Practice For The Diagnostic Exam - Physics and Space Sciences, Florida Institute of Technology

Transcription

1 Three small point-like balls are placed as shown below in the x-y plane. Each ball is charged as shown. Calculate the magnitude and direction of the electric field at the origin of the coordinates due to all three balls. Assume the balls do not move due to any mutual force. The axes are marked off in meters. - 3 nc + 6 nc 5 m 10 m 15 m + 3 nc 1 / 32

2 A water molecule has an intrinsic dipole moment of 6.17 x Cm. Initially, the molecule is orientated in the same direction as an external electric field of magnitude of 4.2 x 10 6 N/C. Suppose the molecule absorbs an amount of thermal energy equal to 1.66 x Joules and all of it goes into rotating the molecule torsionally. Through what angle does it rotate? 2 / 32

3 A molecule with an intrinsic dipole moment is placed between two oppositely charged metal plates as in the figure below. For reference, assume the plates are parallel to the x-y plane and the vertical line represents the z-axis. Then the angle g sweeps vertically down through the z-y plane. The charged metal plates have the same constant, surface charge density. Let q = 2 nc, d = 30 nm and the angle g = 30 degrees. As shown, the molecule s potential energy due to the electric field created by the plates is -5.0 x J. Assume a uniform electric field between the plates. a. Calculate the surface charge density on the plates. b. Determine which plate is positively charged. + q g - q d 3 / 32

4 A thin plastic rod of length L=50 cm is placed on the x-axis as shown. It is charged positively and given a linear charge density of 100 micro-coulombs per meter. a) Show how to derive the general equation for the electric field at any field point P along the positive x-axis as shown. b) Instead of being spread out along the rod, suppose all of the charge on the rod were concentrated into a point at the center of the rod. In this case, what would be the magnitude of the electric field at point P? Assume S = 40 cm and express your answer as a number with appropriate units. L P S 4 / 32

5 A small charged ball is located at the center of an imaginary, hollow, regular tetrahedron. The tetrahedron and the ball are drawn below. The length of each edge is 10 cm and each face is an equilateral triangle. The ball is charged positively to 88.5 nc. Calculate the electric flux through any one flat face of the tetrahedron. Remember to include appropriate units. 5 / 32

6 A metal spherical shell with inner radius, a, and outer radius, b, has a point-particle located at its center. The particle is negatively charged with -35 micro-coulombs. The shell is positively charged with +15 micro-coulombs and it is insulated from its surroundings. Assume a vacuum inside the shell where the positively charged particle is located and outside the shell too. A cutaway view of this is shown below. b a a. Determine an expression for the charge density on the inner surface of the metal shell. b. Determine an expression for the charge density on the outer surface of the metal shell. c. Determine an expression for the magnitude of the electric field in the region a < r < b. d. Determine an expression for the magnitude of the electric field in the region r > b. 6 / 32

7 The drawing below shows an electric potential surface plot of several small charged particles sitting in the x-y plane. The electric potential is graphed on the vertical (z) axis. Suppose each particle is charged to either plus or minus 6 C and are located at x and y coordinates plus or minus 5 meters as shown. Two views of the same potential surface are given for perspective. Part (d) is optional extra credit. The other parts will be graded as usual. a) Give the name of this particular arrangement of charged particles. b) What is the magnitude of the electric field at the origin? c) In one sentence, give a physical argument to explain your answer to the previous question. d) Write one equation in Cartesian coordinates for the entire potential surface, V(x,y). 7 / 32

8 A ball with a positive charge of 12 nc is placed at the origin of a coordinate system. Imagine that gravity is turned off and the charged ball cannot move. A small positively charged test particle (q = 0.1 nc) is brought toward the ball from very, very far away. Calculate the work done to move the particle to a distance 0.5 meter from the ball. Show all your work. 8 / 32

9 A parallel plate capacitor is connected to a power supply set to 20 Volts. A dielectric slab, just large enough to take up all the space between the plates is inserted between the plates. For all parts of this question, the power supply remains connected to the capacitor. For all parts except (b) and (d), the slab is fully inserted into the capacitor. a) A capacitance meter placed across the plates measures 7.09 nano-farads (nano = 10-9 ). The plate area is 1600 cm 2 and the thickness of the slab is 0.4 cm. What is the dielectric constant of the slab? b) What is the strength of the electric field between the plates before ANY dielectric was inserted? c) What is the strength of the electric field between the plates after the dielectric is inserted? d) What is the charge stored on one of the plates before ANY of the dielectric was inserted? Assume the permittivity of free space and that of air are approximately the same. e) What is the charge stored on one of the plates after the dielectric is fully inserted? f) What series resistance would you add to this circuit if you want it to have a time constant of one micro-second (10-6 sec)? 9 / 32

10 A 20 Volt battery is connected to a network of four capacitors as shown below. a) Calculate the equivalent capacitance of the network. b) Calculate the amount of charge stored on the 2 micro-farad capacitor. c) Calculate the voltage across the 8 micro-farad capacitor. 2 μf 0.4 μf 0.6 μf 20V 8 μf 10 / 32

11 A power supply and a resistor are placed in series with one another. An ideal ammeter measures the current in the circuit and an ideal voltmeter measures the voltage of the power supply. A separate knob on the power supply lowers the current and the resulting voltmeter and ammeter readings are plotted on the graph below. a) Draw a circuit diagram of this situation. b) Determine the ideal EMF of the power supply. c) Calculate the internal resistance of the power supply. 11 / 32

12 For the circuit diagram drawn below suppose the voltage source is set to 8 Volts and RA = RB = RC = 20 Ω and RD = 40 Ω. a) Calculate the equivalent resistance of the branch containing resistors A, B and C. b) Find the total resistance of the entire circuit. c) Calculate the current leaving the voltage source. d) What is the voltage drop across resistor D? e) Calculate the current flowing through resistor D. 12 / 32

13 Examine the following circuit diagram. Suppose the unknown EMF is set to 43.1 Volts. a) Apply Kirchhoff s voltage rule to obtain three equations in three unknowns (the currents in each branch of the circuit). You do not need to solve the equations for the currents. b) Apply Kirchhoff s current rule to obtain an equation for the currents entering and exiting the junction immediately above the 48 Ohm resistor. 13 / 32

14 Six identical resistors of resistance R are connected to an ideal battery in such a way that the total resistance of the circuit is 3R. a) Draw a circuit diagram that shows how this is possible. b) If the battery supplies 12 Volts, determine the voltage drop across each resistor. List each voltage drop individually. c) Determine the current that flows through each resistor if R = 10 Ω. d) Calculate the equivalent capacitance of this circuit if all of the resistors were replaced with capacitors having C = 10 µf. 14 / 32

15 A resistor of resistance R Ohms and a capacitor of capacitance C Farads are connected in series with a constant source of EMF. The source of EMF applied zero Volts before t = 0. It was instantly turned ON to V0 Volts at t = 0. a. Use Kirchhoff s voltage rule and write an equation for this single loop circuit. b. Convert all instances of current into electric charge and rewrite this equation as a differential equation. c. Write the solution to this differential equation below. d. How much time (in terms of time constants) does it take for the voltage across the capacitor to reach 75% of its final value? 15 / 32

16 An unseen electromagnet generates a uniform magnetic field throughout all of the space seen in the diagram below. This field has a magnitude of 1.5 Tesla and points in the positive x direction. a) Calculate the magnetic flux through the rectangular face a, e, f, d. Include whether it is positive or negative. b) Determine the net magnetic flux through the entire shaded, triangular prism. c) A proton is injected into this region of uniform magnetic field from an ion source located at (15, 15, 50) in centimeters. The initial velocity of the proton is 3 x 10 7 î x 10 7 ˆk in meters per second. Calculate the radius of curvature of the proton. d) Describe the path of the proton while it is in the magnetic field. 16 / 32

17 A thick wire of radius a carries a constant current of I Amps evenly distributed across its crosssectional area. Suppose the current travels in the positive z direction. a) Starting from Ampere s Law, derive an expression for the magnetic field produced by the current for r < a. Show all steps and include all relevant unit vectors. Express your result as a vector equation. b) Repeat part (a) for r > a. x! r z y 17 / 32

18 An extremely long piece of wire is bent into the shape shown and carries a current of I Amps in the direction of the arrow heads. Hint: Each of the two straight sections of wire are essentially half of an infinitely long wire. Determine an expression for the magnitude of the magnetic field at the center of the semicircle. R 18 / 32

19 An infinitely long wire is bent in the shape of the darker line in the following diagram. A constant current of I Amps flows through the wire in the direction given by the arrowheads. The radius of the semicircle is R meters. a) Determine the correct formula to use to calculate the magnitude of the magnetic field produced by this current at the origin of the coordinates. Express your answer in terms of I, R and relevant constants. b) State the direction of the magnetic field at the origin. y R x 19 / 32

20 A wire carries a current of 0.50 Amp. The wire is wound into a circular coil of 20 turns. The radius of the coil is 0.20 meter. The coil is initially oriented so the plane of the coil is perpendicular to an applied, uniform magnetic field of magnitude 1.0 T. Assume the unit normal vector for the coil s area is initially anti-parallel to the magnetic field. a) What is the change in potential energy of the coil when it rotates 180 degrees about an axis that lies in the plane of the coil and intersects the center of the coil? b) Does it take work for an outside agent to rotate the coil or will the magnetic field tend to rotate the coil on its own? 20 / 32

21 A charged particle (q = - 5 nc) has an initial velocity of 24000î 36000ĵ in meters per second. It enters a region of space where there is an applied magnetic field of B = 0.3ˆk Tesla. Find the initial force on the particle. 21 / 32

22 In a Hydrogen atom s lowest energy state, the electron moves around the nucleus in a circle of radius 5.3 x meters (the Bohr radius). The Bohr magneton is x Joule/Tesla. This number represents the magnetic dipole moment of the orbiting electron. a. Given these values, calculate the magnitude of the electric current the orbiting electron generates. b. Calculate the speed of the electron in its lowest energy state. The mass of an electron is 9.11 x kg and centripetal acceleration is v 2 / r. 22 / 32

23 Two parallel, long straight wires are separated by 1.00 meter and lie in the z-x plane. See the diagram below. The plane of this paper is the x-y plane so the wires are perpendicular to the plane of this paper. The wires carry current in opposite directions as given in the diagram. I1 = 6.00 Amps and I2 = 8.00 Amps. a) Calculate the magnitude of the magnetic field produced by these anti-parallel currents at point S. b) Determine the direction of the total magnetic field at point S. 23 / 32

24 A rectangular wire loop (of sides b-a and L) sits on a table to the side of a very, very long wire. The wire carries a current of I Amps in the positive y direction. The wire loop is being pulled to the right at a constant speed given by the magnitude of the velocity vector. a) Set up and perform an integral to derive an equation for the total magnetic flux through the loop when its left side is a distance a from the long wire. Your answer should be a function of the x-coordinate. b) Use Faraday s Law and obtain an expression for the absolute value of the induced EMF in the left side of the wire loop. Do this again to obtain an expression for the right side of the wire loop. c) Explain why no EMFs are induced in the two sides of the loop parallel to the x axis. You can use a combination of mathematics and words. d) Determine the direction of the induced current in the loop. y I a L! v x b 24 / 32

25 At some point in the future, superconducting wire might operate at room temperature making large-scale magnetic energy storage practical. A gallon of gasoline releases about 30 MJ of thermal energy upon combustion. Typically only 20% of that is converted into a car s kinetic energy. So if a solenoid with a volume of about 1 gallon (say 4 liters = m 3 ) could store 6 MJ, it would be a viable replacement for one gallon of gas. Assuming a ferrite core with a relative permeability of 640µ0, how much current must flow in a 4-liter, cylindrical solenoid to store 6 MJ of energy? Assume the solenoid will be wound with 5,000 turns of wire per meter. 25 / 32

26 E = 74,000ĵ Electric field lines are drawn below. This field is given in N/C by x. A wire loop in the shape of a square of edge 10 cm is oriented as shown. The wire loop s lower left corner is at x = 3 cm. Eidl c. Calculate the circulation of the electric field around the loop. d. If the resistance of the wire loop is 20 Ω, what current flows in the loop? e. What is the direction of the current? f. Around the diagram, draw a distribution of electrically charged particles that could produce such an electric field. y x 26 / 32

27 A circular loop of wire is placed in a spatially uniform magnetic field as seen above. The loop has a radius of meter and a resistance of 4.0 Ohms. The unseen electromagnet that creates the field is connected to a programmable power supply. So the strength of the magnetic field is made to follow this equation: B(t) = t in Tesla. a. Determine the direction of the change in the magnetic field produced by the electromagnet. Use unit vectors to specify all directions. b. Determine the direction of the induced current in the loop. c. Determine the direction of the induced magnetic field created by the induced current. d. Calculate the magnitude of the induced EMF in the loop. e. Calculate the magnitude of the electromagnet s field at t = 5 sec. f. Calculate the current in the loop at t = 5 seconds. 27 / 32

28 A 12 Volt battery is connected in series with a resistor, an inductor and a switch. At t = 0 the switch is closed, completing the circuit. Suppose R = 30 Ohms and L = 0.48 H. Assume each circuit element is ideal and the inductor is initially un-energized. a. At what time will the current be 50% of its maximum value? b. What is the voltage across the resistor after two time constants have passed? c. What is the voltage across the inductor after two time constants have passed? 28 / 32

29 Consider a series circuit containing a signal generator, a resistor, R, an inductor L and a capacitor C. The signal generator applies a sinusoidal alternating voltage ℇ0sin(ωt). The graph provided is a phasor diagram showing the voltage across the capacitor. At this frequency, an oscilloscope measures the voltage amplitude across the inductor and the voltage amplitude across the resistor are each twice that of the capacitor. a) Add a second phasor to the diagram to depict the voltage across the inductor. b) Add a third phasor to the diagram to depict the voltage across the resistor. c) Add a fourth phasor to the diagram to depict the voltage provided by the signal generator d) Show and label the phase angle in the phasor diagram. V C 29 / 32

30 A metal bar of length L slides down a metal inclined plane made from a wire frame. The very thin lines in the diagram are an insulator so the rod forms a complete electric circuit only with the lower part of the inclined plane. A uniform magnetic field exists throughout this region and points vertically downward. Assume zero friction between the rod and the wire rails. Assume the resistance of the bar is R Ohms but the wire frame making up the inclined plane has zero resistance. a) Find the direction of the induced current in the circuit as viewed downward from above the inclined plane. b) Apply the Lorentz Force law again on the current in the bar. In what direction does this magnetic force point? Hint: Redraw the diagram as viewed from the side. c) Determine an expression for the magnitude of the magnetic force mentioned in part (b). Give your answer in terms of R, B, L and the speed of the bar sliding down the incline. d) The bar reaches a terminal speed as it slides down the incline. Determine an expression for the terminal speed of the bar. 30 / 32

31 Write very short answers to each of the following. a) Which of Maxwell s equations states that no magnetic monopoles have ever been discovered? Answer with an equation and a name. b) What term did Maxwell add to Ampere s Law? Answer with an equation and a name. c) Referring to the previous question, what did this additional term help to explain? d) In one short sentence, explain the physical meaning behind Ampere s Law. Do not simply translate the equation into words. e) Repeat the previous question for Faraday s Law. 31 / 32

32 An electromagnetic wave is traveling in the positive z-direction. At a particular position and time, the electric field vector points in the positive x-axis, has a magnitude of 300 V/m and a wavelength of 4 x 10-6 meters. a) Write an equation for the electric field vector as a function of space and time. Use numerical values for as many terms as possible. b) Calculate the (linear) frequency of the electromagnetic wave. c) Extra Credit: What portion of the electromagnetic spectrum does this wave belong in? 32 / 32