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Unformatted text preview: Assignment #6 PHYSICS 354/4-01 ELECTRICITY & MAGNETISM II April. 2010 Assignment #6 Page 1 of 4 Solution to Assignment # 6 3 7 3 . 9.5, p A travelling wave take the form of ). ( vt z f ± At the interface, ) ( ) ( ) ( 2 1 1 t v g t v h t v g T R I- = +- ..................................................................................(1) Continuity of the slope at the interface: = = = ∂ ∂ = ∂ ∂ + ∂ ∂ z T z I z I z g z h z g . And t f v z f v vt z f t f vt z f z f ∂ ∂- = ∂ ∂ ⇒ - ⋅- ∂ ∂ = ∂ ∂ ⋅- ∂ ∂ = ∂ ∂ 1 ) ( ) ( 1 ) ( t g v v t h t g t g v t h v t g v T R I z T z R z I ∂ ∂ = ∂ ∂- ∂ ∂ ⇒ ∂ ∂- = ∂ ∂ + + ∂ ∂- ∴ = = = 2 1 2 1 1 1 1 1 Integrate w.r.t time, C t v g v v t v h t v g T R I +- =-- ) ( ) ( ) ( 2 2 1 1 1 .........................................................................(2) (1)+(2) C v v v g v v v g C v v v g g v v v C g v v g I T T I T I 2 1 2 2 1 2 1 2 2 1 2 2 2 1 2 2 1 2 +- + = ⇒ + + = + ⇒ + + = ............(3) From (1) C v v v g v v v v g C v v v g v v v g g h I I I I T R 2 1 2 2 1 1 2 2 1 2 2 1 2 2 +- +- =- +- + =- = ............(4) (3) and (4) give the amplitude of the reflected and transmitted wave at any z in terms of the incident wave . 2 8 3 . 9.11, p Given ( ) ( ) b a t r k B t r g t r k A t r f δ ω δ ω +- • = +- • = cos ) , ( and , cos ) , ( . The time average of the product is: ( ) ( ) ∫ +- • ⋅ +- • = ⋅ T a a dt t r k t r k AB T g f cos cos 1 δ ω δ ω . ( ) ) cos( ) cos( 2 1 cos cos sin sin cos cos ) cos( sin sin cos cos ) cos( diff sum Y X Y X Y X Y X Y X Y X Y X + = ⇒ + =-- = + ( ) π ω δ δ δ δ ω δ δ δ δ ω 4 2 ), cos( 2 ) 2 2 sin( 2 ) cos( ) 2 2 cos( 2 =- + + +- • =- + + +- • = ⋅ ∫ T AB t r k T AB dt t r k T AB g f b a T b a T b a b a ) cos( 2 1 b a AB δ δ- = ................................................................................................................................(11a) Using complex notation, ) ( ) ( ) ( * ) ( ) ( ~ ~ ~ ~ b a b a b a i t r k i t r k i t r k i t r k i e AB Be Ae g f Be g Ae f δ δ δ ω δ ω δ ω δ ω- +- •- +- • +- • +- • = • = ⋅ ⇒ = = ........(11b) Comparing (11a) and (11b) ] ~ ~ Re[ 2 1 * g f g f ⋅ = ⋅ 396 . 9.18, p (a) The time constant, , σ ε τ = and , 2 n r = ε shown elsewhere. n = refractive index of the medium....
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This note was uploaded on 12/15/2010 for the course PHYS 354 taught by Professor Josephshin during the Winter '10 term at Concordia Canada.

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ASSIGN_6d - Assignment#6 PHYSICS 354/4-01 ELECTRICITY&...

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