Quiz 4 (Discussion Session) Phys 1302W.400 Spring 2018

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1 Quiz 4 (Discussion ession) Phys 1302W.400 pring 2018 This group quiz consists of one problem that, together with the individual problems on Friday, will determine your grade for quiz 4. For the group problem, you are required to write out your answers and all the intermediate steps. The crib sheet is attached. You are allowed to use a non-programmable calculator, i.e., a calculator in which formulae cannot be memorized. No textbooks, notebooks, or personal notes are allowed. Please solve the problem on the sheets that are provided. Additional sheets will be given to you if you need them. Each group only needs to turn in one solution (please also return the question sheets and cover page). On each solution page, please print clearly the names and IDs of all members of your group, as well as the name of your TA.

2 PHY 1302 Cribsheet for quiz 4 Math dx e x = e x + C dx x n = xn+1 + C (n 1) n + 1 dx 1 x = ln x + C dx sin x = cos x + C dx cos x = sin x + C (1 + ɛ) n 1 + n ɛ for ɛ R = x + y R y = R sin θ θ x = R cos θ A B = A B Cos θ A θ A B = A B in θ B Constants k = 1 4πɛ N m2 C 2, ɛ C 2 N m 2, e C µ 0 = 4π 10 7 N A 2, m p m n kg, m e kg g 9.8 m s 2 (gravitational acceleration on Earth s surface) Mechanics Definitions : Laws : v = d r F = m a = d p, a = d v, τ = I α = d L, p = m v, α = d ω Constant acceleration : x(t) = x 0 + v 0 t a t2, v(t) = v 0 + a t, L = r p = I ω, τ = r F Centripetal acceleration : a c = v2 R Elastic force : F m = k x T = 2 π k Work and energy : W = F d l, K = 1 2 m v2 path r, U (r) = ref F d l

3 Electricity Coulomb s law : F 1 q 1 q 2 = 4πɛ 0 r 2 Gauss law : Φ E = Electric field of point charge q : E = 1 4πɛ 0 ˆr E d A = q enc ɛ 0 (infinitesimal flux : dφ E = E d A ) q r 2 ˆr infinite uniform linear charge density λ : E = λ 2 π ɛ 0 r infinite uniform surface charge density σ : E = σ 2ɛ 0 Dipole p = q d in external electric field : Electric potential : V (r B ) V (r A ) = V = U q 0 E = V = î dv dx ĵ dv dy Electric potential of a point charge q : V = 1 4πɛ 0 q r Electrostatic energy for a system of charges : Capacitance : C = q V τ = p E, U = p E U = i (V actually means V) = B A ˆk dv dz q i V i 2 E d l A Parallel plate capacitor : C = ɛ 0 (if no dielectric inside) ; V = E d d Electrostatic energy in a capacitor : U = q V 2 = CV 2 = q2 2 2C 1 Capacitors in parallel : C eq = C 1 + C 2 ; in series : = C eq C 1 C 2 Electric field in a dielectric : E E0 = κ Gauss law in a dielectric : Φ E = Electric current : I = dq (κ > 1 depends on the material) E d A = q enc κɛ 0 Ohm s law : V = RI ; Resistance : R = ρ L A ; Current density : J = qn vd ; I = ; energy density : u = U Vol = ɛ 0 2 E2 J d A

4 Resistors in series : R eq = R 1 + R 2 ; in parallel : Power : P = IV = I 2 R = V 2 /R Kirchhoff s rules : V = 0 ; I in = I out RC circuits loop 1 R eq = 1 R R 2 discharging capacitor (initial charge Q 0 ) : Q(t) = Q 0 e t/(rc) charging capacitor (initial charge 0) : Q(t) = CE ( 1 e t/(rc)) Magnetism Magnetic force : F = q v B (point charge) ; df = Id l B (current element) Cyclotron radius : R = mv qb Magnetic dipole moment : µ = NIAˆn. In external magnetic field : τ = µ B, U = µ B Magnetic field of a point charge : B µ 0 q v ˆr = 4π r 2 Magnetic field of a current element : db = µ 0 Id l ˆr (Biot avart law) 4π r 2 Gauss law for magnetism : B da = 0 Ampere s law : B d l = µ 0 I enclosed Magnetic field of C current loop : B = µ 0 2 R 2 I (z 2 + R 2 ) 3/2 (along loop axis) infinite wire : B = µ 0I 2πr infinite solenoid : B = µ 0 ni ; toroid : B = µ 0 NI 2π r Magnetic flux : Φ B = B da (= NB ˆnA for N planar loops in uniform B) Faraday s law : E = dφ B Magnetic flux through inductor : Φ B = LI (L = µ 0 n 2 la, n = N l, for solenoid) Energy stored in inductor : U = 1 2 LI2 ; energy density : u = U Vol = B2 2µ 0

5 Physics (Greven) Quiz 4 April 19 th, 2018 Name: ID: Name: ID: Name: ID: Name: ID: TA (circle one): Aaron West Dmitrii Torbunov Thomas Peterson Thomas Welle Group Problem (25 points). As shown in the figure below, a conducting rod of resistance R = 16 Ω and length l = 8 cm moves to the left (i.e. ˆx direction) with constant velocity v = 5 m/s in a region of constant magnetic field B = 0.4 T pointing out of the page (i.e. +ẑ direction). The rod moves without friction along conducting rails that have negligible resistance. 1. In what direction does the induced current flow in the conducting rod? What is the magnitude of this current? 2. How much energy is dissipated by the resistor during 1 minute? 3. An external agent must apply a force in order to keep the rod moving at constant velocity. What is the direction and the magnitude of this force? 4. Calculate the work done by the external agent during 1 minute. 5. If the rod is a cylinder of radius 0.3 cm, what is its resistivity?

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