Final_2003Spring

# Final_2003Spring - Physics 8.02 Some(possibly useful...

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Physics 8.02 Final Exam Spring 2003 Some (possibly useful) Relations: enclosed o closed surface Q d ε = ∫∫ E A r r ± closed open path surface d d d dt = − ∫∫ E s B A r r r r ± one turn N d = ∫∫ B A r r LI closed path dI d L dt = − E s r r ± 0 closed surface d = ∫∫ B A r r ± thru o E closed path E open surface d d I dt where d µ µ ε = + Φ Φ = ∫∫ o 0 B s E A r r r r ± 0 2 ˆ 4 µ π × = = s r B B source source Id d r r r r q q q = + × F E v B B r r r r d I d = × F s r r r 0 ˆ 1 4 2 dq r πε = r dE r 2 0 1 ˆ 4 V dq r πε = E r r b a V V(b ) V( a ) d = − E s r r i N i i 1 0 i, q 1 V 4 r πε = = = P / R 2 2 Joule Heating P I R V = = ∆ Q C V = 2 2 1 2 2 C Q U C V C = = 0 1 LC = ω 2 0 1 2 elec u E = ε 2 0 1 2 mag u B = µ 0 0 1 c = µ ε 1 f T = 2 2 f T ω = π = π 2 k = π λ c T f k = λ = λ = ω 0 1 = × µ S E r B r r 0 1 ( , ) ( , ) T x t x T = S S r r t dt 2 0 1 2 sin ( ) 2 T x ct dt T π 1 = λ DOUBLE SLIT: constructive: sin , 0, 1, 2, 3, ... d m m θ = λ = ± ± ± destructive: 1 sin , 0, 1, 2, 3, ... 2 d m m θ = + λ = ± ± ± intensity: 2 max cos sin I I d π = θ λ SINGLE SLIT: destructive: sin , 1, 2, 3, ... a n n θ = λ = ± ± ± intensity: 2 max sin sin sin a I I a π θ λ = π θ λ USEFUL INTEGRALS: b a dx b a = ( ) 2 2 1 2 b a x dx b a = 1 ln b a b dx x a = 2 1 1 b a dx 1 x a b =

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