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MIT 8 02T - Problem Set 7

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MIT Department of Physics8.02 Spring 2014Problem Set 7Due: Tuesday, April 8 at 9 pm.Submit the two online problems in this set (problems 1 and 2) online. Hand in the six writtenproblems in this set (problems 3-8) in your section slot in the boxes outside the door of 32-082 or26-152 depending on which is your classroom. Make sure you clearly write your name and sectionon your problem set.Text: Dourmashkin, Belcher, and Liao; Introduction to Electricity and Magnetism MIT 8.02 CourseNotes Revised.Week Nine: Faraday's Law No Problem Set due this weekW09D1 M/T Mar 31/Apr 1ReadingCreating Fields: Ampere's LawCourse Notes: Sections 9.3-9.4, 9.7, 9.10.2W09D2 W/R Apr 2/3ReadingFaraday's Law; Experiment 3: Faraday's LawCourse Notes: Sections 10.1-10.4W09D3 F Apr 4ReadingProblem Solving 6: Ampere's Law and Faraday's LawCourse Notes: Sections 9.10.2; 10.7, 10.9-10Week Ten: Magnetic Inductionand Energy; DC CircuitsProblem Set 7 Due Tuesday April 8 at 9 pmW10D1 M/T Apr 7/8ReadingInductance & Magnetic EnergyCourse Notes: Sections 11.1-11.3W10D2 W/R Apr 9/10ReadingDC Circuits & Kirchhoff's Loop RulesCourse Notes: Sections 7.1-7.5W10D3 F Apr 11ReadingProblem Solving 7: Building a CircuitCourse Notes: Sections 7.1-7.5, 7.10a) Zero electric and zero magnetic field. b) Zero electric field and magnetic field that is constant in time. c) Zero electric field and time-changing magnetic field. d) Electric field that is constant in time and zero magnetic field. e) Electric and magnetic fields that are both constant in time. f) Electric field that is constant in time and time-changing magnetic field. g) Time-changing electric field and zero magnetic field. h) Time-changing electric field and constant magnetic field. i) Electric and magnetic fields that are both time-changing.Problem 1:This problem must be submitted online.(Part a) The solenoid shown below has a non-constant current running through it which isincreasing at a constant non-zero rate, i.e. . The -axis is into the page.At a point inside the solenoid there is(Part b) A magnet has its north pole pointing upward. A conducting circular loop is movingdownwards beneath the magnet. The induced current in the coil as seen from above the magnet andthe force on the conducting loop due to the magnet are:I(t)dI/dt > 0zPa) current clockwise and force up. b) current counterclockwise and force up. c) current clockwise and force down. d) current counterclockwise and force down. e) current clockwise and zero force. f) current counter clockwise and zero force. g) zero current and zero force.Problem 2:This problem must be submitted online.What is the maximum electromotive force induced in a coil of 4000 turns, average radius cm, rotating at 30 revolutions per sec in the earth's magnetic field where the magnitude of themagnetic field is ?(in V)a = 120.5 × T10-4 =Problem 3: Magnetic Field of a ToroidYou must hand in a written solution to this problem. You may check your answers online ifyou wish, but this is not required.(Part a) A toroid has turns, and an inner radius , outer radius , and height . The toroid has arectangular cross section shown in the figures below.When a current is flowing through the toroid, what are the magnitude and direction of themagnetic field inside the toroid as a function of distance from the axis of the toroid?NabhIr(r < a) =B (a < r < b) =B (r > b) =BProblem 4: SolenoidYou must hand in a written solution to this problem. You may check your answers online ifyou wish, but this is not required.A long solenoid of radius carrying turns per unit length, is looped by a wire of resistance asshown in the figure below.The current in the solenoid is increasing at a constant rate, .(Part a) What is the magnitude of the current that flows through the loop? Indicate the direction ofthe current in the figure.(Part b) While the current is changing in the wire that is wrapped around the solenoid, what is thedirection and magnitude of the electric field inside the solenoid?(Part c) If the current in the solenoid is constant but now the solenoid is pulled out of the loop andreinserted in the opposite direction, what total charge passes through the resistor?anRI(t)dI/dt = h| | =Iinduced| | =E IΔQ =Problem 5: Force and Magnetic FieldYou must hand in a written solution to this problem. You may check your answers online ifyou wish, but this is not required.A very long straight wire carrying current directed into the plane of the figure below is suspendeda distance above a nearly infinite plane surface of thickness with uniform current density directed out of the plane of the page.(Part a) What force per unit length acts on the wire? Assume the current in the wire does not changethe uniform current density .(Part b) The wire is replaced by a particle of charge that is initially traveling parallel to thedirection of the uniform current density and is a distance from the surface, as shown in theabove figure. What is the direction and magnitude of the force acting on the charged particle?(Part c) Sketch the trajectory of the particle. Give the particle a mass . What magnitude of currentdensity is required in order that the charged particle just grazes the infinite surface?IRdJ J =dF dLqJ R=F mJJ =Problem 6: Nested SolenoidsYou must hand in a written solution to this problem. You may check your answers online ifyou wish, but this is not required.Two long solenoids are nested on the same axis, as in the figure below. The inner solenoid has radius and turns per unit length. The outer solenoid has radius and turns per unit length. Eachsolenoid carries the same current flowing in each turn, but in opposite directions, as indicated onthe sketch.Use Ampere's Law to find the direction and magnitude of the magnetic field in the followingregions. Be sure to show your Amperian loops and all your calculations.i) ii) iii) i) ii) iii) R1n1R2n2I0 < r <R1< r <R1R2< rR2=B =B =BProblem 7: Ampere's Law and Superposition PrincipleYou must hand in a written solution to this problem. You may check some of your answersonline if you wish, but this is not required.An infinitely long wire of radius carries a current density , which is uniform and constant. Thecurrent is directed out of the plane of the figure below.(Part a) Calculate the magnitude and direction of the magnetic field for (i) , and (ii). For both cases show your Amperian loop and


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