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, 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)

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

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.

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

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

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)?

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

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

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?

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

B dl  d r dl  B r d

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

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

Field inside a long conductor J

B/Bmax r=R r

Homework: Coaxial cable

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

B L I

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

Toroidal Solenoid

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.

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

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

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

Define Displacement Current between plates so that iD = iC