Electrodynamics Exam 3 and Final Exam Sample Exam Problems Dr. Colton, Fall 2016


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1 Electrodynamics Exam 3 and Final Exam Sample Exam Problems Dr. Colton, Fall 016 Multiple choice conceptual questions 1. An infinitely long, straight wire carrying current passes through the center of a circular loop of wire carrying current, as shown above. The long wire is perpendicular to the plane of the loop. Which of the following describes the magnetic force on the loop? (a) Outward, along a radius of the loop. (b) Inward. Along a radius of the loop. (c) Upward, along the axis of the loop. (d) Downward, along the axis of the loop. (e) There is no magnetic force on the loop.. Two long, identical bar magnets are placed under a horizontal piece of paper, as shown in the figure. The paper is covered with iron filings. When the two north poles are a small distance apart and touching the paper, the iron filings move into a pattern that shows the magnetic field lines. Which of the following best illustrates the pattern that results? a. b. c. d. e. 3. What is true about the BiotSavart Law? (a) It shows that the enclosed magnetic field is proportional to the enclosed electric current. (b) It shows that the net magnetic flux out of any closed surface is zero. (c) It shows how a magnetic field will interact with an electric circuit to produce an EMF. (d) It shows how magnetic fields are generated from steady currents. (e) It show how an Efield is found from the displacement current. 1
2 4. Which of the following is not a source of magnetic fields? (a) A DC current in a wire (b) A permanent magnet (c) A charged disk rotating at uniform speed (d) A time varying electric field (e) None of the above (all are sources of magnetic fields) 5. The work done on a charged particle by a magnetic field is (a) Always positive (b) Always negative (c) Zero (d) Not defined 6. A proton (positive charge) is released from rest in uniform E and B fields. E points towards the top of the page, B points into the page. Which of the paths will the proton initially follow? 7. A wire makes a 90 bend as shown. The current flows to the left, then downwards. What is the magnitude of the magnetic field created by the wire, at the point shown a distance s above the bend? (a) μ 0 I/(πs) (b) μ 0 I/(πs) (c) μ 0 I/(4πs) (d) μ 0 I/(8πs) (e) None of the above 8. A wire loop in a B field has a current I. The magnetic field is localized; it only exists in the hatched region, and is essentially zero everywhere else. Which way must I be flowing to hold the mass in place as shown (gravity points down)? (a) Clockwise (b) Counterclockwise (c) You cannot levitate a mass like this
3 9. The vector potential A due to a long straight wire with current I along the zaxis, will be in which direction? (Assume Coulomb gauge, and don t worry about the sign.) (a) (b) (c) 10. A sheet of current with surface current density K in the ydirection as shown serves as a boundary between two regions of space. If just to the RIGHT of the sheet, in what direction will B be just to the LEFT of the sheet? (The magnetic fields are not solely due to the sheet itself; they are produced in part by additional currents not shown.) (a) +x (b) x (c) +z (d) z (e) +y (f) y (g) None of the above 3
4 Problems Note In a few of the problems below, you need to know the following integral: dx 1 x 3 a x a a x 1. Find the magnetic field at the if a current, I, flows in the directions indicated by the arrows (the is right in the middle of the l segment) l l l. A loop of wire is formed into the shape of a rectangle with sides of length a and b (a > b). Find the magnetic field (both magnitude and direction) at the center of the rectangle if the wire carries a current I that flows in the counterclockwise direction. 3. A current flows counterclockwise in a square loop of side length a that lies flat in the plane of this page. Determine the magnetic field at a point a distance L to the right of the center of the loop where L > a. 4. A loop of wire is formed into the shape of a circle of radius R lying in the x y plane, and carries a current I that flows in the counterclockwise direction when viewed from above. Find the magnetic field (both magnitude and direction) as a function of x and y for an arbitrary point in the xy plane, for distances far away from the loop, x y R. z x y 5. An infinitely long, thick cylindrical shell of inner radius a and outer radius b carries a current given by the current density, J = k/s zhat. Find B everywhere due to this current. 4
5 b 6. An infinitely long, thick cylindrical shell of inner radius a and outer radius b carries a current given by the combined effects of a volume current density, J = k1 s zhat and a surface current density located at s=b: K = kb zhat. (There is no current inside s=a.) Find B everywhere due to these current densities. J K a 7. A circular torus of radius a is wound tightly with N loops of wire each carrying a current I. Find the magnetic field at the center of the torus s cross section (the point marked ). a 8. An infinitely long cylinder of radius R has a magnetization of M = k/s zhat. Find the magnetic field everywhere due to this cylinder (a) via bound currents and Ampere s law for B, and (b) via Ampere s law for H. M 5
6 9. An infinitely long, thick cylindrical shell of inner radius a and outer radius b has a magnetization of M = k/s hat. Calculate the magnetic field as a function of distance from the center of the cylinder. 10. An infinitely long thick cylindrical shell of inner radius a and outer radius b carries a magnetization of M = ks zhat. Inside the cylinder is empty space. (a) Find the magnetic field everywhere using the bound currents. (b) Find the magnetic field everywhere using the Hfield. 11. An infinitely long cylinder of radius R has a magnetization of M = k/s zhat. Find the magnetic field everywhere due to this cylinder. 1. A metal cylinder with a magnetic susceptibility of m is bent into a circular shape and the ends are welded together to form a closed loop in the shape of a torus (a torus is donutshaped object). Wire is wrapped tightly around the surface of the torus so that it is evenly covered by the N windings. If the wire carries a current I, calculate the magnetic field inside and outside the torus in terms of given quantities and fundamental constants. 13. An infinitely long cylinder of radius R centered on the zaxis is magnetized by a free current such that its magnetization is M = k/s hat. The magnetization is such that B = 0 everywhere. Find the free current. What is m? What type of magnetic material is this cylinder made of? 14. A solid infinite cylinder of radius R is placed so that its axis coincides with the zaxis. A free current density is flowing in the cylinder, J f = k s zhat. This current produces a magnetization in the cylinder of s 3 hat. Assuming the cylinder is made of a linear magnetic material: (a) Find the magnetic susceptibility of the cylinder in terms of given quantities and fundamental constants. (b) Find the magnetic field inside the cylinder. 15. An insulated wire is wrapped around a long thin metal cylinder having magnetic susceptibility m. It s wrapped as a solenoid having n turns per length, and a current I is sent through the wire. Use what your book calls the boundary conditions, which is what I ve been calling the discontinuity in B (and/or H), to determine the strength of the magnetic field in the air right outside the end of the cylinder. You may assume that the field inside the solenoid is the same as the case for an infinite solenoid. 16. A circular loop (radius R) is centered at the origin, and originally positioned such that it lies in the xz plane. There is a constant magnetic field present in the area, B = B xhat. The loop is then rotated about the zaxis at a frequency f. Find the induced voltage in the loop as a function of time. 6
7 17. What s the current through the V battery? Which direction is it flowing? 18. For this resistor network, (a) name two resistors that are in series, and (b) two that are in parallel. 19. For this circuit, V b (t) = 1 for t < 0, and e t/10 for t>0. (I.e. the battery starts shutting off at t = 0, but takes some time to shut off completely.) Find the voltage of the capacitor as a function of time. 0. What is the complex impedance of this circuit? (No need to simplify very much.) 7
8 1. The complex impedance of the box is Zbox = i (10000/ω + 10ω) in standard units. What is Vout / Vin for a frequency of 100 Hz? (The circuit uses shorthand notation, where an AV voltage is connecting between in and the ground symbol, and the output voltage is connecting between out and the ground symbol.) Note: the next three problems all involve the displacement current, and are hence not applicable for exam 3 but could be representative of final exam problems from Chapter 7.. A parallel plate capacitor of two circular plates of area A has vacuum between the plates. This capacitor is connected to a battery of constant voltage, V0. The plates are then slowly oscillated so that the distance between them is given by d = d0 + d1sint. Find H between the plates due to the displacement current. 3. A parallel plate capacitor of two circular plates of area A has a dielectric with permittivity between the plates. This capacitor is connected to a battery of constant voltage V0. The plates are slowly oscillating so that the distance between them is given by d = d0 + d1sint. Find H between the plates due to the displacement current. 4. The following fields exist in a region of vacuum where 0 < x < a, E y 0aH0 x sin sinkz wt a kah 0 x sin sinkz wt a x H z H 0 cos coskz wt a H x All other components are zero. Find the free charge density, free current density, and the displacement current in this region. H0, a, k, and are constants and 00 = k + /a. (Hint: Use the differential form of Maxwell's Equations.) 8
9 Answers (let me know if you detect any errors) Multiple choice 1. E. B 3. D 4. E 5. C 6. C 7. C 8. A 9. A 10. G Worked problems 1. B I 0 l into the page 0I 1 a b. B a b b a out of the page I 1 a L a 1 B IR B 4 x y L a L a a L a a L a a L a a 3 0 zˆ a out of the page. 5. B 0k ln( s / a) 0 for s<a; B ˆ s for a<s<b; B 0k ln( b / a) ˆ for s>b s 6. B 0k for s<a; B s a ˆ 4s 0NI 7. B ˆ a k s for a<s<b; b a k b B ˆ for s>b k 8. B 0 zˆ s for s<r; B 0 for s>r (the same answers using either method) 9
10 B 0k for s<a; B ˆ s B 0 for s<a; B ks zˆ 0 for a<s<b; B 0 for a<s<b; B 0 for s>b for s>b (the same answers using either method) k 11. B 0 zˆ s for s<r; B 0 for s>r 1. 0(1 m) NI B ˆ inside, B = 0 outside s 13. I k located at the origin (middle of the cylinder), in the zhat direction; m = 1; it s diamagnetic (but actually, I don t think diamagnetic materials with such a large m exist). 14. m 4 3 k ; B ˆ 0s inside k 4 B m ˆ ni z V R fbsinft ma up 18. R7, R8 are in series; R1, R are in parallel 19. Vc = e t/ e t/10\ 0. R1 i/(c1) + (1/R + ic i/(l)) Final exam only problems. H 0 V 0d d d 0 1 s cos( t ) ˆ sin( ) t 1 3. same as the last problem, but with 0 replaced by 4. free = 0; J 00 ah 0 x free 0; J d sin coskz wtyˆ a 10
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