Exam1_2007FallConflict_Solutions

Exam1_2007FallConflict_Solutions - Physics 8.02 Exam One...

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Physics 8.02 Exam One Conflict Exam Solutions Fall 2007 Please Remove this Tear Sheet from Your Exam 2 3 ˆ 4 4 o o r r πε πε = = E r q q r G G ˆ points source q observer r r = from to r G inside-free o closed surface κ ε = ∫∫ E dA w Q G G points from inside to outside dA G q = F E b G G G = = Δ a a b b to a from moving V V V s d E G 0 = path closed s d E G G W U q V = Δ = Δ point charge 4 o V r q πε = = pairs all j i o j i U r r G G πε 4 q q 2 1 2 o v U E ε = ∫∫∫ x ol dV E x = - V x , E y = - V y , E z = - V z r V E r = − for spherical symmetry Q C V = Δ U = 1 2 C Δ V 2 = Q 2 2C 1 parallel C C = + Circumferences, Areas, Volumes: Circle of radius r : Area = π r 2 Circumference = 2 π r Sphere of radius r Surface Area = 4 π r 2 Volume = 3 4 3 r π Cylinder of radius r and height h Side surface area = 2 π r h End cap surface area = 2 π r 2 Volume = 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 Some potentially useful numbers 2 9 2 0 1 N 9 10 4 C e k πε = = × m Breakdown of air E ~ 3 x 10 6 V/m 2 C 1 2 1 2 series C C C C C = + Speed of light c = 3 x 10 8 m/s Electron charge e = 1.6 x 10 -19 C Avagadro’s number N A = 6.02 x 10 23 mol -1
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