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).