Chapter 29 Sources of Magnetic Field Sources are

Chapter 29: Sources of Magnetic Field Sources are moving electric charges single charged particle: field point ^ r r B v field lines circulate around “straight line trajectory” p 212 c 29: 1

from the definition of a Coulomb and other standards, o= 4 x 10 -7 N s 2/C 2 = 4 x 10 -7 T m/A with o , the speed of light as a fundamental constant can be determined (in later chapters) from c 2 = 1/( o o) p 212 c 29: 2

Consider: Two protons move in the x direction at a speed v. Calculate all the forces each exerts on the other. magnetic: F 1 on 2 = q 2 v 2 x. Bat 2 due to 1 = q 2 v 2 x (k’ (q 1 v 1 xr^12)/(r 12)2) electric F 1 on 2 = (k q 1 q 2)/(r 12)2 ^ r 12 Simple geometry (all 90 degree angles) q 1 Fmagnetic = k’|q 2 v 2 q 1 v 1|/ (r 12)2 = k’ e 2 v 2/r 2 F 1 on 2 ^ r 12 v 1 r 12 B Felectric = k e 2/r 2 v 2 Fmagnetic /Felectric = k’ v 2 /k = ( o/4 v 2/ (1/4 o = v 2 /c 2 but. . . depends upon frame? p 212 c 29: 3

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. p 212 c 29: 4

Current elements as magnetic field sources superposition of contributions of all charge carriers + large number of carriers + small current element field point ^ r r B I dl p 212 c 29: 5

Field of a long straight wire I dl y r^ d. B x p 212 c 29: 6

Example: A long straight wire carries 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)? p 212 c 29: 7

Force between two long parallel current carrying wires (consider, for this example, currents in the same direction) I 2 I 1 F = I 2 Bdue to I 1 L = I 2 ( o. I 1/(2 r))L F/L = o. I 1 I 2 /(2 r) force on I 1 is towards I 2 force is attractive (force is repulsive for currents in opposite directions!) Fdue to I 1 Bdue to I 1 Example: Two 1 m wires separated by 1 cm each carry 10 A in the same direction. What is the force one wire exerts on the other p 212 c 29: 8

Magnetic Field of a Circular Current Loop r^ I dl a d. B r x d. Bx p 212 c 29: 9

p 212 c 29: 10

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? p 212 c 29: 11

Ampere’s Law • Equivalent to Biot-Savart Law • Useful in areas of high symmetry • 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 p 212 c 29: 12

dl B r d p 212 c 29: 13

• Amperian Loop analogous to Gaussian surface • Use paths with B parallel/perpendicular to path • Use paths which reflect symmetry p 212 c 29: 14

Application of Ampere’s Law: field of a long straight wire Cylindrical Symmetry, field lines circulate around wire. r I p 212 c 29: 15

Field inside a long conductor I r R J p 212 c 29: 16

B/Bmax r=R r p 212 c 29: 17

Homework: Coaxial cable I I p 212 c 29: 18

Magnetic Field in a Solenoid Close packed stacks of coils form cylinder I B Fields tend to cancel in region right between wires. Field Lines continue down center of cylinder Field is negligible directly outside of the cylinder p 212 c 29: 19

B L I p 212 c 29: 20

Example: what field is produced in an air core solenoid with 20 turns per cm carrying a current of 5 A? p 212 c 29: 21

Toroidal Solenoid p 212 c 29: 22

Magnetic Materials Microscopic current loops: L electron “orbits” electron “spin” Quantum Effects: quantized L, Pauli Exclusion Principle important in macroscopic magnetic behavior. p 212 c 29: 23

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 p 212 c 29: 24

Ferromagnetism: Greatly increases field M Permanent Magnetization Highly nonlinear, with Hysteresis: Saturation Bo Hysteresis= magnetic record Magnetization forms in Magnetic Domains p 212 c 29: 25

Displacement Current “Generalizing” displacement current for Ampere’s Law conduction current creates magnetic field Amperian loop with surface Ienc = i. C Parallel Plate Capacitor Amperian loop with surface Ienc = 0? ? ? i. C Parallel Plate Capacitor p 212 c 29: 26

Define Displacement Current between plates so that i. D = i. C p 212 c 29: 27