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Chapter 26 Sources of Magnetic Field

Chapter 26 Sources of Magnetic Field. . P. Permeability of free space. . position vector from to the field point P. Biot-Savart Law (P 614 ). Magnetic equivalent to C’s law by Biot & Savart. Magnetic field due to an infinitesimal current:. 2. . P. . B due to full current.

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Chapter 26 Sources of Magnetic Field

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  1. Chapter 26 Sources of Magnetic Field

  2. .P Permeability of free space  position vector from to the field point P Biot-Savart Law (P614) Magnetic equivalent to C’s law by Biot & Savart Magnetic field due to an infinitesimal current: 2

  3. .P  B due to full current 1) Magnitude: 2) Direction: right-hand rule 3) Total magnetic field due to a full current Notice: directions 3

  4. x I x . P a o  Idx r A straight current Example1: Magnetic field of a straight current. Solution: Infinitesimalcurrent 4

  5. I . . . A B P a Discussion: 1) Infinitestraight current 2) Magnetic field at point A or B: No field! 5

  6. Example2: of a square loop at O, A and P. .O I A l l . P . Square loop Solution: BP can also be obtained by the formula 6

  7. I1 I2 d Force between parallel wires (P605) Longparallel straight wires with current I1 and I2 Magnetic field due to I1 : Ampere force on I2 : F per unit length: Operational definitions of Ampere & Coulomb 7

  8. R . x o P Circular current Magnetic field of a circular current on the axis. From the symmetry: 8

  9. R . x o P R R . . o o I I Discussion: 1) Magnetic dipole moment 2) B at the center of a circular / arc current: 9

  10. I b o a R . I o . Combined currents Example3: Magnetic field at point O. (a) (b) Uniform conductor ring 10

  11. z . R o Rotating charged ring Question: A uniformly charged ring rotates about z axis. Determine the magnetic field at the center. 11

  12. Magnetic flux ΦB : Magnetic flux through an area ∝ number of field lines passing that area 1) Uniform field: Unit: Wb (Weber) 2) General case: 3) Closed surface: → net flux 12

  13. Gauss’s law for magnetic field The total magnetic flux through any closed surface is always zero. → Gauss’s law for magnetic field Closed field lines without beginning or end 13

  14. The linear integral of B around any closed path is equal to μ0 times the current passing through the area enclosed by the chosen path. Ampere’s law (1) Ampere’s law 1) Magnetic field is produced by all currents 2) The closed integral only depends on Iin 3) How to count the enclosedcurrent? 14

  15. I I1 I3 I l r I2 l Ampere’s law (2) The sign of enclosed current: right-hand rule 1) Circular path around infinite straight current 15

  16. Ampere’s law (3) 2) Any path around the current in same plane 3) Any path that encloses no current 16

  17. Equations for E & B field Magnetic field is not a conservative field Maxwell equations for steady magnetic field & electrostatic field Applications of Ampere’s law → symmetry 17

  18. I r R Cylindrical current Example4: The current is uniformly distributed over the cylindrical conductor. Determine (a) magnetic field; (b) magnetic flux. Solution: (a) Symmetry of system? 2 r 18

  19. I 2R l R (b) Magnetic flux through the area: 19

  20. × I · I R1 × × · o · · R2 R3 × Coaxial cable Question:Coaxial cable:a wire surrounded by a cylindrical tube. They carry equal and opposite currents I distributed uniformly. What is B ? 20

  21. Solenoid Solenoid: a long coil of wire with many loops → Uniform 21

  22. r Toroid Toroid: solenoid bent into the shape of a circle N loops, current I Outside: B = 0 Inside? Nonuniform field becomes solenoid again 22

  23. Q v . R *B produced by moving charge Equations about Ampere force & Lorentz force: Magnetic field produced by a single moving charge B created by rotating charge? 23

  24. *Ferromagnetic Increase the magnetic field by ferromagnetics Km: relative permeability μ: magnetic permeability Electromagnet Hysteresis: “Hard/soft” 24

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