Chapter 9 Sources of Magnetic Fields 9.1 Biot-Savart Law....................................................................................................9-3 Interactive Simulation 9.1: Magnetic Field of a Current Element.......................9-4 Example 9.1: Magnetic Field due to a Finite Straight Wire ...................................9-4 Example 9.2: Magnetic Field due to a Circular Current Loop ...............................9-7 9.1.1 Magnetic Field of a Moving Point Charge ..................................................9-10 Animation 9.1: Magnetic Field of a Moving Charge ..........................................9-11 Animation 9.2: Magnetic Field of Several Charges Moving in a Circle.............9-12 Interactive Simulation 9.2: Magnetic Field of a Ring of Moving Charges .......9-12 9.2 Force Between Two Parallel Wires ....................................................................9-13 Animation 9.3: Forces Between Current-Carrying Parallel Wires......................9-14 9.3 Ampere’s Law.....................................................................................................9-14 Example 9.3: Field Inside and Outside a Current-Carrying Wire.........................9-17 Example 9.4: Magnetic Field Due to an Infinite Current Sheet ...........................9-18 9.4 Solenoid ..............................................................................................................9-20 Examaple 9.5: Toroid............................................................................................9-23 9.5 Magnetic Field of a Dipole .................................................................................9-24 9.5.1 Earth’s Magnetic Field at MIT ....................................................................9-25 Animation 9.4: A Bar Magnet in the Earth’s Magnetic Field .............................9-27 9.6 Magnetic Materials .............................................................................................9-28 9.6.1 Magnetization ..............................................................................................9-28 9.6.2 Paramagnetism.............................................................................................9-31 9.6.3 Diamagnetism ..............................................................................................9-32 9.6.4 Ferromagnetism ...........................................................................................9-32 9.7 Summary.............................................................................................................9-33 9.8 Appendix 1: Magnetic Field off the Symmetry Axis of a Current Loop............9-35 9.9 Appendix 2: Helmholtz Coils .............................................................................9-39 Animation 9.5: Magnetic Field of the Helmholtz Coils ......................................9-41 Animation 9.6: Magnetic Field of Two Coils Carrying Opposite Currents ........9-43 Animation 9.7: Forces Between Coaxial Current-Carrying Wires......................9-44 9-1 Animation 9.8: Magnet Oscillating Between Two Coils ....................................9-44 Animation 9.9: Magnet Suspended Between Two Coils.....................................9-45 9.10 Problem-Solving Strategies ..............................................................................9-46 9.10.1 Biot-Savart Law:........................................................................................9-46 9.10.2 Ampere’s law:............................................................................................9-48 9.11 Solved Problems ...............................................................................................9-49 9.11.1 Magnetic Field of a Straight Wire .............................................................9-49 9.11.2 Current-Carrying Arc.................................................................................9-51 9.11.3 Rectangular Current Loop..........................................................................9-52 9.11.4 Hairpin-Shaped Current-Carrying Wire.....................................................9-54 9.11.5 Two Infinitely Long Wires ........................................................................9-55 9.11.6 Non-Uniform Current Density...................................................................9-57 9.11.7 Thin Strip of Metal.....................................................................................9-59 9.11.8 Two Semi-Infinite Wires ...........................................................................9-60 9.12 Conceptual Questions .......................................................................................9-62 9.13 Additional Problems .........................................................................................9-62 9.13.1 Application of Ampere's Law ....................................................................9-62 9.13.2 Magnetic Field of a Current Distribution from Ampere's Law..................9-63 9.13.3 Cylinder with a Hole..................................................................................9-64 9.13.4 The Magnetic Field Through a Solenoid ...................................................9-64 9.13.5 Rotating Disk .............................................................................................9-65 9.13.6 Four Long Conducting Wires ....................................................................9-65 9.13.7 Magnetic Force on a Current Loop............................................................9-65 9.13.8 Magnetic Moment of an Orbital Electron..................................................9-66 9.13.9 Ferromagnetism and Permanent Magnets..................................................9-67 9.13.10 Charge in a Magnetic Field......................................................................9-67 9.13.11 Permanent Magnets..................................................................................9-68 9.13.12 Magnetic Field of a Solenoid...................................................................9-68 9.13.13 Effect of Paramagnetism..........................................................................9-69 9-2 Sources of Magnetic Fields 9.1 Biot-Savart Law Currents which arise due to the motion of charges are the source of magnetic fields. When charges move in a conducting wire and produce a current I, the magnetic field at any point P due to the current can be calculated by adding up the magnetic field contributions, dB , from small segments of the wire ds , (Figure 9.1.1). Figure 9.1.1 Magnetic field dB at point P due to a current-carrying element I d s . These segments can be thought of as a vector quantity having a magnitude of the length of the segment and pointing in the direction of the current flow. The infinitesimal current source can then be written as I d s . Let r denote as the distance form the current source to the field point P, and rˆ the corresponding unit vector. The Biot-Savart law gives an expression for the magnetic field contribution, dB , from the current source, Id s , µ Idsr× ˆ dB = 0 (9.1.1) 4π r 2 where µ 0 is a constant called the permeability of free space: −7 µπ0 =×410T⋅m/A (9.1.2) Notice that the expression is remarkably similar to the Coulomb’s law for the electric field due to a charge element dq: 1 dq ˆ dE= 2 r (9.1.3) 4πε0 r Adding up these contributions to find the magnetic field at the point P requires integrating over the current source, 9-3 µ I d s×rˆ BB==d 0 (9.1.4) ∫∫2 wire 4π wire r The integral is a vector integral, which means that the expression for B is really three integrals, one for each component of B . The vector nature of this integral appears in the cross product Ids×rˆ . Understanding how to evaluate this cross product and then perform the integral will be the key to learning how to use the Biot-Savart law. Interactive Simulation 9.1: Magnetic Field of a Current Element Figure 9.1.2 is an interactive ShockWave display that shows the magnetic field of a current element from Eq. (9.1.1). This interactive display allows you to move the position of the observer about the source current element to see how moving that position changes the value of the magnetic field at the position of the observer. Figure 9.1.2 Magnetic field of a current element. Example 9.1: Magnetic Field due to a Finite Straight Wire A thin, straight wire carrying a current I is placed along the x-axis, as shown in Figure 9.1.3. Evaluate the magnetic field at point P. Note that we have assumed that the leads to the ends of the wire make canceling contributions to the net magnetic field at the point P . Figure 9.1.3 A thin straight wire carrying a current I. 9-4 Solution: This is a typical example involving the use of the Biot-Savart law. We solve the problem using the methodology summarized in Section 9.10. (1) Source point (coordinates denoted with a prime) Consider a differential element ds= dx'ˆi carrying current I in the x-direction. The location of this source is represented by ri''= x ˆ . (2) Field point (coordinates denoted with a subscript “P”) Since the field point P is located at (,x y)= (0,a), the position vector describing P is ˆ rP = aj . (3) Relative position vector The vector rr=P −r' is a “relative” position vector which points from the source point to the field point. In this case, r =−aˆj x'ˆi , and the magnitude r==||r a2+x'2is the distance from between the source and P. The corresponding unit vector is given by rjaxˆˆ− 'i rˆ == =sinθ ˆˆj−cosθ i r ax22+ ' (4) The cross product d sr×