Classical Electromagnetism

Transcription

1 Classical Electromagnetism Workbook David Michael Judd

2 Problems for Chapter 33 1.) Determine the number of electrons in a pure sample of copper if the sample has a mass of M Cu = kg. The molecular mass of copper is grams / mol. 2.) Estimate the number of electrons and protons in your body by assuming that your entire mass is composed of H 2 O. 3.) At some time t o, a solid copper sphere of diameter cm is electrically neutral. A short time later, an excess charge of Q = C is placed in the center of the sphere. The charge will come to equilibrium very rapidly. As copper is a very good conductor, the charge will be uniformly distributed over the surface of the sphere. Calculate the surface charge density σ for the sphere. 4.) Calculate the number of electrons in one mole of pure H 2 O. 5.) How many protons would one find in seventy-five grams of pure sodium, if the molecular mass of sodium is 23.0 grams / mol. W 33-1

3 Problems for Chapter 34 1.) We model monatomic hydrogen as a system with a single fixed proton at the center of a circle of radius R about which orbits a single electron, as represented in the diagram below. a) Derive an equation for the magnitude of the electrical force exerted on the electron by the proton. b) Determine the actual value of this force. Recall, q p = q e = e = C, and use R = m. c) As the electrical force is radially directed (i.e. a centripetal force), determine the speed of the electron in its orbit of the proton. d) Calculate the amount of time it would take the electron to go around the proton once. This time interval is called the period, and I will signify it with a lower case Greek letter τ, (tau). v e e e p R 2.) Two point-like electric charges are related by q 1 = 2q 2 = C. Both charges are fixed on the y-axis at positions given by, respectively, r 1 = 3.50 m ĵ, r 2 = 2.25 m ĵ Do the following: a) Using a straight edge, construct a detailed, scaled drawing of this charge configuration. b) Determine the magnitude and the direction of the net electric force that would be exerted on a third point charge q 3 = 2q 1 if it were fixed at the origin of the Cartesian coordinate system in which the positions are measured. W 34-1

4 3.) Four point-like electric charges are related by q 1 = 2q 2 = 4q 3 = 2q 4 = C. The charge configuration is represented in the diagram below. Determine the net electric force, magnitude and direction, exerted on charge q 4. y( ĵ ) s q 1 q 2 s s = 1.50 m q 3 s q 4 x ( î ) 4.) y( ˆ j ) q 2 s s q 1 s q 3 x( ˆ i ) W 34-2

5 Two identical point-like electric charges of q 1 = q 2 = C are fixed at two vertices of an equilateral triangle of side length s = 2.00m, as represented in the diagram above. Determine the initial instantaneous electrical force, magnitude and direction, that would be exerted on a point-like electric charge q 3 = 2q 1 if it were placed at the third vertex of the triangle. 5.) Three electric point charges are related by q 1 = 2q 2 = 3q 3 = C. The point charges are fixed at points in space the positions of which are given by r 1 = 4.00m î m ĵ, r 2 = 6.00m î 5.00m ĵ, and r 3 = 2.00m î 4.00 m ĵ. Do the following: a) On graph paper, using a straight edge, do a detailed drawing of the point charges and the electrical forces acting on charge one. b) Calculate the net electric force, magnitude and direction exerted on charge one. (It would be of benefit to also calculate the net electric force on the other two charges.) 6.) s q 1 q 4 s s q 2 Three identical charges q 1 = q 2 = q 3 = C are located at three vertices of a square of side length s = m, as represented in the diagram above. Find the net electric force exerted on a fourth charge q 4 = 2q 1 located at the intersection of the two diagonals. W 34-3 s q 3

6 7.) The Earth has a mass related to that of the Moon by M = ( ) M M = kg. The magnitudes of the mutual gravitational forces these objects exert on each other is given by F G M = GM M M G 2 = F M d M where G is the universal gravitational constant with a value of G = N m 2 / kg 2, and d M is the average distance between the center of the Earth and the center of the Moon with a value of d M = m. Calculate the following: a) The amount of positive electric charge Q that if placed at the center of the Earth and the center of the Moon would produce an electric force the magnitude of which would equal the magnitude of the gravitational force between the Earth and Moon. b) How many protons would be equivalent to the charge Q. c) The volume of copper that would contain this many protons if the average mass density of copper is ρ Cu = 8,930 kg / m 3., 8.) In the nucleus, protons are separated by distances on the order of m. Do the following: a) Calculate the magnitude of the electrical repulsion a proton would feel if it were located m from another proton. b) Calculate the magnitude of the gravitational interaction between the two protons in part a). c) What is the ratio of the magnitudes of the electrical force to the gravitational force? d) What might one conclude from this ratio concerning electrically charged particles? 9.) Two small spheres, initially uncharged, have identical masses M = kg. Each sphere is connected to a silk thread of length = m. The threads are hung from a common point. If an equal amount of electric charge q is placed on each sphere, the spheres swing apart and come to equilibrium with each thread making an angle of ten degrees to the vertical. Calculate the value of q. 20 M q q M W 34-4

7 10.) Calculate the number of excess electrons that must be placed on each of two identically small spheres that are 3 cm apart if each sphere feels an electric repulsive force of magnitude N. (It is almost always useful to draw a free-body diagram with an appropriated coordinate system.) 11.) atoms of monatomic hydrogen have a mass of one gram. Calculate at what distance the electron on a monatomic atom would have to be from the nucleus so that its electrical force of attraction would equal in magnitude the weight of the atom itself. (It is almost always useful to draw a free-body diagram with an appropriated coordinate system.) 12.) Two positive charges q = 10 9 C are fixed in a vacuum separated by a distance of 8 cm. Calculate the magnitude and the direction of the electric force that would be exerted on a third positive charge q = C if this charge is 5 cm from each of the other two charges. (It is almost always useful to draw a free-body diagram with an appropriated coordinate system.) 13.) A negative point charge of magnitude q is fixed on the y -axis at a point y = a. A positive charge of the same magnitude is fixed at a point y = a. A third charge is located on the x -axis and is positive and also of magnitude q. Do the following: a) Calculate the magnitude and direction of the net electric force on the third charge if it is at a point the coordinate of which is x. b) Calculate the magnitude and direction of the net electric force on the third charge if it is located at the origin. (It is almost always useful to draw a free-body diagram with an appropriated coordinate system.) 14.) A certain metal sphere of volume 1 cm 3 has a mass of 7.5 grams and contains free electrons. Do the following: a) Calculate how many electrons must be removed from each of two such spheres so that the electrostatic force of repulsion between them just balances the force of gravitational attraction. You may treat the spheres as far enough apart as to be treated as point-like charges. b) Express this number as a fraction of the total number of free electrons. (It is almost always useful to draw a free-body diagram with an appropriated coordinate system.) 15.) Three electric point charges are fixed in place on the x -axis, as represented in the diagram below. Calculate the net electric force on: a) Charge one. b) Charge two. 5 µc 2 µc 3µC cm 4 cm x ( î ) (It is almost always useful to draw a free-body diagram with an appropriated coordinate system.) 16.) Three electric point charges are fixed in the configuration represented in the diagram below. Calculate the net electric force on: W 34-5

8 a) Charge two. b) Charge three. You may assume that q = 1 nc. (It is almost always useful to draw a free-body diagram with an appropriated coordinate system.) 2q y( ĵ ) 1 3 cm 5 cm 2 4q 4 cm 3q 3 x ( î ) 17.) y( ĵ ) 2Q 2 3 Q = 3 cm 1 3Q x ( î ) Three electric point charges are fixed at the vertices of an equilateral triangle of side length = 3 cm, as represented in the diagram above, where Q = 2 µc. Find the net electric force on: a) Charge one. b) Charge two. W 34-6

9 Problems for Chapter 35 1.) z( k ˆ ) d Q y( ˆ j ) Q 2e E A parallel plate capacitor carries a positive electric charge of + Q on one of its plates and Q on the other plate. The parallel plates are separated by a short distance d. In between the plates, there is an electric field that is virtually uniform. The direction of the electric field is from the positively charged plate toward the negatively charged plate. The magnitude of this electric field is given by E = Q ε o A = 4π k Q A, where A is the surface area of each plate. A single helium nucleus with an approximate mass M He = kg and positive electric charge of q He = 2e = C is released from rest at the surface of the positively charged plate. If the magnitude of the electric x( ˆ i ) charge on each plate is Q = C, and the dimensions of the capacitor are given by = 10 d = m, then find the following: a) The magnitude and the direction of the electric force exerted on the nucleus. b) The magnitude and the direction of the acceleration of the nucleus. c) The time necessary for the nucleus to reach the other plate. d) The speed the nucleus has at the instant it strikes the other plate. 2.) Initially, a small cork sphere of mass M = kg hangs vertically downward at the end of an insulating string of length = m, as represented in the diagram below. A short time later, a positive electric charge q is placed on the cork sphere and a uniform horizontal external electric field of magnitude E ext = N / C is established in the space around the sphere. The sphere swings into an equilibrium position making an angle ϕ = with respect to the vertical. Determine the magnitude of the electric charge that was placed on the sphere. W 35-1

10 Initial Later ϕ E ext q M M 3.) A single positive electric point charge q = C is fixed at a point in space the position of which is given by r q = 5.00 m î m ĵ. Do the following: a) Using a straight edge, draw a detailed, scaled drawing of the point charge in question and a representation for the electric field it produces at points P 1, P 2, and P 3, the positions of which, respectively, are given below. b) Calculate the electric field, magnitude and direction, produced by charge q at point P 1 the position of which is given by r P1 = 2.00m î m ĵ. c) Calculate the electric field, magnitude and direction, produced by charge q at point P 2 the position of which is given by r P2 = 6.00m î 5.00m ĵ. d) Calculate the electric field, magnitude and direction, produced by charge q at point P 3 the position of which is given by r P3 = 3.00 m î 2.00 m ĵ. 4.) A single negative electric point charge q = C is fixed at a point in space the position of which is given by r q = 3.00m î m ĵ. Do the following: a) Using a straight edge, draw a detailed, scaled drawing of the point charge in question and a representation for the electric field it produces at points P 1, P 2, and P 3, the positions of which, respectively, are given below. b) Calculate the electric field, magnitude and direction, produced by charge q at point P 1 the position of which is given by r P1 = 7.00 m î 2.00 m ĵ. W 35-2

11 c) Calculate the electric field, magnitude and direction, produced by charge q at point P 2 the position of which is given by r P2 = 2.00m î 5.00m ĵ. d) Calculate the electric field, magnitude and direction, produced by charge q at point P 3 the position of which is given by r P3 = 7.00m ĵ. 5.) A single positive electric charge q 1 = C is fixed at a point on the y-axis with a position given by r 1 = 3.00 m ĵ. A single negative charge q 2 = C is fixed at a point on the y-axis with a position r 2 = 4.00 m ĵ. Do the following: a) Using a straight edge, draw a detailed, scaled drawing of the point charges in question and a representation for the electric field it produces at the point given below. b) Calculate the net electric field, magnitude and direction, produced by the two point charges at that point on the x-axis given by r P = 6.00m î. 6.) Two point-like electric charges are related by q 1 = 2q 2 = C. If both of these charges are fixed on the y-axis, and separated by a distance of = 1.250m, then determine the location of a point, other than at infinity, where the net electric field would be zero. Be sure to draw an appropriately detailed drawing of this charge configuration. 7.) A point-like electric charge Q = C is fixed at a point in space the position of which is given by r Q = 2.00 m î m ĵ. Do the following: a) Using a straight edge, draw a detailed, scaled drawing of the point charge in question and a representation for the electric field it produces at the point given below. b) Calculate the net electric field, magnitude and direction, produced by charge Q at that point P the position of which is given by r P = 4 m î + 2 m ĵ. 8.) Two electric point charges are related by q 1 = ( 3 / 2 )q 2 = C. The charges are fixed at positions r 1 = 7 m î 6 m ĵ and r 2 = 4 m î + 3m ĵ, respectively. Do the following: a) On graph paper, using a straight edge, draw a detailed drawing of the point charges in question and a representation for the electric field they produces at the point given below. b) Calculate the net electric field, magnitude and direction, produced by charges q 1 and q 2 at point P the position of which is r P = 3m î 4 m ĵ. 9.) Represented in the diagram below is an electric dipole. Recall that an electric dipole consists of two equal and opposite charges, separated by some distance. Do the following: a) On graph paper, using a straight edge, draw a detailed drawing of a representation for the electric field produced by the dipole at an arbitrary point on the x-axis and an arbitrary point on the y-axis. W 35-3

12 b) Derive an equation for the net electric field, magnitude and direction, produced by the dipole at an arbitrary point P x on the x-axis the position of which is given by r Px = x î. c) Derive an equation for the net electric field, magnitude and direction, produced by the dipole at an arbitrary point P y on the y-axis the position of which is given by r Py = y ĵ, where y > a. d) The equation derived in part b) reduces to what form if x >> a? e) The equation derived in part c) reduces to what form if y >> a? y( ˆ j ) P y y + q a x P x x( ˆ i ) a q W 35-4

13 Problems for Chapter 36 1.) A uniform electric field of magnitude E = 275 N / C is directed at an angle of θ = 25 directly east of vertical. If we set up a Cartesian coordinate system such that east is in the positive x-direction, north in the positive y-direction, and up is in the positive z-direction, then the electric field would be written in vector form as E = E sinθ î + cosθ ˆk ( ). Use this information to do the following: a) Find the electric flux through a circle placed in this electric field if the radius is R = 4.75 m and the plane of the circle is horizontal. To find the direction of the area, assume a counterclockwise sense around the boundary, if viewed from above the circle. up ( ˆk ) θ E north ( ĵ ) east ( î ) b) Find the electric flux through a rectangle placed in this electric field if the plane of the rectangle is vertical and the length and width are related by = 2w = 6.50 m. The long sides of the rectangle are parallel to the z-axis and the short sides are parallel to the y-axis. Assume a clockwise sense if looking due west. ( ) up ˆk θ E north ( ĵ ) w east ( î ) W 36-1

14 2.) A point charge Q = C is placed at the center of a cube of side length s = 2.50 m, as represented in the diagram below. Use Gauss law to find the total electric flux through the cube. s s s Q 3.) R Charge is uniform distributed over the volume of a very long cylindrical plastic rod of radius R, as represented in the diagram above. The charge per unit length of the rod is λ. Use Gauss law to find an equation for the electric field at points a perpendicular distance r from the symmetry axis in the following regimes: a) r < R, b) r > R. W 36-2

15 4.) a b A very long plastic pipe has an inner radius a and outer radius b. Electric charge is uniformly distributed over the region where a < r < b. The amount of charge is λ Coulombs per unit length of the pipe. Use Gauss law to find the electric field in the following regimes: a) r < a, b) a < r < b, c) r > b. 5.) Q a b A thick spherical shell of inner radius a and outer radius b has a charge Q uniformly distributed over its volume as represented in cross-section in the diagram above. Use Gauss law to find the electric in the following regimes: a) r < a, b) a < r < b, c) r > b. W 36-3

16 6.) An early model of the neutron consisted of an inner core of positive charge e uniformly distributed over a sphere of radius a that was surrounded by a concentric spherical shell of radius b. A charge of e was uniformly distributed over the spherical shell. This model is represented below in a cross-sectional view. Use Gauss law to find the electric field in the following regimes: a) r < a, b) a < r < b, c) r > b. e e a b 7.) According to an early incorrect model of the atom proposed by J. J. Thomson, an atom consists of a cloud of positive electric charge within which electrons sit like plums in a pudding. For helium, the cloud would have a uniform distribution of charge 2e while two electrons are symmetrically situated about the center, as represented in the diagram below. The radius of the atom is R. Prove that the separation distance of the two electrons in equilibrium would have to be equal to R. 2e e e R W 36-4

17 Problems for Chapter 37 1.) Two point-like electric charges are related by q 1 = 3q 2 = C. Charge q 1 is fixed at the origin of a Cartesian coordinate system. Charge q 2 is initially located at a point the position of which is r 2,i = 3.00 m î 5.00 m ĵ. Determine the work done by the electrical force if q 2 is moved to a later position given by r 2,f = 6.00m î m ĵ. 2.) An electric dipole consists of two electric point charges of equal magnitude and opposite sign that are separated by a distance = 2a. An electric dipole is represented in the diagram below. Do the following: a) Derive an expression for the electric potential at an arbitrary point P on the y-axis such that y > a. (Note that the y-axis is aligned with the charges.) b) Determine the form this expression takes when y is a number much larger than a, that is, y >> a. c) Using your answer in part a), calculate the potential at y = 3a, if q has a magnitude q = C, and the value of a is a = m. d) Derive an expression for the electric potential at an arbitrary point P on the x-axis. e) (Note that the x-axis is transverse to the alignment axis of the charges.) Determine the form this expression takes when x is a number much larger than a, that is, x >> a. f) Using your answer in part d), calculate the potential at x = 3a, if q has a magnitude q = C, and the value of a is a = m. y( ĵ ) P y a a q q P x ( î ) x W 37-1

18 3.) Three identical electric charges are related by q 1 = q 2 = q 3 = q = C. The charges are fixed at the vertices of an isosceles right triangle for which the legs are of length 2s = m, as represented in the diagram below. Calculate how much work an outfit like FP&L would have to do to move each of these charges closer together so that the final leg length of the final isosceles right triangle would be s = 1.000m. q 1 2s s s q 2 2s q 3 4.) A helium nucleus of charge Q He = 2e and mass M He = kg is released from rest in a region of space where there is a uniform electric field of magnitude E = N / C directed due north. Use work-energy methods to calculate: a) The speed of the helium nucleus at the instant it has moved a distance = 0.765m. b) The change in the electric potential through which the nucleus was displaced. 5.) An electron is released from rest in a region of space where there is a uniform electric field of magnitude E = N / C directed due east. The mass of the electron is M e = kg and its electric charge is Q e = e = C. Use workenergy methods to calculate: a) The speed of the electron at the instant it has moved a distance of one meter. b) The change in the electric potential through which the electron was displaced. 6.) A negative point charge q = C is fixed at a position given by r q = 6.00m î m ĵ. Calculate the following: a) The electric potential due to this charge at a point P in space the position of which is given by r P = 3.00m î 5.00m ĵ. b) The amount of work that would have to be done by an outfit like FP&L to move an identical charge from a location where the electric potential is zero to point P. W 37-2

19 7.) An electric quadrupole is represented in the diagram below. A quadrupole consists of two aligned dipoles. Do the following: a) Derive an equation for the net electric potential at an arbitrary point P on the b) x-axis, where x > a. Determine the form of this equation when the value of x is much larger than the value of a, that is, when x >> a. c) Calculate the net electric potential at point P if Q = C, a = m, x = 2a. d) Calculate the electric potential energy of this configuration of charges. y( ˆ j ) + Q 2 Q + Q P x( ˆ i ) a a x 8.) Three identical charges q = C are fixed at the vertices of a rectangle of dimensions = 3w = m, as represented in the diagram below. Calculate the following: a) The net electric potential at the fourth vertex. b) The amount of work an outfit like FP&L would have to do to assemble such a charge distribution. c) The amount of work done by the electric field in the displacement of a fourth identical charge from infinity to fourth vertex. q q w w q W 37-3

20 9.) One point-like electric charge q = C is fixed at a position given by r q = 3.00 m î m ĵ. Do the following: a) Find the electric potential at point P the position of which is r P = 4.00m î m ĵ. b) The amount of work that would be done by an outfit like FP&L in moving an identical charge from infinity to point P. c) Calculate the amount work done by the electric field in the displacement described in part b). 10.) Three point-like electric charges are related by Q 1 = ( 1 / 2 )Q 2 = Q 3 = C, and fixed at positions, respectively, given by r 1 = 6.00m î m ĵ ; r 2 = 0.00 m î m ĵ ; r 3 =6.00 m î m ĵ. Do the following: a) Find the total electric potential produced by these electric charges at point P the position of which is r P = 2.00 m î m ĵ. b) Calculate the work that would be done by an outfit like FP&L to move a fourth charge from infinity to point P, if Q 4 = ( 1 / 3)Q 1. c) The electric potential energy of the final four charge configuration. 11.) Three identical point-like electric charges q = C are fixed at three vertices of a square of side length s = 1.26m, as represented in the diagram below. Do the following: a) Find the total electric potential produced by these electric charges at the fourth vertex. b) Calculate the work that would be done by an outfit like FP&L to move a fourth identical electric charge from infinity to the fourth vertex. c) The electric potential energy of the initial three charge configuration. d) The electric potential energy of the final four charge configuration. q s s s q s W 37-4 q

21 12.) An electric charge Q = C is uniformly distributed over a sphere of radius R = 0.02 m. An argon nucleus of charge q Ar = 18e and mass M Ar = kg is released from rest a distance of d = 0.08 m from the center of the spherical charge distribution. Do the following: a) Use the conservation of mechanical energy to calculate the speed with which the argon nucleus will strike the surface of the spherical charge distribution. b) Calculate at what distance from the center of the spherical charge distribution the nucleus will be when it has one half of the speed found in part a). c) Calculate the potential difference through which the nucleus moves in its displacement to the surface of the sphere. Q q Ar t o t R d 13.) The electric potential difference between the positive and negative poles of an automobile battery is Volts. In order to recharge the battery completely, the charging device must force Coulombs of electric charge from the negative pole to the positive pole of the battery. Calculate the amount of work the charging device must do in this process. 14.) Three identical point charges q = C are fixed at the vertices of an equilateral triangle of side length s = m. (See the diagram below.) Do the following: a) Calculate the electric potential energy of this three charge configuration. b) Calculate the net electric potential produced by these three charges at point P, the intersection of the angle bisectors of the triangle. c) Calculate the amount of work an outfit like FP&L would do to move an identical fourth charge from infinity to point P. q s P s q s W 37-5 q

22 15.) On days of fair weather, the local atmospheric electric field of the Earth is approximately V / m. The field points vertically downward. Calculate the electric potential difference between a point at the surface of the Earth and a point a distance h = m above the surface of the Earth. (You may treat the Earth s surface as a flat conductor.) 16.) An electric charge configuration is represented in the diagram below. Do the following: a) Derive an equation for the net electric potential at an arbitrary point P on the y-axis. b) Determine the form of this equation when the value of y is much larger than the value of a, that is, when y >> a. c) Determine the net electric potential at point P if Q = C, a = m, y = 2a. P y( ˆ j ) y + Q 2 Q + Q x( ˆ i ) a a 17.) y( ˆ j ) P + Q 2 Q + Q x( ˆ i ) a a x W 37-6

23 An electric charge configuration is represented in the diagram above. Do the following: a) Derive an equation for the net electric potential at an arbitrary point P. b) Calculate the net electric potential at point P if Q = C, a = m, x = y = 2a. 18.) A single proton is fixed at a point in space. A second proton is launched directly toward the first proton with an initial speed v o from an initial distance x o. Using the conservation of the mechanical energy, do the following: a) Derive a general equation for x, the separation distance of the two protons at any arbitrary time after launch. b) Derive a general equation for the speed of second proton at any arbitrary time after launch. c) If the second proton is initially very far away for the fixed proton, at what speed must it be launched so that its closest approach to the fixed proton, x c, is two fermi, ( m ). d) There is a connection between the kinetic energy of particles and their temperature. We can write for the average kinetic energy 1 2 Mv2 = 3 2 k BT, where T is the temperature in Kelvin and where k B is the so-called Boltzmann constant with a value of k B = J / K. The significance of the separation distance of two fermi in part c) is that this is how far apart protons must be to feel the strong nuclear force and fuse together. (Fusion, of course, is the favored process of nature by which to build polyprotonic nuclei. These are the nuclei we need for there to be stuff other than hydrogen.) The core temperature of the Sun is believed to be about K. Use the launch speed you found in part c) and determine the kinetic energy a proton would have with that speed. Determine the temperature a proton must be at in order to have that kinetic energy. e) According to your calculations, is the core temperature of the Sun hot enough for protons to overcome the so-called Coulomb barrier and initiate the fusion process? If your answer is no, then you might give some thought as to how there might be fusion going on in the core of the Sun. p v o v = 0 p x c x o

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25 Problems for Chapter 38 1.) ΔV When a sufficiently high potential difference is applied across a glass tube filled with a gas, the gas is ionized. Electrons move toward the positive electrode and positive ions move toward the negative electrode. a) Calculate the current in a hydrogen discharge if, in each second, electrons and protons move in opposite directions past a common cross section of the tube. b) What is the direction of the current. 2.) In the Bohr model of monatomic hydrogen, an electron orbits a single fixed proton on a circular path of radius R. The magnitude of the electrical force exerted on the electron by the proton is a net radial force and is given by F = ke2 R = Mv2 2 R. a) Determine the speed of the electron as it orbits the proton. b) Calculate the amount of time it would take the electron to go around the proton once. (This time interval is called the period, and I will signify it with a lower case Greek letter (tau), τ.) c) Imagine a small cross-sectional area through which the electron passes. Calculate the average current through this area. d) Calculate the values of a), b) and c) is the radius is the so-called Bohr radius R B = m. p e R v e e W 38-1

26 3.) A circular wire of length = 125 m and diameter D = 2r = 2 mm has a resistivity of ρ = Ω m. a) Calculate the resistance of the wire. b) Calculate the resistance of a second wire of the same material, length, and resistivity as the first wire but with as twice the diameter. 4.) To measure the resistivity of a metal, an experimenter takes a wire of the same metal with a diameter of mm and a length of 1.10 m. A potential difference of 12.0 V is applied across the wire and a current of 3.75 A is established in the wire. Calculate the resistivity of the wire. 5.) A high-voltage transmission line consists of a copper cable of diameter 3.0 cm and length of 250 km. If the cable carries a steady-state current of 1500 A, a) Calculate the electric field intensity in the cable. b) Calculate the drift speed of the free electrons. c) Calculate the time for one electron to travel the entire length of the cable. 6.) According to the National Electrical Code, the maximum permissible current in a #12 copper wire (diameter 0.21 cm ) with rubber insulation is 20 A. a) Calculate the current density in the wire at the maximum allowed current. b) Calculate the drop in voltage across one meter of the wire at the maximum allowed current. 7.) A copper cable connects the positive pole of a 12 V car battery to the starter. The cable is 0.60 m long with a diameter of 0.50 cm. a) Calculate the resistance of this cable. b) When the starter stalls, the current may reach a value of 600 A. With this current, calculate the voltage drop across the cable. 8.) An underground telephone cable, consisting of a pair of wires, has suffered a short, represented by point P in the diagram below, somewhere along the system. The cable is 5 km in length. To find the location of the short, a technician measures the resistance across the terminals AB and then CD. Her first measurement yielded 30 Ω and her second 70 Ω. Where did she calculate the short to be? A P C B D W 38-2

27 Problems for Chapter 39 1.) A parallel plate air capacitor has a capacitance of pf. The magnitude of charge on each plate is 1.00 µc. a) Calculate the potential difference across the plates. b) If the charge is held constant, calculate the potential difference between the plates if the plate separation is doubled. 2.) In the network represented below, capacitances are given by C n = nc 1, where C 1 = 1.00 µf. a) Calculate the equivalent capacitance of the network between points a and b. b) If V ab = 900 V, compute the charge on the capacitors nearest a and b. c) If V ab = 900 V, calculate V cd. a C 3 C 3 C 3 c C 2 C 2 C 3 b d C 3 C 3 C 3 3.) A capacitor consists of two parallel plates of area A = 25 cm 2 separated by a distance d = 2 mm. The volume between the plates is filled with a material of dielectric constant κ = 5. The capacitor is connected to a 300 V battery. a) Compute the capacitance of this capacitor. b) Calculate the magnitude of the charge on each plate. c) Calculate the electric energy in the charged capacitor. 4.) S 12 V r = 0.3 Ω R = 3.7 Ω W 39-1

28 The circuit, represented schematically above, consists of a 12 V battery with an internal resistance of 0.3 Ω connected to a resistor of 3.7 Ω. If the switch is in an open position, determine what a high-resistance voltmeter would read if placed across (a) the battery, (b) the switch, (c) the resistor. Also, if the switch is closed, determine what the voltmeter would read if placed across (a) the battery, (b) the switch, (c) the resistor. 5.) Calculate the equivalent resistance of the network represented schematically below. 8 Ω 16 Ω 20 Ω 16 Ω 9 Ω 18 Ω 6.) If the capacitors are initially uncharged, calculate how much charge passes through switch S when it is closed. a 6 Ω S 6 µf V ab = 18 V 3 Ω 3 µf b W 39-2

29 Problems for Chapter 40 1.) For the circuit represented schematically below, do the following: a) Determine the potential difference between points a and b if switch S is open. b) If switch S is open, identify the correct statement: (i) V a > V b, (ii) V a = V b, (iii) V a < V b. c) When switch S is closed, determine the final potential of point b. d) Calculate the change in charge on each capacitor. 18 V 6 Ω S 6 µf a b 3 µf 3 Ω 2.) For the circuit represented schematically below, calculate the three currents indicated. 5 Ω 8 Ω I 2 1 Ω 1 Ω I 1 12 V 10 Ω 9 V I 3 W 40-1

30 3.) Calculate E 1 and E 2 for the circuit represented schematically below. Also, determine the electric potential difference between points a and b, V ab. 20 V, 1 Ω 6 Ω 1 A a 4 Ω E 1, 1 Ω b 2 A E 2, 1 Ω 2 Ω 4.) For the circuit represented schematically below, do the following: a) Determine the electric potential difference between points a and b, V ab. b) If points a and b are connected by a conducting wire, calculate the current in the 12 V cell. 12 V, 1 Ω 2 Ω 1 Ω a b 10 V, 1 Ω 3 Ω 2 Ω 8 V, 1 Ω 2 Ω W 40-2

31 Problems for Chapter 41 1.) An α -particle of mass M α = kg and electric charge q = 2e is moving with speed v = m / s due east when it moves into a region of space where there is a uniform magnetic field of magnitude B = T directed due north. Calculate the magnetic force, magnitude and direction, that will be exerted on the α -particle at the instant it enters the magnetic field. North ( ˆ j ) B Up( k ˆ ) 2e v East( ˆ i ) M α 2.) In a physics lab somewhere in North America, the Earth s magnetic field has a magnitude of B = Tesla and is directed at an angle of θ = 16 below the northern branch of the north-south axis. Find the magnitude and direction of the magnetic force exerted on an electron moving with speed v = m / s in a direction of due east. (A right-handed coordinate system is represented below.) North ( ˆ j ) Up( k ˆ ) v East( ˆ i ) B θ e e 3.) An α -particle of mass M α = kg and electric charge q = 2e is moving with speed v = m / s due east when it enters a region of space between two parallel plates where there is both a uniform external electric and uniform external magnetic field. The electric field is directed downward with a magnitude of E = 888 N / C. The magnetic field is W 41-1

32 directed north and has a magnitude of B = T. Calculate the net acceleration, magnitude and direction, exerted on the α -particle in this region. (Note: the s in the diagram below signify that the magnetic field is going into the page.) 2e M α v B E Up( k ˆ ) East( ˆ i ) 4.) Two very long straight conductors are parallel to each other and carry steady-state currents in opposite directions as represented in the diagram below. The magnitudes of the currents are related by I T = 2 I B = 4.80 A. Each conductor has a length. The conductors are separated by a distance of d = 2.50 m. Calculate the magnitude and the direction of the force per unit length exerted on the top conductor by its interaction with the bottom conductor. I T Up( k ˆ ) d ( ) North ĵ Into page East( ˆ i ) I B 5.) v M C y 6e d x I A carbon nucleus of charge q C = 6e and mass M C = kg is moving with a speed of v = m / s parallel to a long straight conductor, as represented in the diagram above. The conductor carries a steady-state current I = A. The carbon nucleus W 41-2

33 is a distance d = m from the conductor. Determine the instantaneous magnetic force, magnitude and direction, exerted on the carbon nucleus due to its interaction with the current in the conductor. 6.) A long straight conductor carries a steady-state current I 1 = A due north, as represented in the diagram below. A second conductor of length 2 = m is parallel to the long straight conductor and carries a steady-state current I 2 = 6.33 A in the opposite direction. The parallel conductors are separated by a distance of d = 1.50 m. Find the magnetic force, magnitude and direction, exerted on conductor two by the first. d N( ˆ j ) I 1 I 2 2 E( ˆ i ) 7.) A proton is moving due east with a speed v = m / s when it enters a region of space where there is a uniform magnetic field of magnitude B = 4.00 T directed due south. Calculate the magnitude and direction of the external electric field that must be maintained in this region of space if the proton is to pass through undeflected? 8.) At a specific location in the northern hemisphere, the Earth s magnetic field has a magnitude of B = T. The field has a component directed directly north and a component directed at an angle of θ = 26 below the north-south axis. At this location, an electron is moving with a speed of v = m / s. Determine the magnetic force, magnitude and direction, exerted on the electron if it is moving: a) due north, b) due east, c) due south, d) due west, e) directly upward, f) directly downward. W 41-3

34 North ( ĵ ) Up ( ˆk ) B, down 26 B B, north e e East ( î ) 9.) Two parallel conductors carry equal currents of I 1 = I 2 = 10.00A in the same direction. If the conductors are separated by a distance d = 1.50 m, determine the magnitude of the magnetic force per unit length exerted on each conductor. 10.) A helium nucleus of mass M α = kg and electric charge q α = 2e is released from rest in a region of space where there is a uniform, horizontally directed electric field of magnitude E = 1,000 N / C and no magnetic field. After moving through an electric potential difference of ΔV = 500 Volts, the helium nucleus moves into a region of space where there is no electric field but there is an external magnetic field of magnitude B = 2.50 Tesla that is directed vertically downward. The magnetic force exerted on the nucleus will cause it to move along a circular path. Calculate the radius of that circular path. Helium Source E R E B W 41-4

35 11.) An electron of mass M = kg and electric charge q = e, moves with speed v = m / s when it enters the region of space between the two parallel plates of a capacitor. The uniform electric field between the plates is E = N / C ĵ. Calculate the magnitude and the direction of the magnetic field that must be established between the plates if the electron is to pass through the plates undeflected. e e v y( ˆ j ) x( ˆ i ) 12.) A particle of mass M = kg and electric charge of q = C is moving with a speed of v = m / s east when it enters a region of space where there is a uniform magnetic field of magnitude B = T directed north. The magnetic field and the instantaneous velocity are perpendicular when the particle enters the region of the magnetic field. Calculate the radius of curvature of the particle as it moves into the region of uniform magnetic field. Also, calculate the magnitude of the acceleration of the particle. 13.) Two very long conductors carry steady-state currents of equal magnitude I. Both conductors are parallel to the x-axis, as represented in the diagram below. Each conductor is suspended by cables of length = 0.06 m. The conductors have a linear mass density given by λ m = kg / m. The currents run in such a manner as to cause the conductors to repel each other. If the total angle between the suspending cables is θ = 16, calculate the relative directions of the currents, and the magnitude of each current. y( ˆ j ) θ θ z( k ˆ ) W 41-5 x( ˆ i )

36 14.) A long, straight conductor carries a steady-state current of I 1 = 2.48 A, as represented below. A steady-state current of I 2 = 3.60 A is present in the rectangular loop with the dimensions shown. Find the net magnetic force exerted on the rectangular loop. (Note: c = 3a = 2b = 1.80 m.) y x I 1 I 2 c a b 15.) A positron and an electron chase each other around the circumference of a circle of radius R. Each moves with speed v, as represented on the left in the diagram below. Calculate the instantaneous magnetic force, magnitude and direction, exerted on the positron by the electron. (A positron is the anti-particle to the electron. It has the same properties as an electron except its electric charge is +e.) Also, if two electrons chase each other around a stationary helium nucleus, as represented on the right side of the diagram below, calculate the net magnetic force, magnitude and direction, exerted on each electron. v e v e e e 2e e R 2 p e R e + v e v 41-6

37 Problems for Chapter 42 1.) A long straight conductor lies on the y-axis and carries a steady-state current I = 6.75 A in the negative y-direction. For points P 1 ( 2 m,0,0 ), and P 2 ( 3m,0,0), do the following: a) Set up an appropriate coordinate system. Clearly identify the two points of interest. (Hint: Draw it from a perspective where the current would appear to be coming right at you. That is, look along the conductor in the positive y-direction.) b) Using a compass or circle template, generate circles centered on the wire with the circumferences passing through the points of interest. c) Calculate the magnitude of the magnetic field produced by the current at each point. d) Clearly indicate the direction of the magnetic field at each point. 2.) A long straight conductor carries a steady-state current of I = A due north. Find ( ). The the magnetic field, magnitude and direction, at a point with coordinates P 4m,0,5m coordinates of this point assume a Cartesian coordinate system where east is in the positive x-direction, north is in the positive y-direction, and upward is in the positive z-direction. 3.) The diagram below represents an end view of two parallel, long straight conductors carrying steady-state currents I in opposite directions. The conductors are perpendicular to the x-y plane and separated by a distance 2a. Derive an equation for the magnetic field at an arbitrary point P on the x-axis in terms of x, a, I, and k. Also, for what value of x will the magnetic field be largest? y( ˆ j ) I a a x P x( ˆ i ) I W 42-1

38 4.) A long, straight conductor carries a steady-state current I = A in the positive x-direction. If the conductor lies along the x-axis, determine the magnetic field, magnitude and direction, produced by this conductor at the point P the coordinates of which are P ( 0,4 m,5m ). 5.) A long, straight conductor lies along the y-axis, and carries a steady-state current I = 6.00 Amps in the ˆ j direction. Find the magnitude and the direction of the magnetic field that this current would produce at the point P, the position of which is given by r P = 5 m î + 5 m ˆk. z( ˆk ) P y( ˆ j ) 5m ˆk x( ˆ i ) I 5m î 6.) y( ˆ j ) I 1 y x I 2 10 m x( ˆ i ) P W 42-2

39 Two long, straight conductors are parallel to each other and carry equal steady-state currents of I 1 = I 2 = I = 5.50 A, as represented in the diagram above. Calculate the magnetic field produced by these two currents at the point the position of which is r P = 2.00 m the currents are coming out of the page, and r I1 = 2.00 m ĵ, while r I2 = 8.00 m î. ĵ. Note that 7.) A single, circular loop of wire or radius a = m carries a steady current I = 6.75 A, as represented in the diagram below. The z-axis passes through the center of the circular loop and is perpendicular to the plane of the loop. Do the following: a) Find the magnetic field, magnitude and direction, at point P = P ( 0,0,6a ). b) Redo part a), for N = 350 closely spaced loops of radius a and carrying current I. c) Assume two identical coaxial coils each with N = 350 closely spaced turns, one in the x-y plane, the second coil parallel to the first carrying the same amount of current in the same sense with its center at the point P = P ( 0,0,12a). Find the magnetic field, magnitude and direction at point P. z a x I y 8.) At point A, an electron moves due north with a speed of v o = 10 7 m / s. Do the following: a) Determine the magnetic field, magnitude and direction, necessary to cause the electron to move on a semicircular path of radius 5 cm from point A to point B. b) Calculate the time required for the electron to move from point A to point B. Recall: m e = kg. v o A m e e R W 42-3 B

40 9.) A long straight conductor carries a steady current I 1 in the positive x-direction. A second long straight conductor carries a steady current I 2 in the positive y-direction. Calculate the net magnetic field, magnitude and direction, at an arbitrary point P ( x, y). y x P I 2 y x I 1 10.) In the diagram below, each circle represents a positive charge q moving in the direction indicated with speed v. A uniform magnetic field B is directed in the positive x-direction. Calculate the magnitude and direction of the magnetic force exerted on each moving charge. Recall that F M = q v B. y b d c B = B î z f a e x W 42-4

41 Problems for Chapter 43 1.) Find the magnetic flux through the rectangle represented in the diagram below with side lengths given by = 2w = 56 cm that is situated in a region of space where the magnetic field is uniform and given by B = B ˆB = ( î ĵ ˆk ) T. The rectangle lies in the x-y plane. You may assume that  = ˆk. ( ) z k ˆ w A B ( ) y ˆ j ( ) x ˆ i ` 2.) A triangular prism, as represented in the diagram below, is located in a region of space where there is a uniform magnetic field given by B = 2.00 T î. Find the magnetic flux through each of the five prism faces. (This prism is an example of a closed surface.) y b 40 cm 30 cm a e c 20 cm z d B f x W 43-1

42 3.) A circular disk or radius R = m rotates about the y-axis with a constant angular speed ω in a region of space where there is a uniform magnetic field given by B = T î. At some time t o, the plane of the disk lay in the y-z plane. At a later time t L = t o + ( τ / 36 ), the plane of the disk had rotated through an angle ϕ = 10, as represented in the diagram below. a) Determine the angular speed ω. b) Calculate the magnetic flux through the disk at time t L. Time is measured in s. z ϕ ϕ y A B x 4.) A steady current I is carried in a straight wire of circular cross-section and radius R. The current is uniformly distributed over the cross-sectional area of the wire. a) Derive an equation for the current passing through the circle of radius r. b) Use Ampère's law to derive an equation for the magnitude of the magnetic field at a distance of r from the center of the wire. r R W 43-2

43 5.) I z z I y / 2 R x Two long, identical copper wires are parallel to each other and are in contact with each other along their entire length, as represented in the diagram above. Each wire carries a steady current I. Their current densities are uniform over their circular cross-sectional areas of radius R. a) Use Ampère's law to derive an equation for the magnetic field at any point on the z axis as a function of the distance z from the point of contact between the wires. b) Determine the location at which the magnetic field is a maximum. 6.) r 2 r 1 r 3 x A coaxial cable consists of a long cylindrical copper wire of radius r 1 surrounded by a coaxial cylindrical shell of inner radius r 2 and outer radius r 3, as represented in the diagram above. The wire and shell carry equal and opposite currents uniformly distributed over their volumes. Use Ampère's law to find the magnetic field in the following regimes: r < r 1, r 1 < r < r 2, r 2 < r < r 3, r 3 < r. W 43-3

44 7.) 2R R x A steady current I is uniformly distributed over the volume of a long copper pipe of inner radius R and outer radius 2R, as represented in the diagram above. Use Ampère's law to: a) Calculate the magnetic field at a point a perpendicular distance from the axis of symmetry given by r = ( 3 / 2 )R. b) Calculate the magnetic field at a point a perpendicular distance from the axis of symmetry given by r = 3R. 8.) I Represented above is a solenoid made of one turn of a sheet of copper. The solenoid has a length of 20 cm and a current of 2,000 A flowing through it. If one assumes the current is uniformly distributed over the sheet of copper, and if one treats the solenoid as very, very long, then calculate the magnetic field in the solenoid. 9.) The Crab nebula is a supernova remnant. In the nebula, electrons of linear momentum kg m / s orbit in a magnetic field of intensity B = 10 8 T. Assuming a circular orbit, determine the radius of the orbit. Recall: F M = q v B = F rad = Mv2 R ˆr. W 43-4

45 10.) z y x A large thin conducting sheet coincides with the the x-y plane and carries a steady-state current of σ amps flowing across each meter along the x-axis in the y-direction. Use Ampère's law to calculate the magnetic field at a point above the plane on the z-axis. Hint: the field will be parallel to the plane. 11.) z y x A large conducting sheet coincides with the x-y plane and carries a steady-state current of σ amps flowing across each meter along the x-axis in the y-direction. A second such sheet coincides with the x-z plane and carries a steady-state current of σ amps flowing across each meter along the x-axis in the z-direction. (See the diagram above.) Use Ampère's law, and the results of problem 10.) to calculate the magnetic field at a point in each of the four possible regions. W 43-5

46

47 Problems for Chapter 44 1.) A solenoid of length s = 1.50 m has N s = 12,000 circular turns of radius r s = 1.75 cm. The axis of symmetry of the solenoid is aligned with the z-axis. Coaxial with the solenoid is a circular copper coil of radius r C = 17.5 cm and N C = 120 turns. If the current in the solenoid goes from 0 to A in a time interval Δt = s, then calculate the following: a) The average induced EMF in the coil. b) The average induced current in the coil if R coil = 1.0 Ω. c) The direction the induced current moves through the coil: (Circle the best answer.) i) The induced current passes from A to B. ii) The induced current passes from B to A. N s N C A r C B I s r s z( ˆk ) s 2.) A single circular loop of wire of radius R lies in a horizontal plane. Directly above the center of this loop is a bar magnet that is oriented vertically as represented in the diagram below. If the magnet is suddenly moved away from the loop, using Lenz s law, determine the direction of the induced current in the loop. (Circle the correct answer.) a) The induced current passes from A to B along the shorter path. b) The induced current passes from B to A along the shorter path. W 44-1

48 Up( k ˆ ) v S N North ( ˆ j ) A B East( ˆ i ) 3.) A square loop of copper of side length s o lies in a horizontal plane, as represented in the diagram below. A uniform magnetic field of magnitude B = T ˆk passes through the square. If the square collapses into a smaller square of side length s f = ( 1 / 4 )s o = cm in a time interval Δt = s, then calculate the following: a) The average induced EMF in the winding. b) The average induced current in the winding if the resistance of the winding is R = 2 Ω. c) The direction the induced current moves through the winding: (Circle the best choice.) i) The induced current passes from A to B through the shorter path. ii) The induced current passes from B to A through the shorter path. s o Up( k ˆ ) s f B North ( ˆ j ) s o s f A B East( ˆ i ) W 44-2

49 4.) A metal sphere of radius R = 0.80 m rests on a horizontal insulating surface, as represented in the diagram below. The sphere is in a uniform magnetic field of magnitude B = 3 T directed vertically upward. Determine the following: a) The total magnetic flux through the sphere. b) The total magnetic flux through the sphere if the magnetic field were parallel to the surface rather than perpendicular. (Recall that for closed volumes, A ˆ is positive when directed perpendicular to the surface and outward, while A ˆ is negative when perpendicular to the surface and directed inward.) B R B 5.) In the diagram below, we have represented a solenoid of N S turns. The solenoid is connected to a battery with EMF of value E. Initially, a switch S is open and no current can pass through the solenoid. Next to the solenoid is a coil of N C turns. The coil is connected to a resistor of resistance R. Determine if any induced current will pass through the coil after the switch is closed. (Circle the best answer.) a) There will be no induced current passing through the resistor. b) A brief induced current will pass through the resistor from a to b. c) A brief induced current will pass through the resistor from b to a. d) A steady induced current will pass through the resistor from a to b. e) A steady induced current will pass through the resistor from b to a. N S N C + E S a R b W 44-3

50 6.) A metallic cube of side length s = m rests on a horizontal insulating surface, as represented in the diagram below. A uniform magnetic field of magnitude B = 1.6 T is directed vertically downward. Calculate the following: a) The magnetic flux through the top of the cube. b) The magnetic flux through the bottom of the cube. c) The magnetic flux through each vertical face of the cube. d) The total magnetic flux through the cube. B s s s 7.) B A thin conducting rod of length = 2 m rotates with a constant angular speed ω = 30 rad / s about a vertical axis, as represented in the diagram above. The rod sweeps out an area permeated by a uniform magnetic field of magnitude B = T directed vertically upward. Calculate the following: W 44-4

51 a) The electrical polarity of the tip of the rod and the polarity of the hub. (Viewed from above, the rod rotates in a counter-clockwise sense.) b) The average induced EMF between the tip of the rod and the hub. 8.) A conducting rod of mass M = kg and length = 3 m is released from rest, as represented in the diagram below. The rod is oriented along a north-south line and constrained to slide along two vertical metal conductors with which the rod makes good electrical contact. The rod moves in a region of space where there is a uniform magnetic field of magnitude B = 1.5 T directed due east (into the page in the diagram, represented by the X s ) The total resistance of the rod and vertical conductors is R = 1 Ω. Determine the following: a) An expression for the magnetic flux, Φ M. b) An expression for the change in the magnetic flux, ΔΦ M. c) An expression for the average induced EMF in the rod. d) The terminal speed of the rod. (Hint: Consider the forces acting on the rod and how they affect the acceleration of the rod. You may ignore frictional effects.) Up ( ˆk ) North ( ĵ ) B t o = 0 R W 44-5

52 9.) Up( k ˆ ) w B N( ˆ j ) E( ˆ i ) A rectangle of dimensions = 3w = 12 m is oriented vertically along a north-south line as represented in the diagram above. The plane is in a region of space where there is a uniform magnetic field given by B = T î. Calculate the magnetic flux through this plane. Assume  = î. 10.) A flexible wire is wrapped around an insulating core. The wire is connected to a battery with a control switch initially open. A second winding is connected to a resistor and is coaxial with and located to the right of the first winding, as represented in the diagram below. For a brief time after the switch is closed, there will be an induced current in the second winding. Determine the direction that the induced current will pass through the resistor. ΔV S a R b W 44-6

53 11.) Up( k ˆ ) B ω N( ˆ j ) E( ˆ i ) A thin conducting rod of length = 1.46 m rotates about a vertical axle with a constant angular speed ω = 25 rad / s, as represented in the diagram above. There is a uniform magnetic field B = T ˆk. Use Faraday s law to calculate the following: a) The polarity (positive or negative) of each end of the rod. b) The average induced EMF between the two ends of the rod. 12.) A thin metal conducting rod of length = 2.64 m moves with constant speed v = 20 m / s on top of an elaborate set of horizontal, frictionless, parallel conductors, as represented in the diagram below. A uniform magnetic field of magnitude B = T is directed vertically upward. Calculate the magnitude of the constant force that must be applied to the rod to keep it moving with this speed. You may assume that the resistance of this system of conductors is R = 2.60 Ω and that ˆv = ˆF applied. Recall that M F conductor = I Bext. Up ( ˆk ) B F applied B North ( ĵ ) East ( î ) W 44-7

54 13.) N C I S r S r C N S S A solenoid of N S = 2,500 turns and length S = 1.75 m carries a steady-state current of I S = 4 A, as represented in cross-section in the diagram above. The windings of the solenoid are circular with a radius r S = m. Coaxial with the solenoid is a coil with N C = 1,526 circular windings of radius r C = m. Calculate the following: a) The average induced EMF that would be generated in the coil if the current suddenly were to go to zero in a time interval Δt = s. b) Using the and symbols, indicate the direction the induced current would move in the coil. 14.) Up( k ˆ ) Initial Orientation w ω W 44-8 B N( ˆ j ) E( ˆ i )

55 A rectangular loop of wire of dimensions = 3w = 21 cm is initially oriented vertically along a north-south line, as represented in the diagram above. The plane is in a region of space where there is a uniform magnetic field given by B = 2.25 T î. The loop rotates about the north-south axis with a constant angular speed ω = 2 rad / s in a clockwise sense when viewed looking directly north. Calculate the average induced EMF in the loop for one quarter cycle. Determine the direction the induced current would move in that branch of the loop that lies on the north-south axis. 15.) Up( k ˆ ) B Initial Orientation N( ˆ j ) ω R E( ˆ i ) A circular coil of N tightly wound circles of radius R is initially oriented vertically, as represented in the diagram above. Throughout the region, a uniform magnetic field of magnitude B is directed vertically upward. The coil turns in a clockwise sense around the north-south axis with a constant angular speed ω. Derive an expression for the average induced EMF in the coil when the coil has turned through an arbitrary angle ϕ, measured in the clockwise sense from the vertical. Also, determine the sense in which the induced current moves around the coil. 16.) Calculate a crude estimate of the energy stored in the Earth s magnetic field, by assuming that the magnitude of the field is B = T from the surface to a distance above the surface of one Earth radius, where the Earth s radius is approximated by R = m. 2R R B W 44-9

56 17.) A very long straight wire carries a current that increases at a constant rate. In the plane of the wire is a rectangular loop of wire of length and width a, as represented in the diagram below. Do the following: a) Derive an expression of the rate at which the magnitude of the magnetic field increases at point P, a perpendicular distance r from the wire. b) Derive an expression for the magnitude of the average induced EMF in the loop. Assume that the average magnetic field strength is equal to that at point P. a I P r 18.) A point charge q moves with speed v in empty space. Derive a formula for the magnetic energy density u M at a point P a distance r from q, as represented below. q v r P W 44-10

57 Problems for Chapter 45 1.) A toroid has N = 500 turns wound on a core of relative permeability of κ M = 600. The toroid carries a current I = 0.30 A and has a mean circumferential length of = 50.0 cm. a) Calculate the flux density in the core. b) Calculate the self-inductance of the coil if the cross-sectional area is 8.0 cm 2. I 2.) The dipole moment per unit volume of cobalt is A m 2 / m 3 when at maximum magnetization. Assume this magnetization is due to completely aligned electrons. a) Calculate how many such electrons there are per unit volume. b) Calculate how many such electrons there are per atom. (Note: The average mass density of cobalt is ρ Co = kg / m 3 and the atomic mass is 58.9 grams / mole.) 3.) In iron, two of the electrons of each atom participate in the alignment of spins that gives rise to magnetization. Assume that a cylindrical piece of iron of radius 1.0 cm and length 8.0 cm is completely magnetized along its major axis of symmetry, with every available electron aligned. a) Calculate the number of aligned electrons. b) Calculate the magnetic field produced by this alignment. 4.) The alignment of of electron spins in a ferromagnetic material suggests that the magnetized material has angular momentum. Assume that a cylindrical piece of iron of radius 2.0 cm and length 30.0 cm is completely magnetized along its major axis of symmetry, with every available electron aligned. (See problem 3.)) Now, if the magnetization is suddenly reversed so that spins are now anti-parallel, calculate the change in angular momentum. (Note: The average mass density of iron is ρ Fe = kg / m 3 and the atomic mass is 55.8 grams / mole.) 5.) A long solenoid has a turn density of n = 1200 turns / m. The solenoid is filled with a ferromagnetic material. The value of the magnetic field in this material is B = 2.0 T. Use this information to determine the value of κ M under these conditions. W 45-1