Homework Reading: Chap. 32 and Chap. 33 Suggested exercises: 32.1, 32.3, 32.5, 32.7, 32.9, 32.11, 32.13, 32.15, 32.18, 32.20, 32.24, 32.28, 32.32, 32.33, 32.35, 32.37, 32.39 Problems: 32.46, 32.48, 32.52, 32.53, 32.56, 32.57, 32.60, 32.63, 32.65, 32.67, 32.68, 32.71 (due Fri., Nov. 20) 30 Average = 47.4 Number of Students 25 20 15 10 5 0 20 40 60 80 100 Grade 1
Chapter 32. The Magnetic Field Digital information is stored on a hard disk as microscopic patches of magnetism. Just what is magnetism? How are magnetic fields created? What are their properties? These are the questions we will address. Chapter Goal: To learn how to calculate and use the magnetic field. Chapter 32. The Magnetic Field Topics: Magnetism The Discovery of the Magnetic Field The Source of the Magnetic Field: Moving Charges The Magnetic Field of a Current Magnetic Dipoles Ampère s Law and Solenoids The Magnetic Force on a Moving Charge Magnetic Forces on Current-Carrying Wires Forces and Torques on Current Loops Magnetic Properties of Matter 2
Chapter 32. asic Content and Examples Source of Magnetic Forces Permanent magnet Electromagnet Moving charged particles Elementary particles (electron) S N All the magnetic sources are presented as a dipole, having two poles, North pole and South pole No magnetic monopole has been founded yet! 3
Magnetic Forces Unlike poles attract S N S N Like poles repel S N N S N S S N Magnetic Field Magnetic field A vector Unit: Tesla 4
Magnetic Field Practical definition: Fire a positive charge perpendicular to the magnetic field, the magnitude of the magnetic field is given by: The direction of the magnetic field is given by: Not Copyright that 2008 the Pearson force, Education, field, Inc., publishing and as velocity Pearson Addison-Wesley. are not in the same direction! Magnetic Field Lines Magnetic field can be described by magnetic field lines. 1. Magnetic field lines show the direction of at any point (tangent) 2. Magnetic field lines for a bar magnet come out of the North pole and enter into the South pole Unlike electric field lines, magnetic field lines do not begin or end. They either form closed loops or extend out to infinity. 5
Magnetic Field Lines S N Small magnet All the small magnets aligned S N along the direction of magnetic field Copyright lines: field 2008 Pearson line direction Education, Inc., is publishing the same as Pearson as Addison-Wesley. the magnetic dipole direction. Magnetic Field Lines 6
Magnetic Field Lines Magnetic Field Lines 7
Magnet When you cut bar magnet into two, are you left with one North pole and one South pole? No, since magnetic field lines are continuous both inside and outside the magnet, you get two North poles and two South poles. N S N S + N S Earth Magnetic Field The earth s core is made of molten metal such as ion or nickel. Iron and nickel are very good conductors of electricity and electric currents can flow easily in them. As the earth rotates, large electric currents are built up and these produce the earth s magnetic field. The magnetic South pole is actually geographic North pole. 8
Magnetic Field versus Electric Field -field Electric sources are inherently monopole or point charge sources. E-field Magnetic sources are inherently dipole sources you cannot isolate North and South monopoles. Magnetic Force on Moving Charges 9
Magnetic Force on Moving Charges The Electromagnetic Force If both the electric field and magnetic field exist, the force on a charge is defined as the Lorentz force: The electric force is straightforward, being in the direction of the electric field if the charge q is positive, but the direction of the magnetic part of the force is given by the right hand rule. 10
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A Moving Charge in Magnetic Field In a uniform magnetic field, a constant magnetic force acts on a moving charge with speed v. The direction of the force is perpendicular to the direction of the velocity. So the particle performs a circular movement: F mv r 2 F qv r mv q m q v 13
A Moving Charge in Magnetic Field Other related parameters: The period: T 2r 2 m v q Do not depend on the velocity The frequency: The angular frequency: 1 q T 2 m f 2f q m Only depend on magnetic filed and charge/mass ratio A Moving Charge in Magnetic Field Helical motion v v v cos vsin v v r m vsin q T Charge- field 2 m q Do not depend on the speed pitch v q m p v cos 2 q 14
A Moving Charge in Magnetic Field A Moving Charge in Magnetic Field 15
Magnetic Force on a Current-Carrying Wire For a moving charged particle F qv For a current (a flux of moving charged particles) (Demo) F il F b i dl a F nalev ds Torque F F F 1 2 ia sin il ia ia b F1 sin F 2 iab sin b sin 2 τ ia F 4 2 I F 1 θ F 1 F 2 F 2 τ b a F3 Motor 16
Magnetic Dipole Moment Definition: μ ia Torque: τ μ Example The following figure shows a wire carrying a current i = 6.0 A in the positive direction of the x axis and lying in a nonuniform magnetic field given by ( 2.3T / m) xˆi (2.0T / m) xˆj with in Teslas and x in meters. What is the net magnetic force F on the section of the wire between x = 0 and x = 2.0 m? y i x = 2.0 m x 17
Example (continued) What we know: a current wire (i) in a magnetic field () What we expect: a magnetic force will be exerted on the wire Can we solve the problem directly using the following equation? il F No, since is not uniform! F b i dl a We must mentally divide the wire into differential lengths and using above equation to find the differential force df on each length, then sum these differential forces to find the net force F. Example (continued) y i x = 2.0 m x dx dl df dl dxˆi 18
Example (continued) The differential force df on the length dx of the wire is df idl idxˆi (2.3xˆi 2.0xˆ) j idx[2.3x(ˆi ˆ) i 2.0x(ˆi ˆ)] j idx[0 2.0xkˆ ] 2..0ixdxkˆ The constant 2.0 has the unit of Teslas per meter. Clearly the magnetic force does not depend on the x-component of (this component is parallel to the current). Example (continued) Since we have obtained the expression for df, the total magnetic force will be integrating df from x = 0 to x = 2.0 m: F df 2.0m (2.0T / m) ikˆ (2.0T / m) ixkˆ dx 2.0m 1 2 (2.0T / m)(6.0a)( )(2.0m) kˆ 2 24T A mkˆ 24Nkˆ 0 0 xdx The magnitude of the net force is 24N, and the direction is along the positive direction of z-axis. 19
Activity Activity 20
Activity Activity 21
Example A metal rod having a mass per unit length of 0.01 kg/m carries a current of I = 5.00 A. The rod hangs from two wires in a uniform vertical magnetic field, as shown in the following figure. If the wire makes an angle θ = 45.0 o with the vertical when in equilibrium, determine the magnitude of the magnetic field. θ θ I Example (count.) Let s first think what causes the equilibrium in this system. We need to focus on the rod (the main object!!!). Usually the equilibrium are caused by force equilibrium, torque equilibrium, etc. We have to do a force analysis on the system. What are the forces exerted on the rod? Gravity Tension Magnetic force Magnetic force is directly related to the magnetic field. F N θ N I mg θ 22
Example (count.) Solving the problem in two ways: I. Force equilibrium F N mg cos 45 N sin 45 Since N is unknown, dividing these two equations, F 1 mg tan 45 F mg tan 45 Since we know the mass of the rod, the magnitude of the magnetic force can be obtained. F I mg N 45 o Example (count.) Since the magnetic force is caused by the magnetic field, F IL Since L is perpendicular to, Therefore, F IL mg IL tan 45 mg IL tan 45 ( m / L) g L tan 45 F I mg 2 0.0100 kg m9.80 m s tan 45.0 19. mt 6 5.00 A N 45 o 23
Example (count.) Second method: II. Torque equilibrium oth magnetic force and gravity induce torques. The torque caused by the tension force is zero because the tension force points to the rotating axis O. F The torque caused by the gravity should be balanced by the torque caused by the magnetic force. F d sin 45 F mgd cos 45 mg tan 45 N mg 45 o O Origin of Magnetic Field Oersted s Experiment A current carrying wire generates magnetic field! 24
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Example The following figure shows in horizontal cross section, three wires that are meant to carry current from a lightening rod on top of a house to the ground in case of lightening strikes the rod. The wires are parallel, have a length L = 4.0 m, and are spaced r = 5.0 mm apart. Assume that, during a strike, the current in each wire is I = 5000 A. What are the magnitude and direction of the net force on each wire due to the currents in the other two wires? r r 39
Gauss s Law of Magnetic Field The net magnetic flux through a closed surface is zero. Magnetic monopole has not been discovered yet! Chapter 32. Clicker Questions 40
Does the compass needle rotate clockwise (cw), counterclockwise (ccw) or not at all? A. Clockwise. Counterclockwise C. Not at all Does the compass needle rotate clockwise (cw), counterclockwise (ccw) or not at all? A. Clockwise. Counterclockwise C. Not at all 41
The magnetic field at the position P points A. Into the page.. Up. C. Down. D. Out of the page. The magnetic field at the position P points A. Into the page.. Up. C. Down. D. Out of the page. 42
The positive charge is moving straight out of the page. What is the direction of the magnetic field at the position of the dot? A. Left. Right C. Down D. Up The positive charge is moving straight out of the page. What is the direction of the magnetic field at the position of the dot? A. Left. Right C. Down D. Up 43
What is the current direction in this loop? And which side of the loop is the north pole? A. Current counterclockwise, north pole on bottom. Current clockwise; north pole on bottom C. Current counterclockwise, north pole on top D. Current clockwise; north pole on top What is the current direction in this loop? And which side of the loop is the north pole? A. Current counterclockwise, north pole on bottom. Current clockwise; north pole on bottom C. Current counterclockwise, north pole on top D. Current clockwise; north pole on top 44
An electron moves perpendicular to a magnetic field. What is the direction of? A. Left. Into the page C. Out of the page D. Up E. Down An electron moves perpendicular to a magnetic field. What is the direction of? A. Left. Into the page C. Out of the page D. Up E. Down 45
What is the current direction in the loop? A. Out of the page at the top of the loop, into the page at the bottom.. Out of the page at the bottom of the loop, into the page at the top. What is the current direction in the loop? A. Out of the page at the top of the loop, into the page at the bottom.. Out of the page at the bottom of the loop, into the page at the top. 46
Which magnet or magnets produced this induced magnetic dipole? A. a or d. a or c C. b or d D. b or c E. any of a, b, c or d Which magnet or magnets produced this induced magnetic dipole? A. a or d. a or c C. b or d D. b or c E. any of a, b, c or d 47
Chapter 32. Reading Quizzes What is the SI unit for the strength of the magnetic field? A. Gauss. Henry C. Tesla D. ecquerel E. ohr magneton 48
What is the SI unit for the strength of the magnetic field? A. Gauss. Henry C. Tesla D. ecquerel E. ohr magneton What is the shape of the trajectory that a charged particle follows in a uniform magnetic field? A. Helix. Parabola C. Circle D. Ellipse E. Hyperbola 49
What is the shape of the trajectory that a charged particle follows in a uniform magnetic field? A. Helix. Parabola C. Circle D. Ellipse E. Hyperbola The magnetic field of a point current source is given by A. iot-savart s law.. Faraday s law. C. Gauss s law. D. Ampère s law. E. Einstein s law. 50
The magnetic field of a point current source is given by A. iot-savart s law.. Faraday s law. C. Gauss s law. D. Ampère s law. E. Einstein s law. The magnetic field of a straight, current-carrying wire is A. parallel to the wire.. inside the wire. C. perpendicular to the wire. D. around the wire. E. zero. 51
The magnetic field of a straight, current-carrying wire is A. parallel to the wire.. inside the wire. C. perpendicular to the wire. D. around the wire. E. zero. 52