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Exam2_2003Spring

# Exam2_2003Spring - Physics 8.02 Some(possibly useful...

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Physics 8.02 Exam Two Mashup Spring 2003 Some (possibly useful) Relations: inside closedsurface o Q d κ ε = ∫∫ E A G G w points from inside to outside d A G moving from to b a b b a a V V V = = − E s G G d 0 d = E s G G v many point charges 1 4 N i i o i q V πε = = r r G G V Q C = C Q V = parallelplate o A C d ε = U = 1 2 C V 2 = Q 2 2C series 1 2 1 1 C C = + 1 C parallel 1 2 C C = + C where is the resistivity ρ ρ = E J G G V = i R L R A ρ = 1 R parallel = 1 R 1 + 1 R 2 R series = R 1 + R 2 2 2 ohmic heating V P i V i R R = ∆ = = 2 ˆ 4 o q c r µ π × = < v r B v G G G < 2 ˆ 4 o I d d r µ π × = s r B G G ˆ points source observer r from to through contour o d I µ = B s G G v where I through is the current flowing through any open surface bounded by the contour: through open surface I d = J A G G d s right-handed with respect to d A sgl loop d d N dt dt ε Φ = − = − E ds B dA v ext q = × F v B G G G ext d I d = × F s B G G G 2 cent. F mv = r = perpendicular to loop, right-handed with respect to I A I μ n n G ± ± = × τ μ B G G G z z z dB F 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 Useful integrals: 1/ 2 1/ 2 2 ( ) ( ) d c f c f f θ θ θ = + + 1 ln( ) ( ) d c f c f f θ θ θ = + + 2 1 1 ( ) ( ) d c f f c f θ θ θ = − + + Useful Small Argument Approximations (1 ) 1 for <<1 n n ε ε ε + + ln(1 ) for <<1 ε ε ε +

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