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Exam2_2008Spr_Solutions

# Exam2_2008Spr_Solutions - MASSACHUSETTS INSTITUTE OF...

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2008 Exam Two Equation Sheet Force Laws: ( ) ext ext q = + × F E v B G G G G Source Equations: 2 1 ˆ 4 o dq d r πε = E r G 2 ˆ 4 o I d d r µ π × = s r B G G ˆ r r r = G points from source to field point Current Density and Current: open surface I d = ∫∫ J a G G Gauss’s Law: inside closedsurface / o d Q ε = ∫∫ E A G G w Ampere’s Law: closed path open surface o d d µ = ∫∫ B s J a G G G G v Gauss’s Law for Magnetism: 0 closed surface d = ∫∫ B A G G w Electric Potential Difference: b b a a V V V d = ≡ − E s G G Capacitance: V Q C = 2 2 1 1 2 2 Q U C V C = = Capacitors in Parallel: 1 2 eq C C C = + +⋅⋅⋅ Capacitors in Series: 1 2 1 1 1 eq C C C = + +⋅⋅⋅ Ohm’s Law: V I R = where is the conductivity c c σ σ = J E G G where is the resistivity r r ρ ρ = E J G G Resistors in Parallel: 1 2 1 1 1 eq R R R = + +⋅⋅⋅ Resistors in Series: 1 2 eq R R R = + +⋅⋅⋅ Joule Heating: 2 2 Joule / P I V I R V R = = = ∆ Magnetic Dipoles : = I A μ n G ± Torque: = × τ μ B G G G Force: ( / ) z z z F dB dz µ = Cross-products of unit vectors : ˆ ˆ ˆ ˆ ˆ ˆ 0 × = × = × = i i j j k k ˆ ˆ ˆ × = i j k ˆ ˆ ˆ × = j k i ˆ ˆ ˆ × = k i j Kinematics: 2 / cent v r = a G ; 2 2 / f T ω π π = = Circumferences, Areas, Volumes: 1) Circle radius r : Area π r 2 ; Circumference 2 π r 2) Sphere of radius r : Surface Area 4 π r 2 ; Volume (4 / 3) π r 3 Cylinder of radius r and length l : Area 2 rl π ; Volume 2 r l π Integrals : dr a b = b a ; 1 ln( / ) b a r dr b a = 2 (1/ 1/ ) b a r dr a b = Constants: 9 2 -2 0 1/ 4 9 10 N m C e k πε = = × 7 -1 0 / 4 10 T m A µ π =

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3 8.02 Exam Two Spring 2008 Solutions FAMILY (last) NAME GIVEN (first) NAME Student ID Number Your Section: ____L01 MW 9 am ____L02 MW 11 am ____L03 MW 1 pm ____L04 MW 3 pm ____L05 TTh 9 am ____L06 TTh 11 am ____L07 TTh 1 pm ____L08 TTh 3 pm Your Group (e.g. 10A): ______________ Score Grader Section 1 (25 points) Problem 1 (25 points) Problem 2 (25 points) Problem 3 (25 points) TOTAL

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