p212c28: 1 Chapter 28: Sources of Magnetic Field Sources are moving electric charges single charged particle: v field point B r r ^ field lines circulate around “straight line trajectory” r ^

p212c28: 2 from the definition of a Coulomb and other standards,  o = 4  x10 -7 N s 2 /C 2 = 4  x10 -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 )

p212c28: 3 Simple geometry (all 90 degree angles) F magnetic = k’|q 2 v 2 q 1 v 1 |/ (r 12 ) 2 = k’ e 2 v 2 /r 2 F electric = k e 2 /r 2 F magnetic /F electric = k’ v 2 /k = (  o /4  v 2 / (1/4  o  = v 2 /c 2 but... depends upon frame? 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 xB at2due 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 B v1v1 q1q1 r 12 ^ ^ v2v2 F 1 on 2 ^

p212c28: 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.

p212c28: 5 Current elements as magnetic field sources I dl ^ field point B r r superposition of contributions of all charge carriers + large number of carriers + small current element ^ r

p212c28: 6 Field of a long straight wire x I dldl r ^  dBdB y

p212c28: 7 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)?

p212c28: 8 Force between two long parallel current carrying wires (consider, for this example, currents in the same direction) I1I1 I2I2 B due to I 1 F due to I 1 F = I 2 B due 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!) Example: Two 1m wires separated by 1cm each carry 10 A in the same direction. What is the force one wire exerts on the other

p212c28: 9 Magnetic Field of a Circular Current Loop IdlIdl r ^ r a x dBdB   dB x

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p212c28: 11 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?

p212c28: 12 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. I B dldl

p212c28: 13 B dldl  dd r B dldl  dd r

p212c28: 14 Amperian Loop analogous to Gaussian surface Use paths with B parallel/perpendicular to path Use paths which reflect symmetry

p212c28: 15 Application of Ampere’s Law: field of a long straight wire Cylindrical Symmetry, field lines circulate around wire. r I

p212c28: 16 Field inside a long conductor I J r R

p212c28: 17 B/B ma x r r=R

p212c28: 18 I I Homework: Coaxial cable

p212c28: 19 Magnetic Field in a Solenoid Close packed stacks of coils form cylinder Fields tend to cancel in region right between wires. Field Lines continue down center of cylinder Field is negligible directly outside of the cylinder I B I B

p212c28: 20 B L I

p212c28: 21 Example: what field is produced in an air core solenoid with 20 turns per cm carrying a current of 5A?

p212c28: 22 Toroidal Solenoid Charge trapped in a toroidal magnetic field

p212c28: 23 Magnetic Materials Microscopic current loops: electron “orbits” electron “spin”  L  Quantum Effects: quantized L, Pauli Exclusion Principle important in macroscopic magnetic behavior.

p212c28: 24 Magnetic Materials: Microscopic magnetic moments interact with an external (applied) magnetic field B o and each other, producing additional contributions to the net magnetic field B. Magnetization M =  tot /V B= B o +  o M linear approximation: M proportional to B o  o =>  = K m  o = permeability  m = K m -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

p212c28: 25 Ferromagnetism: Greatly increases field Highly nonlinear, with Hysteresis: Hysteresis= magnetic record Magnetization forms in Magnetic Domains Saturation Permanent Magnetization M BoBo

p212c28: 26 Displacement Current “Generalizing” displacement current for Ampere’s Law conduction current creates magnetic field iCiC Amperian loop with surface I enc = i C Parallel Plate Capacitor iCiC Parallel Plate Capacitor Amperian loop with surface I enc = 0???

p212c28: 27 Define Displacement Current between plates so that i D = i C