#1-25 #26 Phy207 Final Exam (Form1) Professor Zuo Fall 2018 On my honor, I have neither received nor given aid on this examination #27 Total Signature: Name: ID number: Enter your name and Form 1 (FM1) in the scantron sheet. Do all 25 multiple choice questions and 2 essay questions. Multiple choice questions are worth 3 pts each and the essay problem 20 pt each for a total of 115 pt. There is a formula sheet attached at the end. This is a closed book exam, you must work independently! No collaboration is allowed. Prohibited items: any electronic devices including cell phones and calculators, pens, backpacks, notes, books. Anyone found cheating during the exam will automatically receive an F grade for the course and sent to the honor s court.
For the next seven questions, consider two concentric spherical conducting shells with inner radius a and outer radius b for the inner shell and inner radius c and outer radius d for the outer shell. The inner shell carries a charge 2q and the outer shell carries a charge q. 1. Find the surface charge density at r=d A) qq 4ππdd 2 B) 2qq 4ππdd 2 C) 3qq 4ππdd 2 qq D) E) 0 4ππdd 2 2. Find the electric field for r>d A) kkkk 2kkkk B) 3kkkk C) kkkk D) E) 0 3. Find the electric field for b<r<c A) kkkk 2kkkk B) 3kkkk C) D) kkkk E) 0 4. Find the electric potential at r=c, assuming VV(rr = )=0 A) kkkk dd B) kkkk cc C) 3kkkk dd D) 3kkkk cc E) 0 5. Find the electric potential difference between the two shells. A) 2kkkk( 1 bb 1 cc ) B) 2kkkk(1 aa 1 cc ) C) 2kkkk(1 bb 1 dd ) D) 3kkkk(1 bb 1 cc ) E) 0 6. Find the electric field energy contained within a thin shell of radius r (b<r<c) and thickness dr. A) 1 εε 2 oo( 2kkkk )2 2ππππππππ B) 1 εε 2 oo( 2kkkk )2 4ππ C) 1 εε 2 oo( 3kkkk )2 2ππππππππ E) 0 D) 1 2 εε oo( 3kkkk )2 4ππ 7. Find the total electric field energy between the two shells. A) 1 εε 2 oo( 2kkkk )2 4ππ 3 (cc3 bb 3 ) B) 2ππεε oo (2kkkk) 2 ( 1 1 ) bb cc C) 2ππεε oo(2kkkk) 2 ( 1 1 ) aa dd D) 2ππεε oo (2kkkk) 2 ( 1 1 ) aa cc E) 0
8. A very long, solid, conducting cylinder of radius R carries a current along its length uniformly distributed throughout the cylinder. Which one of the graphs shown in the figure most accurately describes the magnitude B of the magnetic field produced by this current as a function of the distance r from the central axis? A) 1 B) 2 C) 3 D) 4 E) 5 9. For the next two questions, consider long, straight conductors with square cross section carrying current I each are laid side by side to form an infinite current sheet, as shown. The conductors lay in the x-y plane and carry current in the +x-axis direction. There are n conductors per unit length along the y-axis direction. What is the direction of the magnetic field above the current sheet? A) xx B) zz C) xx D) yy E) yy 10. Apply Ampere s law, what is the magnitude of the magnetic field? A) BB = 1 2 μμ oonnnn B) BB = μμ oo nnnn C) BB = 2μμ oo nnnn D) BB = 0 E) None of above
For the next four questions, consider a small long solenoid S1 placed inside a large long solenoid S2 as shown. S1 has N1 turns and a length l and S2 has the same length l but with N2 turns with radius r1 and r2, respectively. If S2 carries a time dependent current i2 through it. 11. What is the magnetic field inside S1? A) μμ oo NN 2 ii 2 2 B) μμ oo NN 2 ii 2 4ππ 2 C) μμ oo NN 1 ii 2 ll D) μμ oo NN 2 ii 2 ll E) 0 12. What is the mutual inductance MM = 1 ii 2? A) μμ oo NN 1 NN 2 ll ππππ 1 2 ii 2 B) μμ oo NN 1 NN 2 ll 2 NN ππππ 1 C) μμ 1 NN 2 2 oo ππππ ll 2 D) 0 E) None of above 13. Now if a time dependent current i1 flows through S1 instead of i2 through S2, what is the induced voltage on S2 (M is the mutual inductance)? A) MMii 1 B) MM ddii 1 C) MM ddii 2 D) MMii 2 E) None of above 14. A conducting bar of length 1.0 meter is being moved at a constant velocity of 1.0m/s from left to right on a track by an external force F as shown. A uniform magnetic field B=2.0T is directed out of the plane. The bar makes a good electrical contact and slides without friction. A constant force F=1.0N is applied. What is the resistance of the resistor R? A) 1 Ω B) 2 Ω C) 3 Ω D) 4 Ω E) Cannot be determined
For the next three questions, consider the circuit shown in the figure, the switch has been open for a very long time. 15. What is the potential drop across the 15.0-mH inductor just after closing the switch? A) 150 V B) 0V C) 200 V D) 100 V E) None of above 16. What is the potential drop across the 70.0-µF capacitor after the switch has been closed for a very long time? A) 150.0 V B) 133 V C) 200 V D) 0V E) None of above 17. If the switch S is opened again, what is the current going through the 75 Ω resistor just after it is opened? A) 2 A B) 8/3 A C) 0 A D) 8 A E) None of above For the next two questions, consider a long solenoid with a time-dependent current going through it. The figure shows the cross section of the solenoid with the magnetic field going into the page and the current i going through the coil decreases with time in the form of ii(tt) = II oo ee tt ττ where t is time and II oo and ττ are constants. The solenoid has n turns per unit length and radius R. For the question here, consider point c at r=r/2. 18. Find the magnitude of magnetic field at point c at tt = ττ A) 0 B) μμ oo nnii 0 C) μμ oo nnii 0 /ee D) μμ oo II 0 /2RR E) None of above 19. Find the induced electric field at point c at t= ττ A) 0 B) μμ oonnii 0 RR ȷȷ C) μμ oonnii 0 RR ȷȷ 4ττττ 4ττττ D) μμ oonnii 0 RR ıı 4ττττ E) μμ oonnii 0 RR ıı 4ττττ 20. Which one of the phasor diagrams shown below represents that the total voltage is ahead of current in a series LRC circuit? A) 1 B) 2 C) 3 D) 4 E) 5
21. Consider a serial LRC circuit that has a capacitive reactance (due to its capacitance) of 5 kω, a inductive reactance (due to its inductance) of 2 kω, and a resistance of 4 kω at angular frequency ωω. What is the resonant frequency of the circuit? A) 2 ωω B) 5 ωω C) 5 2 5 ωω D) 2 2 ωω E) None of above 5 22. Which one of the following lists is a correct representation of electromagnetic waves from longer wavelength to shorter wavelength? A) radio waves, infrared, microwaves, UV, visible, X-rays, gamma rays B) radio waves, UV, X-rays, microwaves, infrared, visible, gamma rays C) radio waves, microwaves, visible, X-rays, infrared, UV, gamma rays D) radio waves, microwaves, infrared, visible, UV, X-rays, gamma rays E) radio waves, infrared, X-rays, microwaves, UV, visible, gamma rays 23. If an electromagnetic wave has components Ey = E 0 sin(kx + ωt) and Bz = B 0 sin(kx + ωt), in what direction is it traveling? A) -x B) +x C) +y D) -y E) +z 24. A laser with a power of P has a beam radius of R. What is the peak value of the electric field in that beam? A) 2ccμμ oopp ππrr 2 B) 2μμ oopp ππrr 2 C) 2μμ oopp ccccrr 2 D) 2ccεε oopp ππrr 2 E) 2εε oopp ππrr 2 25. A very small source of light that radiates uniformly in all directions produces an electric field with an amplitude of EE mm at a distance R from the source. What is the amplitude of the magnetic field at a point 3R from the source? A) EE mm 3cc B) EE mm 9cc C) EE mm 3cc D) EE mmcc 3 E) EE mm cc
Do both essay questions below. Please provide detailed steps and answers for each part. 26. Consider a current in a long straight wire AB is decreasing steadily at a rate of di/dt= αα. A) At an instant when the current is i, what is the magnetic flux going through the narrow, shaded strip of length L and width dr at distance r from the wire? B) What is the total flux going through the loop? C) What is the magnitude and direction of the induced current in the loop if the loop has a resistance R? D) What is the total magnetic force acting on the loop due to the long wire? Solution: A) dd = BB(rr) = BB(rr)LLLLLL = μμ ooii bb B) = dd = aa C) II iiiiii = eeeeee RR bb μμ ooii LLLLLL aa 2ππππ = dd RR 2ππππ LLLLLL = μμ ooii 2ππ LL ln bb aa = μμ ooαα LL ln bb, direction of the induced current 2ππππ aa is in the clockwise direction. D) TThee ffffffffff aaaaaaaaaaaa oooo tthee llllllll cccccccccc ffffffff tthee ffffffffffff oooo aaaaaa ffffffff ssssssssss, tthee ffffffffffff oooo tthee tttttt and bottom sides cancel out. The force on the near side (r=a) is given by Fa=Iind LB(a)= μμ ooii LLIind, direction is to the left; the force acting on the far side (r=b) is 2ππaa given by Fb=Iind LB(b)= μμ ooii LLIind, direction is to the right. The next force is 2ππbb Fa-Fb= μμ ooii 2ππ (1 1 )LLIind, direction is toward the wire. aa bb
27. A capacitor consisting of two circular plates of radius R and separation d is being charged with a constant current I. Neglecting the fringing effect (assume uniform electric field inside the capacitor), A) Find the magnetic field at a point between the plates and at a distance r (r<r) from the axis of the plates; B) Find the Poynting vector at the same point (magnitude and direction) when the charge densities on the plates are ±σ; C) What is the rate of energy flow through the surface of the cylinder of radius r and length d? A) The magnetic field inside the capacitor can be calculated using the displacement current dd BBBBBB = BB2ππrr = μμ oo εε oo = μμ ooεε oo ππ, It is easy to see B(r<R)= μμ ooεε oo rr, EE = σσ = εε oo QQ 2, I=dQ/dt, so de/dt= ππrr 2 εε oo ππrr 2 εε oo TThee mmmmmmmmmmmmmmmm ffffffffff cccccc bbbb ssssssssssssssssssss tttt BB(rr) = μμ ooii RR 2 II B) The Poynting vector is defined by SS = EE BB, E= σσ is from the same direction of current, μμ oo εε oo and B is going clockwise. The cross product is pointing toward the center. The magnitude of the Poynting s vector is given by S= εε oorr EE 2 C) The energy flow though the surface the cylinder of radius r and length d equals the area of the cylindrical surface (2ππrrrr) times the poynting vector=2ππrrrr SS = εε oo ππ dd EE = dduu ee = dd( 1 2 εε ooee 2 ππ dd).