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Chapter 29: Sources of Magnetic Field

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”. from the definition of a Coulomb and other standards,

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Chapter 29: Sources of Magnetic Field

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  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. ^ r12 • 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 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. ^ r Field of a long straight wire I  dl y dB x

  7. 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) I2 I1 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. ^ I dl r dB r a   x dBx Magnetic Field of a Circular Current Loop

  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?

  11. 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 I B

  12. dl r  B d dl  B r d

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

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

  15. Field inside a long conductor I R r J

  16. B/Bmax r=R r

  17. Homework: Coaxial cable I I

  18. I B 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 ouside of the cylinder

  19. B I L

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

  21. Toroidal Solenoid

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

  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/VB= Bo + o Mlinear approximation: M proportional to Boo => = Km o = permeabilitym = Km-1 magnetic SusceptibilityTypes 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

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

  25. 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

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

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