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Exam1_2004Fall

Exam1_2004Fall - Physics 8.02 Exam One PLEASE DETACH THIS...

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Physics 8.02 Exam One Fall 2004 PLEASE DETACH THIS SHEET AND USE AT YOUR CONVENIENCE Some (possibly useful) Relations: F = 1 4 πε o q 1 q 2 r 2 2 3 ˆ 4 4 o o q q r r πε πε = = E r G r G ˆ points source q observer r r r = from to G 2 0 1 ˆ 4 V dq r πε = E r G free, inside closedsurface o Q d κ ε = ∫∫ E A G G w pointsfrom inside to outside d A G moving from to b a b b a a V V V = = − E s G G d closedpath 0 d = E s G G v point charge 4 o q V r πε = many point charges 1 4 N i i o i q V πε = = r r G K all pairs 0 4 i j i j q q U πε = r r G G 2 vol 1 2 o U E ε = ∫∫∫ x dV V = −∇ E G or E x = - V x , E y = - V y , E z = - V z Q C V = 2 2 1 ( ) 2 2 Q U C V C = = 1 parallel C C 2 C = + 1 2 1 2 series C C C C C = + Circumferences, Areas, Volumes: 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 4 3 r 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 Definition of trig functions sin is opposite/hypotenuse; cos is adjacent/hypotenuse; tangent is opposite over adjacent; Properties of 30, 45, and 60 degrees ( π /6, π /4, and π /3 radians) : sin( π /6) = cos( π /3) = 1/2, sin( π /3) = cos( π /6) = 2 / 3 ; sin( π /4) = cos( π /4) = 2 / 1 ; Integrals that may be useful a b dr b a = ) / ln( 1 a b dr r b a = = b a dr r b a 1 1 1 2
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