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PHYS 260 Skyline College Properties of Electric Dipoles Quiz Questions

phys 260

Skyline College

PHYS

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Physics 260 – Exam 2 (Take-Home) – Hein Skyline College, Spring 2020 NAME: ______________________ GENERAL INSTRUCTIONS: • • • This is a take-home exam. You may consult with others on this test, but you will turn in an individual solution. The work you present needs to be your own, and may not be copied from others. Your solution may be typed or handwritten. FORMATTING YOUR ANSWERS: • • • • • Justify your work sufficiently, with calculation or explanation, in order to receive credit. Make your answers clear, legible and organized. If I can’t read it, I can’t give it credit. I cannot grade crossed-out or erased work. Be sure before deleting work! I cannot give full credit for a correct response if it is accompanied by another contradicting one. However, “hedging your bets” will earn more credit than not having the correct response at all. Do not forget to include units and express vectors appropriately for full credit. GOOD LUCK! In this exam, you will study various properties of electric dipoles. Electric dipoles are important to describe molecules. Even without a net charge, the charge distribution can be such that they have a dipole moment. This polar nature is, for example, what allows water molecules to bond to each other and support a high surface tension. Before attempting the problem, it is recommended that you read the following chapters in the text: https://openstax.org/books/university-physics-volume-2/pages/5-7-electric-dipoles https://openstax.org/books/university-physics-volume-2/pages/7-3-calculations-of-electric-potential (TO BE FILLED BY INSTRUCTOR) Problem I II III IV Total Points Obtained Maximum Points 30 30 20 20 100 Question #1: Electric field due to a dipole (30 points total) An electric dipole consists of a pair of point charges with equal magnitude and opposite sign, +Q and –Q, and separated by a distance 2a, as shown. Part A (15 points): Show that at all points on the x-axis for '( which |𝑥| ≫ 𝑎, 𝐸 ≈ . . )*+, - Part B (15 points): Show that at all points on the y-axis for which |𝑦| ≫ 𝑎, 𝐸 ≈ '( *+, 0 . . Question #2: Electric Potential due to a dipole (30 points total) Let’s consider the same configuration as Question #1. Part A (10 points): Calculate the electric potential at any point on the x-axis. Part B (10 points): Calculate the electric potential at any point on the y-axis. Part C (10 points): Use your result in Question #2, Part B, to calculate the electric potential on the y-axis, when |𝑥| ≫ 𝑎. Use this result to confirm the answer to Question #1, Part B. Question #3: Torque on an electric dipole (20 points total) The dipole is placed in an electric field, as shown. Part A (5 points): Draw the electric forces acting on each of the charges on the figure. The net force on the electric dipole is zero, but there is a torque that tends to rotate the dipole clockwise. Part B (15 points): We define the electric dipole moment 𝑝. It has a magnitude 𝑝 = 2𝑞𝑎 and its direction is along the dipole axis from the negative charge to the positive charge. Show that the net torque with respect to the center of the dipole is 𝜏 = 𝑝×𝐸. Note: One method for increasing capacitance is to insert a dielectric between the conductors that reduces the voltage because of its effect on the electric field. When the molecules of a dielectric are placed in the electric field 𝐸, their negatively charged electrons separate slightly from their positively charged cores, and dielectric polarization occurs. The molecules acquire an electric dipole moment. We define the polarization vector 𝑃 = 𝑁𝑝, where N is the number of dipoles per unit volume in the material. This polarization is proportional to 𝐸. Question #4: RC circuits (20 points total) Consider the RC circuits shown below. Find the magnitude and direction for the currents at the instant the switch S is closed. Formula Sheet Electric Potential due to a continuous charge distribution Z dq V =k (12) r q1 q2 U =k (13) r12 Physics 260/Spring 2020 Electricity Coulomb’s Law Fe = k Electric Field |q1 ||q2 | r2 ~ = −∇V ~ E (14) ~ = − ∂V ı̂ − ∂V ̂ E ∂x ∂y (15) (1) ~ ~ = Fe E q0 (2) Capacitance q V Capacitance for parallel plate capacitor C= Electric Field due to a point charge ~ = kq r̂ E r2 (3) C= ε0 A d Electric Field due to a continuous charge distribution Capacitors in parallel Z dq ~ E=k r̂ (4) Ceq = C1 + C2 + ... r2 Electric Field due to an infinite plane of charge E= σ 2ε0 Electric Field just outside of a conductor Electric Flux I φE = 1 1 1 = + + ... Ceq C1 C2 U= (7) Q2 1 1 CV 2 = = QV 2 2C 2 C = κC0 (8) p~ = |q|d~ Z ~ ~ ds E. (10) q r (22) Electric Current (11) (23) Potential Energy of Electric Dipole ~ U = −~ p.E Electric Potential due to a Point Charge V =k (21) (9) Torque on Electric Dipole ~ ~τ = p~ × E ∆U =− q0 (20) Electric Dipole Moment Electric Potential ∆V = (19) Capacitor with dielectric Gauss’s Law qenc φE = ε0 Electric Potential Energy Z ~ ~ ds ∆U = −q0 E. (18) Energy stored in a capacitor (6) ~ A ~ E.d (17) Capacitors in series (5) σ E= ε0 (16) I= dq dt (24) (25) Current Density J= Magnetism Magnetic Force on a moving charge I = nqvd A ~ J~ = σ E (26) ~ F~B = q~v × B (41) (27) FB = |q|vB sin θ (42) Resistance Magnetic Force on a current carrying wire V R= I ρL R= A Temperature dependence (28) ~ ×B ~ F~B = I L (29) Magnetic Dipole Moment ~ µ ~ = IA ρ = ρ0 (1 + α(T − T0 )) (44) (30) Torque on Current Loop ~ ~τ = µ ~ ×B Resistors in series Req = R1 + R2 + ... (45) (31) Magnetic Dipole Potential Energy ~ U = −~ µ.B Resistors in parallel 1 1 1 = + + ... Req R1 R2 Electric Power V2 R P = IV = I 2 R = d~l × r̂ r2 (47) ~ s = µ0 Ienc B.d~ (48) Kirchhoff’s Junction and Loop Rules: (34) Magnetic Flux Σclosed ∆V = 0 (35) Z φB = N RC Circuit - Charging Capacitor t q(t) = Q(1 − e− RC ) ε − t e RC R RC Circuit - Discharging Capacitor I(t) = t − RC q(t) = Qe Q − t e RC RC ~ A ~ B.d Gauss’s Law of Magnetism I ~ A ~=0 B.d (37) (49) (36) (50) Faraday’s Law (38) ε=− dφB = dt I ~ s E.d~ (51) (39) Magnetic Field due to a long, straight wire Time Constant τ = RC Z (33) Ampere’s Law I ΣIin = ΣIout (46) (32) Biot-Savart Law ~ = µ0 I B 4π I(t) = − (43) (40) B= µ0 I 2πr (52) Magnetic Field of solenoid LC angular frequency B = µ0 In n= N L (53) ω=√ 1 LC (54) LC energy U = UC + UL = constant Inductance and Inductors Self Inductance N φB L= I Inductance of a solenoid L = µ0 n2 A l dI dt Vmax Imax Irms = √ ; Vrms = √ 2 2 (56) Inductive Reactance XL = ωL (69) 1 ωC (70) XC = (58) Impedance Z= (59) Current Decay in RL Circuit L τ= R Energy stored in an inductor (72) (60) Pave = Irms Vrms cos φ (73) 2 Pave = Irms R (74) (61) RLC Circuit Vrms Z (75) 1 LC (76) N2 V1 N1 (77) I1 V1 = I2 V2 (78) Irms = 1 2 LI 2 (62) Resonance frequency ω0 = √ Mutual inductance ε2 = −M (71) XL − XC ) R φ = tan−1 ( RL time constant dI1 dt (63) Transformers V2 = LC Circuit Q(t) = Qmax cos(ωt + φ) I(t) = p R2 + (XL − XC )2 Average Power t ε I = e− τ R U= (68) (57) Capacitive Reactance RL Circuit dI L + RI = ε dt Current in RL Circuit t ε I = (1 − e− τ ) R (67) (55) AC Circuits Self-induced emf εL = −L (66) dQ = −ωQmax sin(ωt + φ) dt (64) (65) Surface charge density σ Math Stuff Z Q= ~B ~ = |A|| ~ B| ~ cos θ = Ax Bx + Ay By + Az Bz A. ~ × B| ~ = |A|| ~ B| ~ sin θ |A σdA (93) ρdV (94) (79) Volume charge density ρ (80) Z Q= ~×B ~ = (Ay Bz − Az By )ı̂ + (Az Bx − Ax Bz )̂ (81) A +(Ax By − Ay Bx )k̂ (82) Constants Quadratic Formula x= −b ± √ b2 − 4ac 2a k= (83) 1 = 8.99 × 109 N m2 /C 2 4πε0 Permittivity of free space Surface Area of Sphere ε0 = 8.85 × 10−12 C 2 /N m2 4πr 2 (95) (96) (84) Permeability of free space Volume of Sphere 4 3 πr 3 Integrals Z Z µ0 = 4π × 10−7 T m/A (85) Elementary Charge e = 1.602 × 10−19 C dx 1 = ln(k1 x + k2 ) + C k1 x + k2 k1 p dx √ = ln( x2 + a2 + x) + C x2 + a2 Z p xdx √ = x2 + a 2 + C 2 2 x +a Z dx x = √ +C (x2 + a2 )3/2 a2 x2 + a2 Z −1 xdx =√ +C 2 2 3/2 (x + a ) x2 + a2 Z x3 dx x2 + 2a2 √ = +C (x2 + a2 )3/2 x2 + a2 me = 9.11 × 10−31 kg (87) (89) (90) (91) (92) (99) Mass of a Proton (88) Z λds (98) (86) Mass of an Electron Charge Densities Linear charge density λ Q= (97) mp = 1.67 × 10−27 kg (100) c = 3.00 × 108 m/s (101) Speed of Light ...
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