1. Chapter 29: Sources of Magnetic Field
Sources are moving electric charges
single charged particle:
field point B ^ r ^ r r v
field lines circulate around “straight line trajectory”

2. from the definition of a Coulomb and other standards,
o= 4x10-7 N s2/C2 = 4x10-7 T m/A
with o , the speed of light as a fundamental constant can be determined (in later chapters) from
c2 = 1/(o o)

3. Consider: Two protons move in the x direction at a speed v.
Calculate all the forces each exerts on the other.
magnetic: F1 on 2 = q2v2xBat2due to 1
= q2v2x (k’ (q1v1xr12)/(r12)2)
electric F1 on 2 = (k q1q2)/(r12)2 r12 ^ ^ v1 q1
Simple geometry (all 90 degree angles)
Fmagnetic = k’|q2v2 q1v1|/ (r12)2
= k’ e2v2/r2
Felectric = k e2/r2
Fmagnetic /Felectric = k’ v2 /k = (o/4v2/ (1/4o
= v2 /c2 but... depends upon frame?
r12 ^
r12
F1 on 2 B v2

4. Gauss’s Law for magnetism:
Magnetic field lines encircle currents/moving charges. Field lines do not end or begin on any “charges”.
for any closed surface.

5. Current elements as magnetic field sources
superposition of contributions of all charge carriers
+ large number of carriers + small current element
field point B ^ ^ r r r
I dl

6. Field of a long straight wire dl y r ^ dB x

7. Example: A long straight wire carrries a current of 100 A
Example: A long straight wire carrries a current of 100 A. At what distance will the magnetic field due to the wire be approximately as strong as the earth’s field (10-4 T)?

8. Force between two long parallel current carrying wires
(consider, for this example, currents in the same direction) I1 I2
F = I2 Bdue to I1 L
= I2 (oI1/(2r))L
F/L = oI1 I2 /(2r)
force on I1 is towards I2
force is attractive (force is repulsive for currents in opposite directions!)
Bdue to I1
Fdue to I1
Example: Two 1m wires seperated by 1cm each carry 10 A in the same direction. What is the force one wire exerts on the other

9. Magnetic Field of a Circular Current Loop
I dl r ^ a x dB
 
dBx

11. Example: A coil consisting of 100 circular loops
Example: A coil consisting of 100 circular loops .2 m in radius carries a current of 5 A. What is the magnetic field strength at the center?
N loops => N x magnetic field of 1 loop.
At what distance will the field strength be half that at the center?

12. Ampere’s Law Equivalent to Biot-Savart Law
Useful in areas of high symetry
Analogous to Gauss’s Law for Electric Fields
Formulated in terms of:
For simplicity, consider single long straight wire (source) and paths for the integral confined to a plane perpendicular to the wire. dl B I

13. B dl  d r dl  B r d

14. Amperian Loop analogous to Gaussian surface
Use paths with B parallel/perpendicular to path
Use paths which reflect symetry

15. Aplication of Ampere’s Law: field of a long straight wire
Cylindrical Symetry, field lines circulate around wire. r I

16. Field inside a long conductorJ

17. B/Bmax
r=R r

18. Homework: Coaxial cable

19. Magnetic Field in a Solenoid
Close packed stacks of coils form cylinder I B I B
Fields tend to cancel in region right between wires.
Field Lines continue down center of cylinder
Field is negligible directly ouside of the cylinder

20. B L I

21. Example: what field is produced in an air core solenoid with 20 turns per cm carrying a current of 5A?

22. Toroidal Solenoid

23. Magnetic Materials L Microscopic current loops: electron “orbits” L 
Microscopic current loops:
electron “orbits”
electron “spin”
Quantum Effects: quantized L, Pauli Exclusion Principle important in macroscopic magnetic behaviour.

24. Magnetic Materials: Microscopic magnetic moments interact with an external (applied) magnetic field Bo and each other, producing additional contributions to the net magnetic field B. Magnetization M = tot/V B= Bo + o M linear approximation: M proportional to Bo o => = Km o = permeability m = Km-1 magnetic Susceptibility Types of Materials Diamagnetic: Magnetic field decreases in strength. Paramagnetic: Magnetic field increases in strength. Ferromagnetic: Magnetic field increases in strength! Diamagnetic and Paramagnetic are often approximately linear with  m

25. Greatly increases field Highly nonlinear, with Hysteresis:
Ferromagnetism:
Greatly increases field
Highly nonlinear, with Hysteresis:
Hysteresis= magnetic record
Magnetization forms in Magnetic Domains M Bo
Saturation
Permanent
Magnetization

26. Displacement Current
“Generalizing” displacement current for Ampere’s Law
conduction current creates magnetic field
Amperian loop with surface
Ienc = iC iC
Parallel Plate
Capacitor
Amperian loop with surface
Ienc = 0??? iC
Parallel Plate
Capacitor

27. Define Displacement Current between plates so that iD = iC