Chapters 27 and 25 (excluding 25.4)

1. Chapters 27 and 25 (excluding 25.4)

2. Magnetism • Magnetism known to the ancients • Most Famous Magnet: Earth • North=South! (today) • Seems to have flipped several times • Based on orientation of magnetic layers in the earth • Is Moving! • From 1580 to 1820, compass changed by 35o • ||Bearth|| = 8 x 1022 J/T S N

3. Geomagnetism: It’s a life saver! • Sun and other galactic radiation sources emit charged particles • Magnetic fields divert charged particles • Astronauts can get large radiation doses • Geomagnetic anomaly off of Tierra del Fuego

4. Origin of Geomagnetism • Uranium and other radioactive materials provide heat through alpha decay • This heat keeps the earth’s core (mostly iron) hot • The molten iron circulates

5. Broken Symmetry There are no magnetic monopoles i.e the simplest magnetic system is a north pole-south pole system Simplest Magnetic System Simplest Electric System

6. A magnetic field does not diverge, its’ field line circulate Gauss’s Law for Magnetic Fields

7. Magnetic Fields exerts a force on charged particles • Force is proportional to the charge,q, the velocity of the charge,v, and the strength of the magnetic field,B • Since v, B, F are vectors • We need a way to multiply a vector by a vector and get a vector: cross-product • F=qv x B • ||F||=qvB sin f where f is the angle between v and B

8. Direction of Force

9. Units • Units of B = newtons/(coulomb* meter/second) • Called Tesla (T) • Coulomb/second called Ampere (A) • T=N/(A*m) • cgs units are gauss (G) • where 1 T = 104 G • Earth’s magnetic field at any point is about 1 G • Largest magnetic field is 45 T (explosion-induce about 120 T)

10. Magnetic Flux • Magnet flux through a closed surface=0 • This is the field lines through a surface • Units=weber (Wb) and 1 Wb=1 T*m

11. Motion of Charged Particles in a Magnetic Field • Since F is perpendicular to v, there is no acceleration but it does change the direction • A particle moving initially perpendicular to B remains perpendicular to B • Particle’s path is a circle traced out with a constant speed, v

12. Mathematically w is the angular frequency of the particle f is called the cyclotron frequency R is the radius of the charged particles path

13. Combined Force: Lorentz Force • If there is a static electric field, E, and a static magnetic field, B, a force is exerted on the particle equivalent to

14. Velocity selector • Let E and B be perpendicular as shown below. • We will solve for the velocity of particles are in equilibrium (F=0).

15. Leaving Electrostatics • Electrostatics meant charges did not move • We will consider “steady” currents • Steady currents are constant currents • Current: a stream of moving charges

16. Units • Ampere (A) = Coulomb/second (C/s) • 1 A in two parallel straight conductors placed one meter apart produce a force of 2x10-7 N/m on each conductor

17. Can’t we all get along? (Blame Benjamin Franklin) • For physicists: • The current arrow is drawn in the direction in which the positive charge carriers would move • Positive carriers move from positive to negative • For engineers: • The current arrow is drawn in the direction in which the negative charge carriers would move • Negative carriers move from negative to positive • A negative of a negative is a positive so at the end of the day, we should all agree. • (Technically speaking, the engineers have it right.)

18. Current Density q q q q If the current is uniform and parallel to dA then i=JA or J=i/A A

19. At the speed of what? • When a conductor has no current, the electrons drift randomly with no net velocity • When a conductor has a current, the electrons still drift randomly but they tend to drift with a velocity, vd in a direction opposite of the electric field • Drift speed is TINY (about 10-5 to 10-4 m/s) compared to the random velocity of 106 m/s So if the electrons only move at 0.1mm/s then why do the lights come on so fast?

20. Charge carrier density • Let n=number of charge carriers/volume • If wire has cross-sectional area, A, and length, L, then volume = AL • Total number of charges, q=n(AL)e • Let t be the time that the charges traverse the wire with drift velocity, vd, this must be t=L/vd Charge carrier current density

21. Resistivity and Ohm’s Law • Each material has a property called resistivity, r, which is defined as • r=E/J where E is the electric field and J is the charge density (actual definition of Ohm’s law) • Units: (V/m)/(A/m2)=W*m • The reciprocal of resistivity is conductivity, s. • J=sE • Materials are “ohmic” when r is constant • If materials do not depend on this simple relation, then the material is non-ohmic

22. Resistance • “resistance” to current flow • How much voltage required to make current flow • Units: ohm =V/A (W) • Symbol

23. Relationship between Resistance and Resistivity

24. Ohm’s Law • A current through a device is always proportional to the potential difference applied i i V V Both obey V=iR but the resistor obeys Ohm’s law while the diode does not diode resistor

25. Power in resistors

26. Conduction Band Conduction Band Conduction Band Band Valence Band Valence Band Valence Band Theory of Solids • Electrons are restricted to certain energy levels: they are “quantized” • “quantized” think “pixilated” • Electrons can occupy any level but cannot have an energy between levels • Proximity of the atoms squeezes these levels into a few bands Band represents many energy levels in close proximity Band Gap Insulator Conductor Semiconductor

27. Force Law from current perspective • q=i*t • For a length of wire, L, with drift velocity vd, then t=L/vd so q=i*L/vd • F=qv x B or F=qvB sin q • In the case of the wire, v=vd so • F=(i*L/vd)*vdBsin q • F=iL x B • Where ||L|| is the length of the wire and the direction of L points in the direction of current flow • For each infinitesimal piece of wire dL, has a force, dF exerted on it by B : dF=I dL x B

28. Force and Torque on a Current Loop • While this seems an academic exercise, its importance cannot be overstated. • This is the basis of both: • Electrical motor • Power generation • Thus, its results impact us immensely • We would die without it.

29. Diagram B

30. Forces • F=iL x B • For sides length a • Always perpendicular to B (out of page) • F=iab • Because of this: a a the forces have opposite directions on opposite sides F+ F- • For sides length b • Their angle w.r.t. to B changes as the loop moves • F=ibBsin(900-f)=ibBcosf B b f

31. Directions • For length a, the forces are in the x-direction (+x-hat and –x-hat) • For length b, the forces are in the y-direction • So the net FORCE is zero But not the net TORQUE!

32. Torques • Recall t=r x F • For length b sides, their line of common action is through the center and thus, their net torque is zero.

33. Sides of length, a, have a net torque • As shown in the figure on the right, the vector torques for both sides of length a are in the +y-direction • The torque is rFsinf • Where ||r||=b/2 • F=iaB • t=2(iaB)(b/2)sinf • Area=a*b=A • t=iABsinf f r F f

34. Magnetic Moment, m • The product of iA is called the magnetic moment and is a vector quantity • m =i A n • Where n is normal to the area of the current loop • Since t = m x B, this behavior is similar to that of an electric dipole (t = p x E) • Thus, m is sometimes called a magnetic dipole • You might expect that the potential energy would have the form of U=-mB

35. Magnets on an atomic level • Think of an electron as a charge orbiting the nucleus • This is a charge moving through space at a constant angular velocity so essentially i=q*v where v=rw .and r=electron orbital radius. • So this is a small current loop with area=p*r2 • Thus atoms can experience torques and forces when subjected to magnetic fields

36. Hall Effect • Assume a current i is flowing in the positive x direction along a copper strip (as shown on the right) • A static magnetic field is directed into the page • B forces the negative charge carriers to the right • Eventually, the right side is filled with negative charges and the left side is depleted which sets up a potential difference • An electric field is produced • The electric field is proportional to the magnetic field which produces it and the current • In the next chapter, we will learn how the Hall effect is used to measure currents. i