Phys Final Exam F09 Instructions. December 7, 2009

Transcription

1 Phys Final Exam F09 Instructions December 7,

2 INSTRUCTIONS & FORMULAS FINAL EXAM F09 PHYS2306 This is a closed book exam. No books, notes, computers or other paper may be brought into the exam. All cell phones must be shut off. The honor system is in effect. No one should seek from or give any aid to another and should report instances of cheating if observed. You may use a calculator and these instruction sheets on this exam. Bring a sufficient quantity of number 2 pencils to work with. These instruction sheets have been sent to you before the exam and you will find a hard copy stapled to the exam questions. Thus do not bring another copy of these instructions to the exam. Use bathrooms before the exam because you can t return after leaving the exam room. Make sure your family name, given name, VT id number are on the answer sheet. Be sure to indicate the form, A-E, that you are using or you ll get an incorrect mark that is usually detrimental. You must also put this information on the instruction and question sheets. Needless to say, your exam will only be graded, if your answer sheet, question sheets and instruction sheets are returned. This protects the integrity of the makeup exam. The exam will last 2 hours. For scratch paper you may use the backside of the instruction and question sheets. If you mark your answer sheet incorrectly, we can look at the scratch sheets to correct your grade. The relevant formulas from chapters 5, 6, and 2-34 of the text are below. If you think any others would be of help, you can memorize or understand how to derive them. If your calculation method is correct and you keep sufficient significant digits, your answer should not differ from a correct choice by more than 3%. velocity v 2 = (λf) 2 wavelength frequency () medium items v 2 = F/µ = B/ρ = Y/ρ (2) 2 y(x, t) 2 y(x, t) = 0 d wave eq displacement (3) x 2 v 2 t 2 y(x, t) = A cos 2π(x/λ ± f t) sinusoidal wave (4) P av = 0.5 µf (2πfA) 2 power (5) I /I 2 = (r 2 /r ) 2 intensity distance relation(6) I = p 2 max/(2ρv) sound wave (7)

3 p max = BA2π/λ pressure sound wave (8) β = (0 db) log(i/0 2 ) (9) y = y + y 2 superposition (0) standing wave f n = (, 2, 3,...)v/(2L) stringends fixed () = (, 2, 3,...)v/(2L) open pipe (2) = (, 3, 5,...)v/(4L) stopped pipe (3) f beat = f f 2 (4) f L = f S v + v L v + v S Doppler (5) sin α = v/v S shock wave (6) Constructive,destructive interference when waves arrive in phase,λ/2 out of phase. F = q E(r ) (7) E(r) = E j (r) j superposition from charges q j (8) r r j E(r) = q j charges q 4πɛ 0 j r r j 3 j at r j (9) = dq r r continuous q density (20) 4πɛ 0 r r 3 dq = dr λ(r ) = da σ(r ) = dv ρ(r ) (2) E axial,ring (x) = (λ2πr)x(x 2 + R 2 ) 3/2 i 4πɛ 0 λ uniform (22) E line (x) = (Lλ) i 4πɛ 0 x(x 2 + (L/2) 2 ) /2 (23) E axial,disk (x) = σ x [ ]i 2ɛ 0 x2 + R2 σ uniform (24) T dipole = p E torque in E (25) U dipole = p E potential energy in E (26) E nda = Q en /ɛ 0 Gauss Law vacuum (27) 2

4 Electric field magnitudes for various geometries and charges using R for radius of sphere or cylinder: E distance r from point charge q, E 2 distance r from conducting sphere with charge q, r>r, E 3 distance r from conducting sphere with charge q, r<r, E 4 distance r from insulating sphere with charge q spread uniformly, r>r, E 5 distance r from insulating sphere with charge q spread uniformly, r<r, E 6 distance r from infinite, conducting cylinder with charge/length λ, r>r, E 7 distance r from infinite, conducting cylinder with charge/length λ, r<r, E 8 anywhere from an infinite plane of uniform charge/area σ E = q 4πɛ 0 r 2 (28) E 2 = E (29) E 3 = 0 (30) E 4 = E (3) E 5 = qr 4πɛ 0 R 3 (32) E 6 = λ 2πɛ 0 r (33) E 7 = 0 (34) E 8 = σ 2ɛ 0 (35) The work done by the electric force on a point charge Q in moving Q from a to b is independent of the path and so can be expressed in terms of a potential energy U. W a b = b a QE dr = U a U b (36) V = U/Q potential (37) V = (U a U b )/Q = V a = = b a E dr (38) q i point charges (39) 4πɛ 0 i r a r i dq charge distribution (40) 4πɛ 0 r a r 3

5 E x,y,z = V x, y, z (4) C = KC 0 = Q/ V = Kɛ 0A plates (42) d = (43) C series i C i C parallel = C i (44) i U = C V 2 2 U/vol = Kɛ 0E 2 2 = Q2 2C (45) (46) KE nda = Q free,en /ɛ 0 Gauss law dielectric (47) I = dq dt = n q v da current (48) J = nqv d current density (49) J = E/ρ Ohm s law ρ constant (50) R = V/I = ρl/a (5) V ab = E Ir power source with internal r (52) P = I V = I 2 R = ( V ) 2 /R power (53) R eq = R i series (54) i = parallel (55) R eq i R i I = 0 Kirchoff junction rule (56) V = 0 Kirchoff loop rule (57) q = Q f ( exp t ) RC circuit charging (58) RC q = Q 0 exp t RC circuit discharging (59) RC F = qv B magnetic force on a charged particle (60) F = IL B magnetic force on a current carrier (6) 4

6 Φ B = B nda n normal to surface (62) µ = IAn I clockwise n into paper (63) τ = µ B (64) U = µ B (65) B(r) = µ 0 qv (r r ) 4π r r 3 r is q s position (66) db(r) = µ 0 IdL (r r ) 4π r r 3 r is dl s position (67) B(r) = µ 0I 2πr outside long wire(cylinder) (68) B(x) = µ 0 Ia 2 2(x 2 + a 2 ) 3/2 axis of loop (69) B(r) = µ 0Ir inside long cylinder (70) 2πR 2 B(r) = µ 0 ni inside solenoid near center 0 outside (7) B(r) = µ 0NI inside toroid 0 outside (72) 2πr EMF = dφ B acts to oppose what causes it (73) dt = E dl (74) dl = clockwise n for Φ B into paper (75) I D = ɛ dφ E dt displacement current (76) B nda = 0, closed surface bounding a volume (77) B dl = µ(i + I D ) (78) EMF (2) = MdI 2 /dt, (79) EMF () = MdI /dt, mutual inductance circuits, 2, (80) M = N 2 Φ B2 /I = N Φ B /I 2, (8) EM F = LdI/dt, self inductance, (82) L = NΦ B /I. (83) 5

7 For non-ac sources of EMF U B = 0.5LI 2, (84) u = U/V ol = 0.5B 2 /µ. (85) I = (V/R)( exp( Rt/L)), current growth RL circuit,(86) I = I(0) exp( Rt/L), current decay RL circuit, (87) Q = Q(0) cos /LCt + (I(0)/ω) sin /LCt, LC circuit, (88) Q = exp( at)[q(0) cos bt + B sin bt], RLC seriescircuit, (89) a = R/2L, b = /LC (R/2L) 2, (90) B = (I(0) + RQ(0)/2L)/b. (9) For AC circuits, reactances X behave like resistances for DC circuits. ±V sin ωt = Im(V exp(±iωt)), (92) V cos ωt = Re(V exp(±iωt)), (93) X R = R, X L = ilω, X C = i/cω, (94) X(total) = Z exp iφ, (95) V (X) = Im(I(X)X) or Re(I(X)X), (96) P (X) = Im(V (X)I(X)) or Re(V (X)I(X)). (97) To get the root mean square value of a quantity from the above instantaneous values, square the quantity, calculate the average over one period, and then take the square root of the average. For V sin ωt it is V/ 2. V I = V 2 I 2, V /V 2 = N /N 2 for transformers. For electromagnetic waves with veclocity V i, with V = /µɛ and polarization j, E i, B i, E B, (98) E(kx ωt) = je cos(kx ωt), (99) B(kx ωt) = k(e/v ) cos(kx ωt), (00) S = E B/µ, (0) U E /V ol = 0.5ɛE 2, U B /V ol = 0.5B 2 /µ, (02) P (radiation) = S/V. (03) 6

8 Light traveling, but not interacting, is an electromagnetic wave. λ = 2π/k = , nm in free space, (04) f inc = f refracted, (05) θ inc = θ reflected, (06) n inc sin θ inc = n refracted sin θ refracted, (07) n inc sin θ critical =, (08) n = c/v. (09) For spherical mirrors of radius R, in the small angle approximation, rays parallel to the optic axis are reflected through the focal point (R/2), rays through the focal point are reflected parallel to the optic axis, and rays through the center of curvature R are reflected back through that point. For a thin lens in the same approximation, rays parallel to the optic axis are focussed or appear to be coming from a focus at one of the focal points, rays passing through a focal point are focussed parallel to the optic axis, and rays passing through the center of the lens are unaltered. If there is more than optical element, the image of element m becomes the object for element m+. 7