Final Exam : Physics 2113 Fall 2014 5:30PM MON 8 DEC 2014 Name (Last, First): Section # Instructor s name: Answer all 6 problems & all 8 questions. Be sure to write your name. Please read the questions carefully. You may use only scientific or graphing calculators. In particular you may not use the calculator app on your phone or tablet! You may detach and use the formula sheet provided at the back of this test. No other reference materials are allowed. You may not answer or use cell phones during the exam. Please note that the official departmental policy for exams is as follows: During your test, the only electronic device you may have with you at your seat is a scientific or graphing calculator. You may not have your cell phone, tablet, smartphone, PDA, pager, digital camera, computer, or any other device capable of taking pictures or video, sending text messages, or accessing the Internet. This means not just on your person, but close enough to you that you could reach it during the test. Any student found with such a device during a test will be assumed to be violating the LSU Honor Code and will be referred to the Dean of Students for Judicial Affairs. The simplest remedy is to bring nothing to this test but the calculator, and leave your backpack or purse at home. If you have brought your cell phone or tablet with you, please leave it at the front of the room under the watchful eye of your instructor. Some questions are multiple choice. You should work these problems starting with the basic equation listed on the formula sheet and write down all the steps. Although the work will not be graded, this will help you make the correct choice and to determine if your thinking is correct. On problems that are not multiple choice, be sure to show all of your work since no credit will be given for an answer without explanation or work. These will be graded in full, and you are expected to show all relevant steps that lead to your answer. Please use complete sentences where explanations are asked for. For numerical answers that require units you must give the correct units for full credit. YOU GET 120 min (2 hrs) 1
1.(Question) [10 points] In the figure, an electron e travels through a small hole in plate A and then toward plate B. A uniform electric field in the region between the plates then slows the electron without deflecting it. (a)[2 points] What is the direction of the electric field? Four other particles similarly travel through small holes in either plate A or plate B and then into the region between the plates. Three have charges +q 1, +q 2, and -q 3. The fourth (labeled n) is a neutron, which is electrically neutral. (b)[8 points] Does the speed of each of those four other particles, in the region between the plates (circle the correct answer): (i) +q 1 increase decrease or remain the same (ii) +q 2 increase decrease or remain the same (iii) -q 3 increase decrease or remain the same (iv) n increase decrease or remain the same 2.(Question) [10 points] An isolated conductor of arbitrary shape has no net charge. Inside the conductor is a cavity within which is a point charge q = +3.1 10-6 C. (a) [5 points] What is the charge on the cavity wall? (i)+3.10 x 10-6 C (ii) +1.0 x 10-5 C (iii) -3.10 x 10-6 C (iv) -1.0 x 10-5 C (v) 0 C (b) [5 points] What is the charge on the outer surface of the conductor? (i)+3.10 x 10-6 C (ii) +1.0 x 10-5 C (iii) -3.10 x 10-6 C (iv) -1.0 x 10-5 C (v) +1.31x10-5 C 2
3. (Question) [10 points] In the figure to the right, a wire that carries a current consists of three sections of the same material but with different radii. Rank the sections according to the following quantities, greatest first: a) [5 points] current, i, (i)i B > i A > i C (ii) i A > i B = i C (iii) i B > i C > i A (iv) i A = i B = i C (v) i C > i B > i A b) [5 points] magnitude of current density, J, (i)j B > J A > J C (ii) J A > J B = J C (iii) J B > J C > J A (iv) J A = J B = J C (v) J C > J B > J A 4.(Question) [10 points] The figure below shows three long straight, parallel, equally spaced wires with identical currents either going into or out of the page. (a) [5 points] Rank the wires according to the magnitude of the force on each due to the currents in the other two wires, greatest first. (i)f a > F b > F c (ii) F c > F b > F a (iii) F a > F c = F b (iv) F b > F a > F c (v) F a = F b > F c (b) [5 points] What is the direction of the net force on current a due to the currents in the other two wires? (i) (ii) (iii) (iv) (v) (vi) 3
5.(Question) [10 points] The figure to the right shows three oscillating LC circuits with identical inductors and capacitors. a) [5 points] Rank the circuits according to the time taken to fully discharge the capacitors during the oscillations, greatest first. b) [5 points] Rank the circuits according to the angular frequency of the oscillations, greatest first. 6.(Question) [10 points] A magnet, shown in the figure to the right, is in the form of a cylindrical rod. It has a length L = 5.30 cm and a diameter D = 1.00 cm. It has a uniform magnetization of 4.70 10 3 A/m. Circle the correct answer below. (a) [5 points] What is the magnitude of its magnetic dipole moment? (i) 0.0196 J/T (ii) 1.2 J/T (iii) 0.038 J/T (iv) 0.001 J/T (b) [5 points] What is the direction of its magnetic moment? (i) Into the page, (ii) Out of the page, (iii) Down, (iv) Up, 4
7.(Question) [10 points] The figure shows light reaching a polarizing sheet whose polarizing direction is parallel to the y axis. We shall rotate the polarizing sheet 40 clockwise about the light's indicated line of travel. During this rotation, does the fraction of the initial light intensity passed by the polarizing sheet increase, decrease, or remain the same if it is initially polarized as follows? (Select all that apply.) (a) [5 points] unpolarized: Increases decreases remains the same (b) [5 points] polarized parallel to the x axis: Increases decreases remains the same 8.(Question) [9 points] Consider the figure to the right. Circle either True or False. (a) [3 points] The drawing in part (a) is a physically possible refraction. True or False? (b) [3 points] The drawing in part (b) is a physically possible refraction. True or false? (c) [3 points] The drawing in part (c) is a physically possible refraction. True or false? 5
9. Problem [20 points] Figure to the right shows an arrangement of four charged particles at the corners of a square. Use that q 1 = q 2 = q 3 = q 4 = 2.0 10 10 C, q 1 and q 2 are negative and q 3 and q 4 are positive, and a = 20 cm. (a) [4 points] Find the electric potential in the center of the square. Show your work to clarify how you arrived at your answer. (b) [4 points] Draw the direction of net electric field in the center of the square and label it. (c) [8 points] Calculate the magnitude of the net electric field in the center of the square. (d) [4 points] All of the four particles in the figure above have mass m = 0.05 kg. Now, in addition, assume a neutron in the center of the square (m n = 1.68x10-27 kg and zero charge). Calculate the net gravitational force on the neutron, due to the other four particles. NOTE: Only gravitational forces are important now. 6
10. (Problem) [20 points] As shown in the figure to the right, two semicircular arcs of radii R 1 and R 2 form part of the circuit ADFGA carrying current i. a) [7 points] Find an expression for the magnetic field at point C, which is the common center of the semicircular arcs, due to the two straight segments AD and FG. Explain your answer. b) [7 points] Derive an expression for the magnitude of the magnetic field at point C due to the complete circuit ADFGA. Express your answers in terms of R 1, R 2, i, and any constants as needed. c) [6 points] What is the direction of the magnetic field at point C? Explain your choice. 7
11.(Problem) [20 points] In the figure to the right a 20 turn coil of radius 4.0 cm and resistance 10 is coaxial with a solenoid with 5000 turns/m and radius 2.0 cm. The current in the solenoid increases linearly from 0 A to 50 A in a time interval t = 15 ms. a) [4 points] Calculate the magnitude of the magnetic field in the interior of the solenoid at the end of the 15 ms interval. b) [5 points] Calculate how much magnetic flux (magnitude) does the current in the solenoid produce, within the interior of the coil, at the end of the 15 ms interval. c) [6 points] Calculate how much current (magnitude) is induced in the coil during the 15 ms interval. d) [5 points] Indicate the direction of the current which is induced in the windings of the coil on the picture. Make sure to show the direction on both top and bottom part of the coil. Explain the reason for your choice. 8
12. (Problem) [20 points] In the figure to the right, R = 14.0 Ω, C = 6.60 µf, L = 54.0 mh, and the ideal battery has emf = 34.0 V. The switch is thrown to position a, kept in position a for a long time, and then thrown to position b. (a) [5 points] Calculate the initial current through the resistor just after the switch is thrown to position a. (b) [5 points] Calculate the charge on the capacitor after the switch is on position a for a long time. (c) [5 points] Calculate the angular frequency of the LC oscillations after the switch is thrown from a to b. (d) [5 points] Calculate the amplitude of the oscillations of the current. 9
13. (Problem) [21 points] In the figure to the right, a parallel-plate capacitor has square plates of edge length L =1.4 m. A current of 2.0 A charges the capacitor, producing a uniform electric field perpendicular to the plates. (a) [4 points] What is the displacement current i d through the whole region between the plates? (b) [6 points] Calculate de/dt in this region. (c) [6 points] Calculate the displacement current, (i d ) enc, enclosed by the square dashed path of edge length d = 0.60 m. (d) [5 points] Calculate around the square dashed path. 10
14. (Problem) [20 points] An airplane flying at a distance of 40 km from a radio transmitter receives a signal of intensity 30 µw/m 2. The transmitter radiates uniformly over a hemisphere, it is not a point source. (a) [5 points] Calculate the amplitude of the electric component of the signal at the airplane. (b) [5 points] Calculate the amplitude of the magnetic component of the signal at the airplane. (c) [5 points] Calculate the transmission power of the transmitter. (d) [5 points] Given the airplane s tail fin has an area of 10 m 2 and is made of totally reflective metal, calculate the radiation force on the fin due to the incident signal. Assume the fin is perpendicular to the signal. 11
Formula Sheet for LSU Physics 2113, Final Exam, Fall 14 Constants, definitions: g = 9.8 m s 2 R Earth = 6.37 10 6 m M Earth = 5.98 10 24 kg G = 6.67 10 11 m 3 kg s 2 R Moon = 1.74 10 6 m Earth-Sun distance = 1.50 10 11 m M Sun = 1.99 10 30 kg M Moon = 7.36 10 22 kg Earth-Moon distance = 3.82 10 8 m ɛ o = 8.85 10 12 C2 Nm 2 k = 1 = 8.99 10 9 Nm2 4πɛ o C 2 e = 1.60 10 19 C c = 3.00 10 8 m/s m p = 1.67 10 27 kg 1 ev = e(1v) = 1.60 10 19 J dipole moment: p = q d m e = 9.11 10 31 kg charge densities: λ = Q L, σ = Q A, ρ = Q V Area of a circle: A = πr 2 Area of a sphere: A = 4πr 2 Volume of a sphere: V = 4 3 πr3 Area of a cylinder: A = 2πrl Volume of a cylinder: V = πr 2 l Units: Joule = J = N m Kinematics (constant acceleration): v = v o + at x x o = 1 2 (v o + v)t x x o = v o t + 1 2 at2 v 2 = v 2 o + 2a(x x o) Circular motion: F c = ma c = mv2 r, T = 2πr v, v = ωr General (work, def. of potential energy, kinetic energy): K = 1 2 mv2 F net = m a E mech = K + U W = U (by field) W ext = U = W (if objects are initially and finally at rest) Gravity: Newton s law: F = G m 1m 2 r 2 Gravitational Field: g = G M r ˆr = dv g ( ) 2 dr 4π 2 Law of periods: T 2 = GM Potential Energy of a System (more than 2 masses): Gauss law for gravity: g d S = 4πGM ins Electrostatics: S Gravitational acceleration (planet of mass M): a g = GM Gravitational potential: V g = GM r r 3 Potential Energy: U = G m 1m 2 r 12 ( U = G m 1m 2 + G m 1m 3 + G m ) 2m 3 +... r 12 r 13 r 23 Coulomb s law: F = k q 1 q 2 Force on a charge in an electric field: F = q E r 2 Electric field of a point charge: E = k q r 2 Electric field of a dipole on axis, far away from dipole: E = 2k p Electric field of an infinite line charge: E = 2kλ r Torque on a dipole in an electric field: τ = p E Potential energy of a dipole in E field: U = p E z 3 r 2
Electric flux: Φ = E d A Gauss law: ɛ o E d A = q enc Electric field of an infinite non-conducting plane with a charge density σ: E = σ 2ɛ o Electric field of infinite conducting plane or close to the surface of a conductor: E = σ ɛ o Electric potential, potential energy, and work: V f V i = E = V, f i E d s E x = V x, In a uniform field: V = E s = Ed cos θ E y = V y, E z = V z Potential of a point charge q: V = k q r Potential of n point charges: V = Electric potential energy: U = q V U = W field Potential energy of two point charges: U 12 = W ext = q 2 V 1 = q 1 V 2 = k q 1q 2 r 12 Capacitance: definition: q = CV Capacitor with a dielectric: C = κc air Potential Energy in Cap: U = q2 Capacitors in parallel: C eq = C i Current: i = dq dt = Definition of resistance: R = V i n V i = k i=1 n q i r i=1 i Parallel plate: C = ε A d 2C = 1 2 qv = 1 2 CV 2 Energy density of electric field: u = 1 2 κε o E 2 Capacitors in series: J d A, Const. curr. density: J = i A, Charge carrier s drift speed: v d = Definition of resistivity: ρ = E J 1 C eq = 1 C i J ne Resistance in a conducting wire: R = ρ L A Temperature dependence: ρ ρ = ρ α(t T ) Power in an electrical device: P = iv Power dissipated in a resistor: P = i 2 R = V 2 R Definition of emf : E = dw dq Resistors in series: R eq = R i Resistors in parallel: 1 R eq = 1 R i Loop rule in DC circuits: the sum of changes in potential across any closed loop of a circuit must be zero. Junction rule in DC circuits: the sum of currents entering any junction must be equal to the sum of currents leaving that junction. RC circuit: Charging: q(t) = CE(1 e t τc ), Time constant τ C = RC, Discharging: q(t) = q o e t τc Magnetic Fields: Magnetic force on a charge q: F = q v B Lorentz force: F = q E + q v B Hall voltage: V = v d Bd = i B d = width to field and i, l = thickness to field and to i nle
Circular motion in a magnetic field: qv B = mv2 r Magnetic force on a length of wire: F = i L B with period: T = 2πm qb Magnetic Dipole: µ = Ni A Torque: τ = µ B Potential energy: U = µ B Generating Magnetic Fields: (µ 0 = 4π 10 7 T m A ) Biot-Savart Law: d B = µ 0 id s r 4π r 3 Magnetic field of a long straight wire: B = µ 0 2i 4π r Magnetic field of a circular arc: B = µ 0 i 4π r φ Force between parallel current carrying wires: F ab = µ 0i a i b 2πd L Ampere s law: B d s = µ 0 i enc Magnetic field of a solenoid: B = µ 0 in Magnetic field of a toroid: B = µ 0iN 2πr, Magnetic field of a dipole on axis, far away: B = µ 0 2π Induction: Magnetic Flux: Φ = B d A µ z 3 Faraday s law: E = dφ dt Induced Electric Field: (= N dφ for a coil with N turns) dt E d s = dφ dt Definition of Self-Inductance: L = NΦ i EMF (Voltage) across an inductor: E = L di dt Motional emf: E = BLv Inductance of a solenoid: L = µ 0 n 2 Al RL Circuit: Rise of current: i = E R (1 e tr L ), Time constant: τl = L R, Decay of current: i = i 0e tr L Magnetic Energy: U B = 1 2 Li2 Magnetic energy density: u B = B2 LC circuits: Electric energy in a capacitor: U E = q2 2C = CV 2 2 Magnetic energy in an inductor: U B LC circuit oscillations: q = Q cos(ωt + φ) (i = dq dt, q = Cv) ω = 1 LC ( R Series RLC circuit: q(t) = Qe Rt/(2L) cos(ω t + φ) where ω = ω 2 2L Transformers: Transformation of voltage: V s V p = N s N p Maxwell s Equations: E d A = q enc ɛ 0 B d A = 0 Turns ratio: N p N s E d s = dφ B dt ) 2 T = 2π ω 2µ 0 = Li2 2 f = 1 T Energy conservation: I p V p = I s V s B d s = µ 0 ɛ 0 dφ E dt + µ 0 i enc
Displacement current: i d = ɛ 0 dφ E dt Magnetization: M = µ volume Electromagnetic Waves: Wave traveling in +x direction: E = E m sin(kx ωt) B = B m sin(kx ωt) where E B, the direction of travel is E B, E m /B m = c, fλ = c, λ = 2π/k Velocity of light in vacuum = c = 1 µ0 ɛ 0 Energy flow: S = 1 µ 0 E B I = 1 2cµ 0 E 2 m E rms = E m 2 I = P Area Radiation force and pressure: total absorption: F r = IA c, p r = I c total reflection: F r = 2IA c, p r = 2I c Polarizing Sheets: Unpolarized polarized: I = 1 2 I 0 Polarized polarized: I = I 0 cos 2 θ Reflection/refraction: Law of reflection: θ i = θ r Law of refraction: n 2 sin θ 2 = n 1 sin θ 1 Total internal reflection (critical angle): θ c = sin 1 n 2 n 1 Polarization by reflection (Brewster s angle): θ B = tan 1 n 2 n 1