Physics 2220 Fall 2010 George Williams THIRD MIDTERM - REVIEW PROBLEMS Solution sets are available on the course web site. A data sheet is provided. Problems marked by "*" do not have solutions. 1. An electron is accelerated from rest by a potential difference of 525 volts. The electron is moving horizontally in a vacuum and enters a region where -4 there is a magnetic field of 1.32 10 T at right angles to its velocity. After it travels 12.5 cm (measured along its actual path) in the magnetic field the electron strikes a screen. Calculate the deflection of the electron from the path it would have followed in the absence of the magnetic field. 2. Electrons are incident from the left on a small hole. There is a uniform electric field of 15750 V/m over the entire region in the vertical direction, as shown. If the electrons cross the region -7 from wall to wall in 2.00 10 s, calculate the magnetic field necessary for the electron to travel across in a straight line. (A numerical value, plus a clear statement of direction, is needed.) 3. Take the Earth's magnetic field as being along a N-S direction. What is the direction of the force on a wire carrying current from E to W? Calculate the maximum value of the torque on a circular coil of wire carrying 11.0 A, if the coil has 17.0 turns and radius of 1.5 cm. The magnetic field is 1700 gauss. A capacitor of C = 175 F charged to 100 V is discharged through a 11,000 resistor. Calculate the voltage on the capacitor after 2 time constants have passed (from the beginning of discharge). Calculate the magnitude of the magnetic flux, in Wb, through the lecture table in 101 JFB. Assume the table is horizontal. Take the Earth's field as 0.500 gauss, at an angle of 70 from the horizontal. The table is 1.00 m wide and 6.00 m long. (e) 3 Calculate the radius of the orbit of an electron of v = 1.2 10 m/s, in a magnetic field of 150 gauss, if the field is perpendicular to the plane of the orbit. 4. Mercury ions (singly charged mercury atoms) are passed through a velocity selector, as discussed in class, with E = 125,000 V/m and B = 0.850 T. They pass through a slit and follow a curved path in a field B = 0.850 T perpendicular to the paper. Calculate the distance, d, for two mercury isotopes 202 and 200. Take the masses as 202 and 200 times the mass of the proton (not precisely true).
5. Calculate the cyclotron frequency for electrons in a magnetic field of 0.010 T. Protons are measured to travel in a circular path of radius 6.00 cm in a magnetic field of 1.50 T. Calculate their velocity. If a DC power line is carrying a steady current of 12,500 A in the earth's magnetic field which is assumed to be 0.500 gauss, what is the maximum possible force on 1000 meters of this wire? A square coil is constructed with 250 turns of wire. Each side is 3.75 cm. Calculate the torque on this coil which has a current of 1.24 A, a magnetic field of 575 gauss, and the direction of the field is in the plane of the loop. (e) If each resistor has a value of R, calculate the effective resistance between a and b. 6. An electron is accelerated from rest through a potential of 500 volts. Calculate the radius of its circular path in the earth's magnetic field (assumed to be exactly 1.00 gauss). Calculate the cyclotron frequency, in Hz, for a proton in a magentic field of 0.525 T. For the circuit shown calculate the current in the resistor 2.10 time constants after the switch is closed. = 250 V; R = 350 ; C = 1.75 F (e) 5 If a bolt of lightning has a current of 1.2 10 A and a radius of 3.75 cm, what is the average current density? 3 3 If copper has a density of 8.97 10 kg/m find the magnetic field needed to balance the weight of a copper rod whose diameter is 3.25 cm. The rod is carrying a current of 10,000 A and is horizontal. The magnetic field is horizontal and perpendicular to the rod. 7. Calculate the cyclotron frequency (in Hz) for electrons in a magnetic field of 1.75 T. A galvanometer is built with a plane circular coil of radius 3.75 cm and 750 turns of wire. If the magnetic field is in the plane of the coil, calculate the torque (in N m) for a current of 175 milliamperes and a field of 0.450 T. Determine the drift velocity for electrons in a round copper wire of radius 0.75 mm. The current is 15.0 A, the wire is 6.00 m long, the density of copper is 8.50 grams/cc. The atomic mass of copper is 65.0. A 12.0 volt battery has an internal resistance of 1.50. What is the power it can deliver to a load of 1.00? (e) -4 If the earth's magnetic field is 1.00 10 T, calculate the force on 10.0 m of wire which is perpendicular to the field and carries a current of 1,750 A. 8. Electrons are run through a velocity selector as discussed in class with the following values of the fields: E = 125,000 V/m and B = 0.350 T. Calculate the velocity of electrons selected by this system; Protons are incident on the same system. Find the radius of the orbit of these protons after they leave the velocity selector and pass through a region with B = 0.350 T perpendicular to the velocity.
9. Calculate the time constant for a circuit containing a 11.0 pf capacitor and a 22,500 ohm resistor in series. 7 Electrons from the sun arrive at earth with a velocity of 1.00 10 m/s. What is the radius of their orbit in a magnetic field of 1.00 gauss? What is the equivalent resistance between a and b if all resistors have the value R. (e) If the internal resistor of a 12.0 volt car battery is 0.01 ohm, calculate the power dissipated in the internal resistance when the battery is connected to a load of 0.02 ohms. Calculate the drift velocity for electrons in a copper wire whose diameter is 1.20 mm if the wire is 28-3 carrying a current of 7.75 A. [n = 8.47 10 e /m ] 10. 5 Find the radius of the orbit of a proton with speed = 3.86 10 m/s in a perpendicular magnetic field of 3650 gauss. Calculate the cyclotron frequency for electrons in a magnetic field of 250 gauss. Calculate the time constant for a single loop circuit containing a 150 V battery, a 75000 ohm resistor -12 and a 3.25 10 F capacitor. 18 3 Determine the drift velocity in a semiconductor with 2.79 10 charge carriers/cm. The current is 4.75 A in a circular wire with a diameter of 0.132 mm. (e) All resistors have the same values, R. Calculate the effective resistance between and. 11 Calculate the cyclotron frequency for electron in a magnetic field of 367 gauss (in Hz). 17 3 Calculate the drift velocity in a semiconductor with a carrier density of 3.46 10 carriers/cm. The current is 3.27 A in a wire with circular cross section and diameter 2.75 mm. All resistors have the same values, R = 100. Calculate the effective resistance between and. Numerical answer. A 12.3 volt battery has an internal resistance of 0.75 ohm. What power will it deliver to an external load of 1.25 ohms? (e) Calculate the time constant for charging the capacitor if the resistor R = 17.500 and the capacitance is 6.25 F.
12. A circular coil of wire with 175 turns and a radius of 1.75 cm is suspended in a horizontal magnetic field. The magnetic field is in the positive x-direction. The current in the coil is 0.750 A. Calculate the magnitude of the torque on the coil. Find the magnitude of the magnetic moment of the coil. In the top view shown, the torque is observed to move the coil in a counterclockwise direction. What is the direction of? Express this as an angle measured counterclockwise from the positive x- direction. 13. A and B are two very long, straight wires. A carries a current of 27.0 A into the paper, and B carries a current of 33.0 A out of the paper. If a = 17.0 cm, calculate the magnetic field, magnitude and direction, at point P. Show clearly with a drawing how you define the direction. 14. Given an equilateral triangle of wire of side a carrying a current I. Calculate the magnetic field (magnitude and direction) at the center of the triangle, point P. 15. A long copper pipe with thick walls has an inner radius a = 0.75 cm and an outer radius b = 2.24 cm. It carries a current, uniformly distributed, of 9500 A, into the paper. Calculate the magnitude of the magnetic field at R = 3.00 cm. Calculate the magnitude of the magnetic field at R = 1.75 cm. On a drawing indicate the direction of the magnetic field in. 16. Two semi-infinite wires are in the same plane. The wires make an angle of 45 with each other, and they are joined by a curved section of wire that is an arc of a circle of radius R. If the wires carry a current I, find the magnitude of the magnetic field at the center of the arc of the circle (point P).
17. Four long, straight wires are arranged in a square perpendicular to the paper as shown. (+ means current out of the paper, - means current into the paper.) The sides of the square have length a = 2.75 cm. Calculate the force per unit length (magnitude and direction using the coordinates shown), on wire 4. I = +3.00 A I = -4.75 A 1 3 2 4 I = +6.25 A I = +5.25 A 18. Three very long wires are arranged in the configuration shown. The two lower wires are fixed in position and carry identical currents out of the paper. The upper wire has a mass density of 1.50 kg/m. It has the same current as the lower wires, but in the opposite direction. Calculate the magnitude of the current I that will support the upper wire in the position shown. 19. Use the Biot-Savart law to calculate the magnetic field at point P (in ^x, ^y, ^z notation), due to a current of 5.00 amperes in the direction shown by the arrows. The round portion of the wire is circular (R = 6.00 cm), and P is at the center. For ease in grading, label the infinite straight segments (1) and (2) as shown in the figure. 20 Calculate the cyclotron frequency (in Hz) for electrons in a magnetic field of 0.275 T. A circular loop of wire has 35 turns and a diameter of 11.0 cm. If the current in the wire is 7.25 A, calculate the magnitude of the magnetic field at the exact center of the loop. Three long, straight wires are in a plane. They are each a distance a apart. The currents are in the direction of the arrows, and have the magnitudes, I 1 = 4.00 A, I 2 = 3.00 A and I 3 = 1.00 A. Calculate the magnitude of the magnetic field at point P, which is in the same plane as the wires. Take a = 1.00 cm. (e) Electrons are accelerated from rest by a potential difference of 475 V. Calculate the magnitude of their velocity. 12 (f) Three charges are arranged as shown. If Q = 4.0 10 C and 4 a = 3.00 10 m, calculate the electric potential at point P, including sign, using the usual choice for the zero of potential. 21 A long, hollow, cylindrical copper pipe, with outer radius R o = 2.75 cm and 4 inner radius R i = 0.50 cm, carries a current of I = 1.46 10 A. The current is uniformly distributed. Calculate the magnitude of the magnetic field at r = 0.45 cm from the center of the cylinder. Calculate the magnitude of the magnetic field at r = 3.75 cm. Calculate the magnitude of the magnetic field at r = 1.25 cm.
22 Find the magnetic field at the center of a circular loop consisting of 25 turns of wire, with each turn carrying 13.2 Amps. The radius of the loop is 0.25 m. Find the magnetic field in the interior of an ideal solenoid consisting of 1750 turns carrying a current of 0.375 Amps. The solenoid is 10.0 cm long and has an inside diameter of 1.2 cm. Find the cyclotron frequency of an electron in a magnetic field of 0.52 gauss. A cable carries a current of 1000 Amperes. Find the force on 250 m of the cable if there is a magnetic field of 0.50 gauss (about the Earth's field) at right angles to the cable. (e) Find the magnetic field at a distance of 10 cm from a bolt of lightning carrying a current of 30,000 A. (A typical peak current.) 23. Find the magnetic field 15 m from a long straight wire carrying a current of 125 A. Given a circular coil of radius 15 cm and 17 turns. If it carries a current of 1.25 A, find its magnetic dipole moment. Two long parallel wires carry a current of 18 A. If they are 3 cm apart, calculate the force per meter on each wire. In the circuit shown, calculate the time constant. = 10 V C = 1000 pf R 1 = 10 4 R = 2 10 4 2 24 The drawing shows the cross section of three very long straight wires. They are arranged in the form of an equilateral triangle of sides a. + means a current out of the paper and - is a current into the paper. Calculate the magnetic field, both magnitude and direction at point P. Show on a clear drawing how you define any angle used to indicate the direction of the field. I 1 = + 5.00 A I 2 = - 3.25 A I 3 = - 2.75 A a = 1.25 cm 25 The diagram shows a cross section of long, straight wires perpendicular to the paper. + indicates currents out of the paper. - indicates currents into the paper. Find the magnitude and direction of the magnetic field at point P. The four wires form a square. P is at the center of the square. 26. Given three long straight wires, A, B and C that are perpendicular to the plane of the paper. Calculate the magnetic field, magnitude and direction, at P. Use the x-y axes shown. Positive (+) current means current coming out of the paper, negative (-) is current going into the paper. I = +8.00 A; I = - 4.00 A; I = - 11.00 A A B C 27 Consider modeling a bolt of lightning as a long cylinder with a current density given by j = j o(1 - R ), where is a constant and j = 0 at the outside radius R o where R o = 4.50 cm. The total current is 65,500 Amperes (a typical number). 3 Calculate (numerical value with units). Find j o (numerical value with units). What is the magnetic field a distance R = 2.50 cm from the center of the current distribution? [If you cannot do and do this symbolically.]
28. In the diagram + means currents out of the paper, means currents into the paper. There are two long straight wires, A and B perpendicular to the paper. A, B and P are at 3 corners of a square. Calculate the magnetic field, magnitude and direction, at point P. Measure angles counterclockwise from the positive x-direction. A third wire is placed at P perpendicular to the paper. Determine the force per meter, magnitude and direction, on the third wire if its current is 4.75 A. 29. Three long, straight wires are perpendicular to the plane of the paper. Wire A has current into the paper. Wires B and C have current out of the paper. Each wire has a current of magnitude 1.50 A. [a = 5.65 cm] Find the magnetic field, magnitude and direction, at point P which is in the center of the square of which A, B and C are corners. Calculate the force per unit length, magnitude and direction, on a fourth wire at Q, the remaining corner of the square. The current in Q is 1.50 A out of the paper. The wire Q is parallel to A, B and C. 30. A long, straight, copper pipe has an inside radius a and an outside radius b. The pipe carries a current I o of 3 4.75 10 A out of the paper. [a = 1.75 cm; b = 2.15 cm] Calculate the magnitude of the magnetic field a distance 1.60 cm from the center of the pipe. Calculate the magnitude of the magnetic field a distance 2.25 cm from the center of the pipe. Calculate the magnitude of the magnetic field a distance 2.05 cm from the center of the pipe. In, is the magnetic field direction clockwise or counter-clockwise? 31. Calculate the cyclotron frequency for an electron in a magnetic field of 1.22 T. Take the direction of the earth's magnetic field as being exactly N-S. If its value is 0.550 gauss, what is the force, magnitude and direction, on one meter of wire carrying 150 amperes from west to east (the direction of the conventional current). Calculate the radius of the orbit of a proton in a magnetic field of 0.300 T, if the velocity of the proton 4 is v = 2.4 10 m/s. The plane of the orbit is perpendicular to the magnetic field. Calculate the time constant for a simple RC circuit if R = 450 k and C = 670 pf. (e) Calculate the magnetic field, in Tesla, needed for a velocity selector for electrons if the velocity desired 3 4 is 2.00 10 m/s and the electric field is 4.00 10 V/m. 32. For the circuit shown the capacitor is initially uncharged. Initially S 1 is closed. At t = 0 S is closed and S is still closed. 2 1 Calculate the charge on the capacitor at t =, Find the voltage across the capacitor when t = 1.50. Calculate the current in R 1 at t = 0. With the capacitor fully charged S 1 is opened. Calculate the time constant for discharging C. (e) With S 1 and S 2 closed, calculate (numerical value) the time constant for charging C. (No short cuts from advanced courses allowed.) = 100 V; R 1 = 150 k ; R 2 = 175 k ; R 3 = 125 k ; C = 800 pf
33. Calculate the time constant for charging the capacitor in the circuit shown. (e) Calculate the magnitude of the magnetic field, in Tesla, at the exact center of a circular loop of wire carrying a current of 14.5 A. The diameter of the loop is 17.0 cm. Assume the earth's magnetic field is exactly 1.00 gauss. Calculate the radius of the orbit of an electron 6 moving perpendicular to this field at a velocity of 2.1 10 m/s. Two parallel wires carry a DC current of 55.0 amperes. If the wires are 4.00 m apart, calculate the magnitude of the force on each meter of wire. Calculate the capacitance of a parallel plate capacitor if the plates are circular with diameter 17.0 cm. The plates are 0.25 mm apart (in air). 34. A velocity selector is created with the fields E = 1200 V/m and B = 0.300 T. For protons incident on this selector, what velocity is passed? What is the cyclotron frequency for protons in the 0.300 T magnetic field? What is the radius of the cyclotron orbit for protons passed by the velocity selector in in the magnetic field of 0.300 T? 35. Three long parallel wires are arranged perpendicular to the paper at three corners of a square as shown. Positive (+) currents are out of the paper. Calculate the magnetic field, magnitude and direction, at point P, the fourth corner of the square. I 1 = + 3.30 A; I 2 = + 4.75 A; I 3 = - 6.00 A; a = 7.00 cm 36. Calculate the cyclotron frequency for an electron in a magnetic field of 0.81 Tesla. Calculate the time constant for a simple series RC circuit where R = 600 and C = 325 pf. Calculate the magnetic field in the exact center of a circular loop of wire carrying a current of 3.75 A. The diameter of the loop is 6.75 cm. 3 4 If the observed velocity of an electron is 3 10 m/s and the electric field is 5.00 10 volts/meter, find the magnetic field, in Tesla, needed for a velocity selector. (e) Two long parallel wires each carry a current of 75 A (DC current). If the wires are 0.75 m apart, calculate the magnitude of the force on each meter of wire. 37. In the drawing, A, B and P are at corners of a square of side a. + currents are out of the paper; - currents are into the paper. A and B are long wires perpendicular to the paper. Calculate the magnetic field, magnitude and direction, at point P. Use the coordinate system shown. a = 2.50 cm; I = -7.00 A; I = +5.00 A A B 38. Shown is the circular cross section of a long cooper rod of diameter 1.00 cm. The rod carries a current of 75 A into the paper. Assume the current is uniformly distributed. Calculate the magnitude of the magnetic field at point A, a distance a = 2.00 cm from the center of the rod. Calculate the magnitude of the magnetic field at point B, where b = 0.25 cm from the center of the rod.
39. Three very long wires are arranged in the configuration shown. The two lower wires are fixed in position and carry identical currents out of the paper. The upper wire has a mass density of 2.50 kg/m. It has the same current as the lower wires, but in the opposite direction. Calculate the magnitude of the current I that will support the upper wire in the position shown. means out of and means into the paper. (a = 10.0 cm) 40. Three long wires,,, and, are perpendicular to the paper. They carry currents with magnitude given with directions indicated. is out and is into the paper. Calculate the magnetic field, magnitude and direction, at point P. Use the x-y coordinates shown. State clearly the direction. I 1 = 10.00 A; I 2 = 5.00 A; I 3 = 8.00 A; a = 10.0 cm 41. 4 Take the earth s magnetic field as 1.27 10 T at 80 from the horizontal in Salt 2 Lake City. Calculate the magnetic flux through 6.25 m of the physics parking lot. Given a circular loop of wire of R = 2.25 cm. There is a magnetic field 4 perpendicular to the paper which can be described by B(t) = (945 + 1.75 10 t) Tesla. Calculate the magnitude of the emf that appears in the loop. Given a circular loop of wire. There is a magnetic field into the paper and its magnitude is decreasing. What is the direction of the current in the loop clockwise or counterclockwise? Calculate the cyclotron frequency, in Hz, for an electron in a magnetic field of 0.333 T. (e) Calculate the magnetic field at the surface of a large wire (diameter = 6.25 cm) carrying a current of 4750 A. 5 42. At (A) a velocity selector is operated with an electric field of 1.62 10 V/m in the direction shown. Calculate the magnitude of the magnetic field necessary to select the velocity v = 5 1.27 10 m/s for singly positively charged particles moving to the right. What is the direction of this magnetic field into or out of the paper? Explain. The same magnetic field exists in region (B). There is no electric field in region (B). Calculate the value of R for protons. Calculate the displacement, in meters, at point (C) between nuclei of U-235 and U-238 through the same velocity selector. Take the mass numbers as exact, which they are not in nature.
43. Three long, straight wires are perpendicular to the paper at points, and. Positive (+) currents are out of the paper, negative ( ) currents into the paper. Calculate the magnetic field, magnitude and direction, at point P. The direction should be specified as an angle measured from the positive x-axis. Point P is directly above. I 1 = +3.27 A; I 2 = 1.75 A; I 3 = 1.10 A 44. Show in detail how to calculate the magnetic field at point P using the Biot- Savart law and evaluate the answer. Consider the wire to be infinite at both ends.. If the magnitude of the current in the wire is 5.50 A, calculate the magnitude of the field at P, if a = 2.63 cm What is the direction of the magnetic field at P (into or out of the paper)? 45. An ideal solenoid with circular cross section has 4750 turns in a length of 10.2 cm. The radius of its cross section is 0.350 cm. When the current is 11.2 A, what is the outward force per meter on th wire? A coil with 3 turns and a diameter of 2.75 cm is rotated about the vertical axis in a horizontal magnetic field of 850 gauss. The frequency of rotation is 300 Hz. Write a complete expression, with all quantities numerically evaluated for the voltage at A, in the form = D cos t. (e) Calculate the magnetic field at the surface of a big fat wire carrying 17,500 A. The wire has a circular cross section with a diameter of 3.50 cm. Assume that you tried to cancel the earth s magnetic field with a coil wrapped around the equator. If the coil had 1000 turns, what current would be needed to created a magnetic field of 1.00 gauss at its center? 17 3 If bismuth has 4.7 10 charge carrier/cm and these carriers all have the electron charge (not, in fact, true), calculate the Hall voltage for a long strip of Bi 1.00 cm wide and 0.100 cm thick, in a magnetic field perpendicular to the wide face, of 3.25 T if the current is 0.115 A. 46. In the drawing, y is the vertical direction. The currents I 1 and I 2 are in the direction given. The wire carrying I 1 is fixed, the wire carrying I 2 can move up and down in the y direction only. The wire carrying I 2 has a mass density of 0.035 g/meter. If I 1 = 10.0 A, what must the current I 2 be to suspend I 2 at a = 1.25 cm? If I = I, what must the value be to suspend I at a = 1.25 cm? 1 2 2
47. Three long straight wires are carrying current perpendicular to the paper. means current out of the page, means current into the page. Using the current values given, calculate the magnetic field at point P. Express the magnetic field in î, notation. Calculate the magnitude and direction of the field. Express the direction as an angle measured counter clockwise from the positive x-direction. Show on a drawing how you define this angle. a = 4.50 cm; I 1 = +10.25 A; I 2 = -5.75 A; I 3 = - 2.30 A 48. In the drawing, the horizontal lines represent conducting rails. The rod, A, has a mass of 75.0 g. The switch A is closed at t = 0. There is a magnetic field perpendicular to the paper, and INTO the paper, whose strength is B = 0.930 T. (e) Which direction, right or left, would the rod move? Just after t = 0, what is the force on the rod? Just after t = 0, what is the acceleration of the rod? If the rod moves without friction, what is its maximum speed? What is the time constant for the approach to this maximum speed? = 75.0 V; R = 2.00 ; = 9.50 cm 49. Calculate the magnetic field (in Tesla) 0.015 m from the center of long, straight wire carrying a current of 16.2A. Calculate the force due to the earth s magnetic field on 150 m of electric power line when the current is 1670 A if the earth s is 0.75 gauss and is perpendicular to the power line. 27 Find the radius of the path of a proton (m = 1.67 10 kg) in a magnetic field of 0.11 gauss, if its 5 velocity perpendicular to the field is 4.20 10 m/s. If R = 375 ohms, calculate the power dissipated by the top resistor if a battery of 155 volts is applied between a and b. (e) Calculate the charge on the capacitor 1.50 time constants after the switch is closed. = 175 V; R = 275 ; C = 15 pf 50. Three long straight wires are perpendicular to the paper at the points labeled A, B, C. The currents are given below. + is out of the paper, is into the paper. Calculate the x-component of the magnetic field at P (with sign). Calculate the y-component of the magnetic field at P (with sign). Calculate the magnitude of the resulting magnetic field at P. Calculate the direction of the magnetic field at P, as an angle measured co unter clockwise from the positive x-axis. I A = +15.2 A; I B = +12.7 A; I C = 17.2 A; a = 3.65 cm
51. A velocity selector is set up using an electric field of 10,000 V/m. 4 Calculate the magnetic field (in Tesla) necessary to select a velocity of 7.50 10 m/s. Calculate the difference in the radii of two atoms at this velocity in a magnetic field perpendicular to their velocity of 0.912 T. The two atoms are and. Each has 27 one electron missing. (1 amu = 1.66 10 kg) 52. A long, straight wire carries a current given by I = Ioe, where I o and k are constant. Nearby is a rectangular loop of wire with the dimensions shown. Both the straight wire and the rectangular loop are in the plane of the paper. -kt Calculate an expression for the flux through the rectangular loop as a function of time. 53. Given a long, straight solenoid of inside diameter 3.75 cm, length 55.0 cm, consisting of 8700 turns of very fine wire. Inside the solenoid is a coil of 7 turns of wire, whose plane is oriented as shown. The radius of the inner coil is 0.15 cm. If the current in the solenoid is 1.75 A, calculate the flux through each turn of the inner coil. 54. A long cylindrical conductor of radius R carries current I of uniform density 2 J = I/ R. Find the magnetic flux per unit length through the area indicated in the drawing. This area is a long plane. One side is the center of the cylinder, the other is the outside radius. 556. Two long, straight wires are parallel and a distance d apart. Calculate the magnitude of the magnetic flux through the rectangular region shown. Each wire carries a current i, in the direction shown by the arrows. o 56 Consider a toroid with a rectangular cross section and dimensions shown. The toroid has 975 turns of wire and carries a current of 2.75 A. Calculate the magnetic field within the toroid at any value of r between r = a and r = b. (This requires a numerical answer, except for r.) Calculate the flux crossing a section of the toroid of width c, between r = a and r = b/2. (This requires a numerical answer.) a = 2.50 cm b = 12.50 cm c = 4.00 cm
57 In the drawing shown the wire carrying current and the rectangular loop are both in the plane of the paper. The wire is long. The current in the wire is given by I = I o cos t. The positive direction is shown. Determine the magnetic flux through the rectangular loop in terms of the current in the wire and the geometry. Find the current in the resistor (magnitude) as a function of time. Clearly explain which direction (clockwise or counter-clockwise) the current in the rectangular loop is going at t = 0. 58 4 Calculate the radius of the path of an electron moving at 7.50 10 m/s perpendicular to a magnetic field of 0.35 T. Determine the magnitude of the force on a horizontal electric power line 200 meters long carrying a current of 1,750 A. Assume the earth's magnetic field is vertical and has a value of 0.450 gauss. For the circular loop shown the magnetic field is perpendicular to and into the paper. If the loop is a conductor and the magnetic field is decreasing, is the induced current clockwise or counterclockwise? A magnetic field in a region of space has B = B o sin t, with = 266 rad/s and B o = 0.0150 T. Find the EMF that appears in a coil of 7 turns whose plane is perpendicular to this field. The coil has a diameter of 27.5 cm. (e) If the earth's magnetic field is at an angle of 70 from the horizontal as shown, what is the magnetic flux through an area 7.00 2.70 m in the Physics parking lot. Take the earth's field as 0.800 gauss and assume the parking lot is level. 59. A long straight wire carries a current I in the direction shown. Calculate the magnetic flux through the rectangular area shown. The long sides of the rectangle are parallel to the wire, the short sides are perpendicular to the wire. 60. 6 Electrons are moving with a velocity of 3.30 10 m/s perpendicular to a magnetic field of 0.119 T. Calculate the magnitude of the electric field perpendicular to their path such that they move in a straight line. Show is a top view of a circular coil of 12 turns and radius 3.27 cm. If the magnetic field can be represented as B = B o cos t, calculate an expression for the EMF induced on the coil. The drawing shows a rectangular cross section of a toroid of 375 turns with a current of 1.75 A. The inner radius of the toroid is 12.65 cm. Calculate the flux crossing the hatched area. a = 1.15 cm; b = 2.35 cm 4 If the earth's magnetic field is 1.10 10 T and is directed at an angle of 67.0 from the horizontal, calculate the flux through an area 10.0 m 15.0 m of the physics parking lot. (e) A wire carries a current from west to east horizontal to the ground. What is the direction of the force on this wire due to the earth's magnetic field (assumed to be in a north-south direction).
61. 2 Calculate the magnetic flux (in T m ) through a section of the physics parking lot that is 5 m 10 m. Take the earth s field as 0.75 gauss, and its direction as 70 from the horizontal. Find the magnitude of the magnetic force per unit length on a power line if the current is 150,000 A and the earth s magnetic field is 0.75 gauss and is perpendicular to the wire. Calculate the cyclotron frequency, in Hertz, for electrons in a magnetic field of 0.75 gauss. A circular loop of wire has 11 turns, carries a current of 1.50 A, and has a diameter of 7.50 cm. Calculate the magnetic field at the exact center of the loop. (e) For the circular loop shown the magnetic field is perpendicular to and out of the paper. If the loop is a conductor and the magnetic field is increasing, is the induced current clockwise or counterclockwise? (B = means out of the paper.) 62. The long vertical wire carries an upward current given by I = I o e kt where I o is the current at t = 0. The conducting loop is in the plane of the paper. Calculate the magnetic flux through the rectangular loop at t = 0. Find the current in the rectangular loop as a function of time. Explain clearly why this current in the rectangular loop will be either clockwise (CW) or counter clockwise (CCW). 63. In the drawing, the horizontal lines represent conducting rails. The rod, A, has a mass of 75.0 g. The switch A is closed at t = 0. There is a magnetic field perpendicular to the paper, and INTO the paper, whose strength is B = 0.930 T. (e) Which direction, right or left, would the rod move? Just after t = 0, what is the force on the rod? Just after t = 0, what is the acceleration of the rod? If the rod moves without friction, what is its maximum speed? What is the time constant for the approach to this maximum speed? = 75.0 V; R = 2.00 ; = 9.50 cm 64. Given a rectangular toroid with the cross section shown and N turns of wire, calculate its inductance. If N = 360, a = 2.75 cm, b = 7.25 cm and c = 1.25 cm, calculate the numerical value for the inductance. When the current is 1.65 A, calculate the magnetic energy stored between R = 1.25 cm and R = 4.85 cm. 65. Given a long, straight copper wire of circular cross section. The radius of the wire is 1.50 cm. (It is a big wire.) The wire carries a current, uniformly distributed across the cross section, of 12500 A. Find an expression for the magnetic field at an arbitrary point inside the wire. Find an expression for the magnetic energy density at an arbitrary point within the wire. Find the magnetic energy stored within a 10 m length of the wire.
66. A long straight wire carrying a current is in the same plane as a rectangular loop of wire with the dimensions shown. The wire has a resistance, R, as shown. If the current in the wire is given by I = I o sin t, calculate the current through R as a function of time. Assume the positive direction of the current in the wire is as shown by the arrow, and that the positive current in the rectangle is clockwise. Now the current in the long wire has a constant value I (new situation), and the shape of the rectangle is changed by changing the value of c at a steady rate of dc/dt = A m/s. Calculate the current in the resistor, including its sign, using the convention in. 67. A large (R = 3.65 m) circular loop of wire (1 turn) carries a current given by I = 35.0 cos 400 t Amperes. A very small circular coil of 15 turns and radius r = 2.25 mm is at the exact center of the loop. The time is indicated by t. If the plane of the small coil makes an angle of 15.0 with the plane of the loop, calculate the magnitude of the emf induced in the small coil as a function of time. If the current in the loop is fixed at 45.0 A and the small coil rotates about the axis A at an angular speed of 275 rad/s, calculate the magnitude of the emf generated in the small coil as a function of time. The diameter of the small coil is parallel to A. 68. Given a long, straight wire carrying current to the right as shown. A rectangular loop of wire is placed near the wire as shown. If the current in the wire is given by I = I o sin t, find an expression for the voltage across the resistor R as a function of time. 69. A conducting rod of mass M and length L slides without friction on two long, parallel, horizontal rails. A uniform magnetic field B fills the region in which the rod is free to move. B is out of the paper. A battery is attached to the rails and supplies a constant emf through a resistor R. The rails and rod have negligible resistance. Find the direction of the current (clockwise or counterclockwise) such that the rod moves to the left. When the rod moves with velocity v, what is the magnitude of the emf generated in the rod? Find a differential equation for the speed of the rod as a function of time. (Hint: You have seen this equation before.) Find an expression for the speed of the rod as a function of time. 70. A conducting, resistanceless rod slides down a pair of resistanceless frictionless rails inclined at an angle of 25.0 above the horizontal. The entire region is in a uniform B field of 0.365 T that is in the vertical direction. The resistance of the resistor is R = 10.0. The distance between the rails is = 0.650 m. The mass of the rod is 8.00 grams. Find the steady state (or terminal) velocity of the rod as it slides down the rails. Find the time constant for the approach to steady state motion.
71. A conducting rod of mass 1.75 kg slides without friction on two horizontal conducting rails that are 10.0 cm apart. It starts from rest at t = 0. At t = 0 a steady current I = 25.0 A, is turned on. The rod travels 2.09 m to the right in the first 3.00 seconds. Find the magnitude and direction (up or down) of the magnetic field (assumed uniform and vertical). Calculate the emf generated in the rod as a function of time. 72. A very long straight wire carrying a constant current I o is in the same plane as two conducting rails. A rod of resistance R and mass M can slide without friction on the rails. The rod experiences a constant external force F in the direction shown. The rod remains perpendicular to the rails. Calculate the emf generated in the rod as a function of its velocity. Calculate the acceleration of the rod (in terms of I o,, a, x, v, M as needed). Calculate the time constant for the approach of the motion to a steady state. 73. A conducting rod moves on two parallel frictionless conducting rails. The only resistance is that shown as R. There is a battery in series with R. The rod has a -2 mass m of 1.35 10 kg. The magnetic field is in the vertical direction, and has the magnitude of B = 4.75 T. and R have the values shown. The rod starts at rest (t = 0). At t = 0, what is the acceleration of the rod? Take the direction up the incline as positive. Assume the rails are infinitely long in both directions and B is the same everywhere. What is the velocity of the rod at very long times? What is the time constant for the approach of this system to the steady state? 74. A bolt of lightning bolt is modeled as I = I o sin At between t = 0 and t = /A. Calculate the EMF that appears around the cross hatched loop of metal fence with the dimensions shown, when the lightning strikes the metal post. 75. Calculate the magnetic field at the center of a circular coil of 45 turns carrying a current of 1.75 A. The coil has a diameter of 35.0 cm. Calculate the cyclotron frequency (in Hz) for electrons in a magnetic field of 4750 gauss. 2 Imagine that the magnetic field in some region of space varies as B = B o + at + bt, where B o, a and b are positive constants. Calculate the maximum emf that is induced at t = 10 s in a coil of 45 turns and a diameter of 35.0 cm if placed in this region of space. 3 If the capacitor shown has a charge of 45.0 10 C at t = 0, calculate the voltage across the capacitor at t = 5.00 s. = 4500 F, R = 4800 ).
(e) A wire carries a current in the direction shown. The wire and the loop are in the plane of the paper. If the magnitude of the current in the wire is decreasing with time, is the induced current in the loop clockwise or counterclockwise? (f) A bolt of lightning with a peak current of 65,000 A strikes a tall metal post stuck in the ground. Calculate the peak value of the magnetic field 10.0 cm from the post. 76. The toroidal coil shown has a rectangular cross section and is wound with 953 turns of wire. The distances a and b are measured from the center of the doughnut. TOP VIEW SIDE VIEW (cross section) Calculate the value of the magnetic field inside the coil for any r between a and b (a < r < b) and for any value of the current I. kt If I = Ioe where k is a constant, calculate the current at any time t in the single rectangular loop of wire inside the toroid. The loop has dimension a/2 and c/3 as shown, and its left edge, as in the picture, is at r = a. 77. A coil, which has 15 turns, is in a region of space where the magnetic field is given by 5 B = B o cos t, where B o = 1.50 10 T and = 266 rad/s. The coil has a diameter of 5.25 cm. The wires from the ends of the coil are close together and outside the magnetic field there is a resistor R = 150 between the coils ends. If the coil is in the paper, B is perpendicular to the paper. Calculate all possible numbers. Find the magnitude of the EMF generated in the coil as a function of time. Calculate the magnitude of the current that flows through the resistor as a function of time. Determine the power dissipated in the resistor as a function of time. 78. In the drawing shown, everything is in the plane of the paper. The positive direction for current is given by the arrow. The rectangle is made of a conductor. For a current I, calculate the magnitude of the flux through the rectangle. If the current is given by I = I o sin t, calculate the magnitude of the voltage as a function of time across R. For the current given in explain clearly the direction of the induced current (clockwise or counterclockwise) in the rectangle at time t = 0.
79. A conducting rod moves on two parallel conducting rails that are a distance w apart. The only resistance is the R shown. The magnetic field is perpendicular to the paper, and into the paper, and is everywhere in the drawing. If the rod is moved to the right with constant velocity v, calculate the current through R. If the rod is moved to the right with constant force F, calculate the velocity of the rod at a very long time. (the rails are infinitely long.) 80. A rectangular conducting wire has the shape and dimensions as shown. The upper wire and the rectangle are in the plane of the paper. If the current in the upper wire has a steady value I o, calculate the magnetic flux through the rectangular loop. If the current in the upper wire is given by I = I o sin t, calculate the current in the rectangular loop as a function of time. If the positive direction of the upper current is to the right, as shown, calculate the direction of the current in at t = 0, clockwise or counterclockwise. For full credit, state clearly your reasoning. 81. A conducting rod of mass M and length L slides without friction on two long, parallel rails. As shown in the side view, the rails are at 15 from the horizontal. A uniform vertical magnetic field B fills the region in which the rod is free to move. B is out of the paper in the top view. A battery is attached to the rails and supplies a constant emf through a resistor R. The rails and rod have negligible resistance. Find the direction of the current from the battery (clockwise or counterclockwise) such that the rod moves to the right. When the rod moves with velocity v, what is the magnitude of the induced emf generated in the rod? What is the minimum voltage for (battery) such that the rod will move to the right? Find the differential equation for the speed of the rod as a function of time. (e) Find an expression for the speed of the rod as a function of time. Express the result in terms of, R, M, L, B, and angles, as needed. Side View 82. Two long conducting rails are in the horizontal plane. There is a uniform magnetic field, B, perpendicular to the paper and out of the page. Starting at rest, a constant force F is applied to a conducting rod of mass M, as shown. The rails are as long to the right as needed. Find the direction, clockwise or counter clockwise, for the induced current. Determine the final velocity of the rod. Obtain a formula for the velocity of the rod as a function of time. Show all details. Top View
83. In the drawing the horizontal lines represent conducting rails that are very long. The conducting rod, mass m, slides (no rolling) without friction along the rails. It is acted on by a constant horizontal force, F. When the speed of the rod is v, calculate the current in the resistor. Obtain an expression for the limiting speed at which the rod will move. Obtain an expression for the time constant for the approach to the limiting speed. 84. Calculate the speed of electrons that move in a straight line through a region that has a vertical E-field of 1750 v/m and a perpendicular magnetic field of 1.35 T. Protons are moving perpendicular to a magnetic field of 1.32 T. The radius of the circle they move in is found to be 1.75 cm. Calculate their velocity. An ideal solenoid has 6750 turns of wire, and a total length of 0.725 m. Calculate the magnetic field at its center. A DC electric power line (they do exist) is carrying a current of 10,000 A. Calculate the force on 20.0 m of this wire due to the earth s magnetic field. Assume the component of the earth s field perpendicular to the wire is 0.750 gauss. (e) Assume that a wire carrying a current of 1.00 A is wrapped around the earth s equator. How many turns would be needed to generate a field of 10.0 gauss at the earth s center? Take the radius of the earth as 3 6.38 10 km. 85. The drawing shows three long straight wires perpendicular to the paper. Each carries the same current, 1.75 A. + means out of the paper; means into the paper. a = 1.65 cm Calculate the magnitude of the magnetic field at A due to each of the three wires. Calculate the magnetic field vector at A due to all three wires, magnitude and direction (not just components). 86. The drawing shows three long straight wires perpendicular to the paper. Wire has a current of 5.00 A out of the paper, wire has an equal current out of the paper. a = 2.75 cm If wire has a current of 10.0 A, calculate the magnitude of the net force per/meter on. What must be the direction of the current in be so that the net force is upward? Explain clearly. 87. The drawing shows a long straight wire carrying current to the right. If the current in the wire is I o, calculate the magnetic flux through the rectangle beside the wire. Do the calculations; don t just put down a formula. If the rectangle is a conductor with the resistance given, calculate an expression for the current in the rectangle as a function of time if the current in the straight wire is kt given by I(t) = I o e. What will the direction of the current (clockwise or counter clockwise) in the rectangle be? Explain.
88. In the circuit shown, the switch is closed for a long time and then opened at t = 0. -6 Calculate the current in the inductor at t = 1.32 10 s. -6 Calculate the voltage across R 2 at t = 0.87 10 s. -6 Calculate the voltage across R 2 at t = -0.100 10 s. Write three equations governing the behavior of the circuit with the switch closed. Use the current labels given. Solve these to obtain the differential equation governing the time rate of change of current in the inductor, and from this calculate a numerical value for the time constant. = 350 V, R 1 = 250, R 2 = 275, R 3 = 125, L = 4.25 mh 89. In the figure shown, E = 100 volts, R 1 = 10 ohms, R 2 = 20 ohms, R 3 = 30 ohms, and L = 2 henry. Find the value of i 1 and i 2 immediately after S is closed. Find the value of i 1 and i 2 a long time later. Find the value of i 1 and i 2 immediately after S is opened again. Find the value of i and i a long time later. 1 2 90. In the drawing shown the network initially has no current. The switch S is closed for two time constants. Then the switch is opened. Call this t = 0. = 65.0 V Calculate the time constant for the circuit after the switch is opened. -6 Calculate the current in R 3, 1.80 10 seconds after t = 0. Calculate the current in R at long times if the switch is closed. 4 6 4 1 2 4 R 3 = 50.0 k = 5.00 10 R = 1.26 m = 1.26 10 3 3 1 2 91. In the circuit shown = 9.00 volts, R = 155 10 ohms, R = 195 10 ohms, and L = 3.00 mh. The switch S is closed at t = 0. R = 23.0 k = 2.30 10 L = 13.0 mh Find the time constant for the circuit. Find the value of the voltage across R 1 when t = 2. Find an expression for the voltage across R as a function of time. 2 92. The switch S is closed at t = 0. Find the current in R 2 as a function of time. Write the differential equation for the current in L as a function of time. After the current in R 2 has reached a steady state S is opened. Call this a new t = 0. Find the current in R 1 as a function of time. Specify its direction.
93. A toroidial inductor has 1742 turns and the physical dimensions shown in cross section. Derive the formula for the inductance of a toroid and calculate its value for this case. Calculate the energy stored between R = 5.00 cm and R = 6.00 cm when the current is 0.275 A. 94. Given the circuit shown. 95. In the circuit shown the switch is closed at t = 0. Calculate the current through the inductance a long time after the switch is closed. The inductance is assumed to have zero resistance. Calculate the time constant for the approach to equilibrium after the switch is closed. Show your work. You must show that you know how to do this calculation from the beginning. After the switch is closed for a long time, it is opened at t = 0. -4 Calculate the voltage across R at t = 6.25 10 s. = 125 V, R 1 = 1000, R 2 = 3000, R 3 = 2000, L = 1.25 H 2 Calculate the current in the inductance after 1.75 time constants have elapsed. The switch is opened after being closed for 1.75 time constants in. Calculate the time constant for the decay of the current in the inductance. Calculate the time constant for the growth of the current in the inductance with the switch closed. You must develop the differential equation governing the current as function of time. For convenience in grading use the branch current labels and directions given in the drawing. No shortcuts. = 150 V, R 1 = 8,500, R 2 = 15,700, R 3 = 15,700, L = 3.25 mh 96. In the circuit shown, the switch is closed for a long time. Calculate the magnitude of the current in the inductance. Calculate the magnitude of the current in R 3. If the switch is now opened at t = 0, calculate the magnitude of the current in R 3 2.50 time constants after t = 0. Calculate the time constant with the switch closed. Show ALL details, as done in class. = 125 V; R 1 = 75 ; R 2 = 100 ; R 3 = 150 ; L = 13.0 mh 97. A long straight wire carries a current of 4.75 A. Find the magnitude of the magnetic field at a distance 2 r = 1.75 10 m from the wire. Calculate the magnetic energy stored in a region of space that is a tube parallel to the wire of inner radius a = 3.20 cm and outer radius b = 7.70 cm with a length of 2.75 m.