1. Chapter 26 Sources of Magnetic Field

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

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

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

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

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

7. Force between parallel wires (P 605 ) 7 Long parallel straight wires with current I 1 and I 2 I1I1 I2I2 d Magnetic field due to I 1 : Ampere force on I 2 : F per unit length: Operational definitions of Ampere & Coulomb

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

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

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

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

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

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

14. Ampere’s law (1) 14 1) Magnetic field is produced by all currents Ampere’s law The linear integral of B around any closed path is equal to μ 0 times the current passing through the area enclosed by the chosen path. 2) The closed integral only depends on I in 3) How to count the enclosed current?

15. 15 The sign of enclosed current: right-hand rule l I1I1 I2I2 I3I3 l I I 1) Circular path around infinite straight current Ampere’s law (2) r

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

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

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

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

20. Coaxial cable 20 Question: Coaxial cable: a wire surrounded by a cylindrical tube. They carry equal and opposite currents I distributed uniformly. What is B ? × × × × I · · · R3R3 o I R1R1 R2R2 ·

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

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

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

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