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
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
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
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
A magnetic field does not diverge, its’ field line circulate Gauss’s Law for Magnetic Fields
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
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)
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
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
Mathematically w is the angular frequency of the particle f is called the cyclotron frequency R is the radius of the charged particles path
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
Velocity selector • Let E and B be perpendicular as shown below. • We will solve for the velocity of particles are in equilibrium (F=0).
Leaving Electrostatics • Electrostatics meant charges did not move • We will consider “steady” currents • Steady currents are constant currents • Current: a stream of moving charges
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
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.)
Current Density q q q q If the current is uniform and parallel to dA then i=JA or J=i/A A
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?
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
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
Resistance • “resistance” to current flow • How much voltage required to make current flow • Units: ohm =V/A (W) • Symbol
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
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
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
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.
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
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!
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.
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
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=-mB
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
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