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Final_2004Spring

Final_2004Spring - Physics 8.02 Some(possibly useful...

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Physics 8.02 Final Exam Spring 2004 Some (possibly useful) Relations: enclosed o closed surface Q d ε = ∫∫ E A G G w total closed open path surface external self open surface d d d dt d d dt = − = − + ∫∫ ∫∫ E s B A B B G G G G G G G v A 0 closed surface d = ∫∫ B A G G w closed path thru o E E open surface d d I dt where d µ µ ε = + Φ Φ = ∫∫ o o B s E A G G G G v 2 0 1 2 elec u E = ε 2 0 1 2 mag u B = µ 0 2 ˆ 4 µ π × = = s r B B source source Id d r G G G q q q = + × F E v G G G B G B d I d = × F s G G G 0 1 ˆ 4 2 dq r πε = dE r G 2 0 1 ˆ 4 V dq r πε = E r G 2 1 2 1 P 2 1 P The electric potential at point P minus that at point P is given by V( P ) V( P ) d = − E s G G i N i i 1 0 i, q 1 V 4 r πε = = = P V IR = 2 2 Joule Heating P I R V = = ∆ / R Q C V = 2 2 capacitor 1 2 2 Q U C V C = = self field self one turn N d = ∫∫ B A L I G G 2 inductor 1 2 U L = I 0 1 LC = ω 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 G B G G 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 = ± ± ± SINGLE SLIT: destructive: sin , 1, 2, 3, ... a n n θ = λ = ± ± ± Areas, Volumes, etc.: 1) The area of a circle of radius r is π r 2 Its circumference is 2 π r 2) The surface area of a sphere of radius r is 4 π r 2 . Its volume is 3 3 r 4 π 3) The area of the sides of a cylinder of radius r and height h is 2 π r h. Its volume is π r 2 h USEFUL INTEGRALS: d c dx d c = ( ) 2 2 1 2 d c r dr d c = 1 ln d c d dr r c = 2 1 1 d c dr r c 1 d =
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