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Practice Problems β Part 2 TF: Widagdo Setiawan, [email protected] 1. Logic problem a. Write down the truth table b. Write down the Boolean expression for H(A,B) 2. Give the Boolean expression for the following circuits 3. Write down the Boolean expression for the following truth table A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 F(A,B,C) 1 0 0 0 1 1 0 1 G(A,B,C) 0 0 0 1 0 1 1 1 4. Write and XOR in terms of AND, OR, and NOT 5. Parity bits We would like to create a device that takes three inputs, π΄, π΅, πΆ and gives one output, π, the parity of the inputs. In other words, the output is one if there are an odd number of ones among the inputs, and zero otherwis a. Write down the truth table for this device b. Draw a logic diagram showing an implementation of the parity bit generator 6. A mass spectrometer accelerates doubly ionized atoms over a potential difference π before they enter a uniform magnetic field π΅ which is perpendicular to the direction of motion of the ions. If π is the radius of the ionsβ path in the magnetic field, what is the mass of one ion? 7. A current πΌ goes through a rectangular wire in the direction shown with the arrows in the figure. The dimensions of the rectangle are π and π as shown. A uniform magnetic field of strength π΅ is in a direction perpendicular to the paper as shown in the picture. What is the torque on the rectangular loop? 8. Same as previous problem, but this time, the magnetic field is pointing to the left. 9. Charged particle velocity selector An electron with velocity π£ to the right enters a parallel plate capacitor with separation between the plates π (see picture below). At the capacitor, there is magnetic field going out of the page with magnitude π΅. The capacitor is connected to a battery. What battery voltage π do we have to use so that the electron can travel through the capacitor without being deflected? 10. True or false questions: ββ force a. Charged particles in a magnetic field cannot experience an acceleration due to the π£β × π΅ ββ force b. Charged particles in a magnetic field cannot change speed due to the π£β × π΅ c. A positive charge and a negative charge turn in the opposite direction under uniform magnetic field d. e. f. g. Magnetic field lines start at the north pole of a magnet, and end at the south pole of the magnet The magnetic south pole of the earth is located near the earthβs north pole Magnetic field lines go from the south pole to the north pole inside a magnet. NASAβs tether experiment generated electric power by moving a very long conducting wire (attached to the shuttle) through the Earthβs magnetic field. The electric energy that was generated was at the expense of the kinetic energy (thus speed) of the shuttle. h. When a magnetic flux through a conducting loop is zero, there cannot be an induced EMF in that loop 11. An electron is traveling with a constant velocity π£ as shown in the picture below. The electron enters a region with uniform magnetic field π΅. Which of the following trajectories best describes the electronβs motion in this region 12. An electron is traveling with a constant velocity π£ as shown in the picture below. The electron enters a region with uniform magnetic field π΅. Which of the following trajectories best describes the electronβs motion in this region 13. Two wires, each carrying current of 10 π΄ in the same direction are separated by 1 π. Calculate the force per unit length on these wires. 14. Electromagnetic pump on molten metal. (Irodov 3.261) In an electromagnetic pump designed for transferring molten metals, a pipe section with molten metal inside is located in a uniform magnetic field π΅. A current πΌ is made to flow across this pipe section in the direction perpendicular both to the vector π΅ and to the axis of the pipe. Find the pressure difference produced by the pump. π΅ = 0.1π, πΌ = 100 π΄, π = 2 ππ. 15. Rail Gun Consider the following setup A pair of conducting rails with negligible resistance is placed horizontally. A bar of mass π = 1 ππ with resistance = 10 Ξ© , and length πΏ = 2 π, that can freely move is placed on top of these rails, initially at rest. A magnetic field of magnitude π΅ = 5 π is applied on the entire setup with the direction shown on the picture above. A battery with voltage π = 1000 π is then connected to the rails as shown in the figure. Ignore Faradayβs law in this problem. Assume that there is no friction. Ignore the magnetic field generated by the rails and the bar. a) What happen with the bar qualitatively after the battery is connected? b) What is the initial acceleration of the bar just after the battery is connected? c) Write down the relevant differential equation to describe the system above involving the position of the bar d) Calculate the position of the bar as a function of time e) Calculate the velocity of the bar as a function of time f) How long does the rail have to be in order for the bar to escape the earthβs gravitational field? 16. Rail gun with Faradayβs law. Same problem as before, but this time, you cannot neglect Faradayβs effect. You can still neglect the induced magnetic field. This problem captures a lot of things that you learn for the past few weeks: current, voltage, magnetic field, magnetic force, EMF. The math is similar to the capacitor charging problem. This could be a nice but extremely hard exam problem. a) What happen with the bar qualitatively after the battery is connected? b) What is the initial acceleration of the bar just after the battery is connected? c) Write down the relevant differential equation to describe the system above involving the position of the bar d) What is the terminal velocity of the bar? e) Calculate the velocity of the bar as a function of time f) Calculate the position of the bar as a function of time g) Can the bar ever escape the earthβs gravitational field? If yes, what is the minimum length of the rail? If not, what parameter should you change (and to what value), such that the bar can escape the earthβs gravitational field. 17. A simple electric generator (as shown below) is rotating about the y-axis with a frequency of π. There is a uniform magnetic field π΅ in the +π§ direction. The rotor consists of a coil of π windings each with an area of π. The generator, through slipping contacts, is powering a light bulb whose resistance is π (see the figure). The ohmic resistance of the coil is negligibly small compared to that of the light bulb. You may also assume here, for simplicity, that the self-inductance of the coil is negligibly small. a. What is the maximum value magnetic flux? b. What is the maximum value πΌπππ₯ of the induced current? (Note that the current changes as a function of time). Also indicate in the figure one of the two positions of the coil when this maximum current occurs. 18. A Conducting bar of length D rotates with angular frequency π about a pivot P at one end of the bar (see the figure). The other end of the bar is in slipping contact with a stationary conducting wire in the shape of a circle (we only show a smaller part of that circle to keep the drawing simple). Between point P and the circular wire, there is a resistor R as shown. Thus the bar, the resistor and the arc form a closed conducting loop. The resistance of the bar and the circular wire are negligibly small. There is a uniform magnetic field B everywhere, it is perpendicular to the plane of the paper as indicated. What is the induced current in the loop? Express your answer in terms of π·, π, π , and π΅. 19. In the figure below, a uniform magnetic field points into the page. (The magnetic field vectors are indicated by the β¨βs ). Four particles with the same mass and different electric charges follow the paths shown as they pass through this magnetic field with identical, constant speed. Ranks the charges from the least to the greatest? (Remember, any negative number is smaller than any positive number) 20. MRI Concept questions: a. What is Larmor frequency? b. How do you select the plane that you want to image? c. When should we turn the π΅0 on/off? d. When should we turn the π΅1 on/off? e. What is the oscillation frequency of π΅0 ? f. What is the oscillation frequency of π΅1 ? g. What particle does typical MRI machine image? h. What are the differences between MRI imaging and x-ray imaging? 21. Spherical Mirror (Past exam) An object is placed on the axis of a spherical mirror. The radius of curvature of the mirror is π . As a function of the distance π₯ from the mirror to the object, a. Sketch the distance π¦π from the image position to the mirror. Indicate the meaning of any relevant signs. Indicate on the plot the focal point and the role of the radius R, and show where π¦π = π₯ b. Sketch the magnification of the image. Indicate the meaning of any relevant signs. Indicate on the plot the focal points and the role of the radius π . Label any points where the magnification is 1, 0, or β1. 22. Mirascope Mirascope, can be purchased for $6.83 at amazon.com) is constructed by having two concave mirror facing each other as shown in the figure above (shiny sides are on the inside). We then put an object on the bottom of this device (represented by the purple arrow). The two mirrors create images sequentially, creating the final image above the mirascope itself represented by the green arrow. The final image is inverted. We also know that the two mirrors have the same curvature. Finally, we also know that the image looks weird but awesome. a. b. c. d. Is the image real or virtual? Which one is the first mirror and which one is the second mirror? What is the focal length of the two mirrors given the information that we have? Is this a hologram? 23. Light sail Consider a mirror with mass π = 1 gram floating in space initially at rest (itβs not mounted to anything). A 100 Watt laser beam shines at the mirror. How long do we have to wait until the mirror reach 100 ππ/βππ’π? Note that the mirror reflects the laser beam. 24. Slowing down atom A Sodium atom moves at 1500 m/s. We shine a laser beam at 589 nm from the opposite side to slow down this atom. a. How many photons does the atom need to absorb until the atom does not move anymore? b. If the atom can absorb 6 million photons per second, how long does it take to slow down this atom? c. How far does the atom travel during the slowing process? 25. Photodiode You have a photo diode which is a device that converts photon into electron. You shine 1 mW laser beam at 589 nm on the photodiode. What current do you measure coming out of the photodiode? 26. Photoelectric effect Consider a setup shown in figure above. We shine a laser on the bottom plate. We then measure the current on the Ampere meter. a. Set V=0. You have 2 lasers, one at 300 nm and one at 900 nm (both are invisible). You do not know which one is which. Letβs call the laser A and B. You shine laser A, then you measure the current. The current is zero. When you shine laser B however, you measure some current. What wavelength is laser A? b. You shine laser B on the bottom plate and you measure some current. What do you have to do with V to make the current goes to zero? (apply negative V or positive V)? c. You increase the magnitude of V slowly while measuring the current. When the magnitude of V is exactly 1 V, you see the current goes to zero. What is the work function of the bottom plate?