Phys-7 Lectue 13 Magnetism Magnetic foces
Chaged paticle motion in a constant field - velocity in plane to. Suppose we have a magnetic field given by 0 and a paticle stats out at the oigin moving in the +x diection. The paticle will move in a cicle with the adius R and angula velocity ω: θ y v R F m a ω R qv mv q v R q m zˆ v m R x magnetic foce povides centipedal foce independent of and v.
Motion in a constant magnetic field If the velocity of the chage paticle is pependicula to the field the motion is cicle with adius Rmv/q. If the velocity is not pependicula, the motion is a helix. In this case we beak up the velocity into components pependicula v and paallel to the field v //. F qv 0 Theefoe have unifom motion in diection. Path looks like extended sping, v // constant along the diection and v moves in a cicle. R (of helix) mv /q
LHC@Geneva, Switzeland Cicumfeence 7 km π R mv p R q q
The KEK Collide (Tsukuba, Japan) SCC RF(HER) elle detecto 8 x 3.5 GeV beam enegies Wold ecod: L.1 x 10 34 /cm /sec ARES(LER) Aes RF cavity e + souce 3.1 km aound
Applications Electic/Magnetic Velocity Selecto A chaged paticle entes a egion with pependicula electic and magnetic fields. The electic and magnetic foces will cancel if the velocity is just ight. Paticles with this velocity will go though undeflected. Othes will be deflected. FE qe Fm q E q v 1 v E qv Choosing a paticula E/ atio will select the desied velocity. Example, E10,000V/m and 0.001T selects v10,000,000 m/s
Mass Spectomete (Chemisty) Measue m to identify substances: 1.) Ionize atoms by hitting them with acceleated electons..) Aceleate ions though known potential V. qe if singly ionized. UqV 3.) Velocity select. 3.) Pass ions though known field and measue R. R v 1 m mv mv q q mr ; qv q R V v q m R
Mass Spectomete (Chemisty) Applications: Think CSI Paleoceanogaphy: Detemine elative abundances of isotopes (they decay at diffeent ates geological age) Space exploation: Detemine what s on the moon, Mas, etc. Detect chemical and biol. weapons (neve gas, anthax, etc.).
Yet anothe example Measuing cuvatue of chaged paticle in magnetic field is usual method fo detemining momentum of paticle in moden expeiments: e.g. mv mk R q q - chaged paticle e + + chaged paticle e - End view: into sceen
Thomson s chage/mass atio of electon measuement E and fields act as a velocity selecto 1 E mv v ev ev v Found single value of e/m m ev m e m E V independent of cathode mateial. This led to discovey of electons. e/m 1.7588 x 10 11 C/kg
Cycloton, the fist Atom Smashe Acceleates potons and deuteons to high enegies to bombad atomic nuclei. Allows: study of nuclea physics. poduction of adioactive isotopes cance teatment. Thee ae thousands of cyclotons in use. E. O. Lawence and M. S. Livingston, invented paticle acceleato, called cycloton. Won 1939 Nobel Pize.
Cycloton Enest Lawence, South Dakota, 19 fist cycloton Led to giant acceleatos nea Chicago. And Lage Hadon Collide (LHC) nea Geneva.
Cycloton Electomagnet causes paticles to move in a cicle. Oscillating voltage between Dees acceleates paticles each time paticle cosses gap. Vey impotant to opeation of cycloton fequency of electic field does NOT depend on v. ω q 1 ; f m T q π m Maximum enegy detemined by size of cycloton. Limit imposed by size and cost of magnet. v K max max qr m 1 mv max max q m R max
Magnetic foce on cuent caying wie Suppose we have a staight wie with cuent whee chages q ae moving upwads and thee is a field pointing into this slide. Thee is a F qv foce tending to push the wie to the left. F q v L Recall that cuent is I n q v A (Y&F eqn 5.) with n#chage/vol, vvelocity and Aaea. In a length, L, the #chages n L A, so the total foce magnitude is, F ( nla) qv ( nqva) L IL A If we conside a small line segment, dl, we can wite the vecto foce eqn. as, df Idl Moe geneal.
E+M Rail Gun Suppose we have a movable pojectile between bas supplying cuent flow and a unifom pependicula magnetic field. Assume, 4T, I5000A, pojectile width10cm and mass of ba0.1kg. A) What is the foce on the pojectile? F Il 5000A 0.1m 4T 000N ) How fast will be the be moving afte it moves a distance 1m? a F / m 000 N / 0.1kg 0000 m / s v v + a x 0 v (0000 m / s )(1 m) 00 m / s
Toque on a ectangula cuent loop The loop will feel a toque which tends to otate the loop, back into plane Impotant application: electic moto Conside a wie loop dimensions a x b whose plane is an angle φ elative to a constant field. Thee will be a net toque whose magnitude on this loop is given by, τ Iab sinφ I ( aea) sinφ
I Magnetic Moment, µ, of a ectangula cuent loop Definition ; µ cuent aea I A µ µ is vecto quantity, whose diection is nomal to loop plane, use ight hand ule to define diection. We can define vecto toque moe conveniently in tems of vecto magnetic moment cossed by field τ µ Anothe name fo a cuent loop is magnetic dipole
Deivation of Foces on wies of length b ae collinea and cancel. Foces on wies of length a cancel but ae not collinea. Thee will be a toque. F F I l I l Ia τ µ b τ F ( sinφ)( Ia) τ I( ab)sinφ τ µ sinφ µ τ µ Toque on a ectangula cuent loop ϕ b/ ϕ F
Clicke: F v I L A squae loop of wie is caying cuent in the counteclockwise diection. Thee is a hoizontal unifom magnetic field pointing to the ight. ) What is the foce on section a-b of the loop? a) zeo b) out of the page c) into the page ab: F ab 0 F cd since the wie is paallel to. 3) What is the foce on section b-c of the loop? a) zeo b) out of the page c) into the page bc: F bc IL RHR: I is up, is to the ight, so F points into the sceen. 4) What is the net foce on the loop? a) zeo b) out of the page c) into the page y symmety: F da F bc F F F F F n e t a b + b c + c d + d a 0
Magnetic Moment, µ, of abitay loop Definition ; µ cuent aea I A We can moe geneally define magnetic moments only by aea and do not need to know the actual dimensions. This is geneal µ µ µ I I I Fo any shape, µ cuent aea I A and τ τ AI sinθ µ Note: if loop consists of N tuns, µ NAI
Toque on loop in diffeent angula positions µ τ µ I Net toque is zeo when is paallel to µ I Net toque is maximum when is pependicula to µ
a Magnet Analogy You can think of a magnetic dipole moment as a ba magnet: µ N In a magnetic field they both expeience a toque tying to line them up with the field As you incease I of the loop stonge ba magnet We will see that such a cuent loop does poduce magnetic fields, simila to a ba magnet.
Application; galvanomete uses toques on coiled loops in magnetic field Toque is poduced about the needle axis and this counte acts the estoing sping and enables the needle to otate Cuent inceased µ I Aea inceases Toque fom inceases Angle of needle inceases Cuent deceased µ deceases Toque fom deceases Angle of needle deceases This is how almost all dial metes wok voltmetes, ammetes, speedometes, RPMs, etc.
Example: Loop in a -Field A cicula loop has adius R 5 cm and caies cuent I A in the counteclockwise diection. A magnetic field 0.5 T exists in the negative z-diection. The loop is at an angle θ 30 to the xy-plane. y z x x x x x x x x x x x x x x I x x x x x x x x x x x x x x z y X µ θ x X x What is the magnetic moment µ of the loop? µ π I.0157 Am The diection of µ is pependicula to the plane of the loop as in the figue. Find the magnitude of the toque on the loop : τ µ Β sin (180-30) 1.98x10-3 (Am )(N/Am) 1.98x10-3 Nm
Electic Dipole Analogy +q F τ F τ F F p qe F. -q E x θ θ µ F IL p qa µ NAI F F. (pe tun) τ p E τ µ
Clicke: A squae loop of wie is caying cuent in the counteclockwise diection. Thee is a hoizontal unifom magnetic field pointing to the ight. 6) What is the net toque on the loop? a) zeo b) up c) down d) out of the page e) into the page τ µ µ points out of the page (cul you finges in the diection of the cuent aound the loop, and you thumb gives the diection of µ). Use the RHR to find the diection of τ to be up.
Potential Enegy of Dipole Wok must be done to change the oientation of a dipole (cuent loop) in the pesence of a magnetic field. Define a potential enegy U (with zeo at position of max toque) coesponding to this wok. F x θ µ F. U θ 90 τdθ U θ 90 µ sin θdθ Theefoe, U µ[ cosθ] θ 90 U µcosθ U µ
Potential Enegy of Dipole µ x x µ x µ τ 0 U -µ τ µ X U 0 τ 0 U µ negative wok positive wok
Clicke: Two cuent caying loops ae oiented in a unifom magnetic field. The loops ae nealy identical, except the diection of cuent is evesed. 8) What is the toque on loop 1? a) clockwise b) counte-clockwise c) zeo Loop 1: µ points to the left, so the angle between µ and is equal to 180º, hence τ 0. 9) How does the toque on the two loops compae? a) τ 1 > τ b) τ 1 τ c) τ 1 < τ Loop : µ points to the ight, so the angle between µ and is equal to 0º, hence τ 0. 10) Which loop occupies a potential enegy minimum, and is theefoe stable? a) loop 1 b) loop c) the same U µ Loop 1: U 1 +µ Loop : U µ U is a minimum.
The Hall Effect Is cuent due to motion of positive o negative chages? Positive chages moving CCW expeience upwad foce Uppe plate at highe potential Negative chages moving clockwise expeience upwad foce Uppe plate at lowe potential Equilibium between electostatic & magnetic foces: VH Fup qvdift Fdown qeinduced q VH vdift w "Hall Voltage" w This type of expeiment led to the discovey (E. Hall, 1879) that cuent in conductos is caied by negative chages (not always so in semiconductos). Can be used as a -senso.