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Exam1_2008Spr - Physics 8.02 Equation Sheet Exam One Please...

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1 Physics 8.02 Equation Sheet Exam One Please Remove this Tear Sheet from Your Exam 2 3 ˆ 4 4 o o q q r r πε πε = = E r r G G ˆ points source q observer r r r = from to G many point charges 3 1 ( ) 4 N i i i o i q πε = = E r r r r G G K G K 2 0 1 ˆ ( ) 4 source dq πε = E r r r r G G G G q source q = F E G G enc o closed surface Q ε = ∫∫ E dA G G w points from inside to outside dA G 0 closed path = E ds G G v b moving from a to b b a a V V V Δ = = − E ds G G U q V Δ = Δ point charge ; ( ) 0 4 o q V V r πε = ∞ = many point charges 1 ; ( ) 0 4 N i i o i q V V πε = = ∞ = r r G K 0 1 ( ) ; ( ) 0 4 source dq V V πε = ∞ = r r r G G G ; ( ) 0 4 i j all pairs o i j q q U U πε = ∞ = r r G G 2 1 2 o vol all space U E dV ε = ∫∫∫ E r = − V r for spherical symmetry, V = −∇ E G G x V E x = − y V E y = − z V E z = − C = Q Δ V U = 1 2 C Δ V 2 = Q 2 2 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 (4 / 3) π r 3 3) The area of the sides of a cylinder of radius r and height h is 2 π rh . Its volume is π r 2 h Integrals that may be useful dr a b = b a ln( / ) b a dr b a r = 1 r 2 dr a b = 1 a 1 b Some potentially useful numbers k e = 1 4 πε 0 = 9 × 10 9 N m 2 C 2
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