Motion of Chaged Paticles in Magnetic Fields Atos and olecules ae paticles that ae the building blocks of ou uniese. How do scientists study the natue of these sall paticles? The ass spectoete shown in Figue 1 is an instuent scientists use to study atos and olecules. Mass spectoetes ae used to define the eleental coposition of a saple o olecule, to deteine asses of paticles, and to eeal the cheical stuctues of olecules. 8.4 Figue 1 A scientist insets a saple into a ass spectoete. How does a ass spectoete wok? Iagine a billiad table with a billiad ball olling acoss the table fo you left to you ight. If you hit the ball with a sideways foce, the ball will oe away fo you. Now suppose a bowling ball olls acoss the table in the sae diection. If you apply the sae sideways foce on the bowling ball, it will also oe away fo you but not as fa. The asses of the billiad ball and bowling ball deteine the distance they will be deflected by the foce. If you know the aount of foce, the speeds of the balls, and the cue of thei paths, you can calculate the ass of each ball. The less deflection thee is, the heaie the ball ust be. In a siila way, a ass spectoete uses a agnetic field to deflect electically chaged paticles. Atos ae coneted into ions and then acceleated into a finely focused bea. Diffeent ions ae then deflected by the agnetic field by diffeent aounts, depending on the ass of the ion and its chage. Lighte ions ae deflected oe than heaie ones. Ions with oe positie chages ae deflected oe than ions with fewe positie chages. Only soe ions ake it all the way though the achine to the ion detecto, whee they ae detected electically. If you ay the agnetic field, diffeent types of ions will each the detecto. Scientists use the ass spectoete to identify unknown copounds, to deteine the stuctue of a copound, and to undestand the isotopic akeup of olecula eleents. The ass spectoete has applications in the edical field, the food industy, genetics, cabon dating, foensics, and space exploation. CAREER LINK NEL 8.4 Motion of Chaged Paticles in Magnetic Fields 397
Chages and Unifo Cicula Motion To undestand how a ass spectoete woks, we fist need to undestand how a diectional foce affects the otion of an object in this case, a chaged paticle. Conside the diection of a agnetic foce F > M and how this foce affects the otion of a chaged paticle. We know F M q sin u. Fo siplicity, we assue the agnetic field, >, is unifo, so the agnitude and diection of > ae the sae eeywhee. Figue 2(a) shows a chaged paticle, 1q, oing at elocity > paallel to the diection of >. In this case, the angle u between > and > is zeo. The facto sin u in F M q sin u is then zeo, so the agnetic foce in this case is also zeo. If a chaged paticle has a elocity paallel to >, the agnetic foce on the paticle is zeo. Figue 2(b) shows a chaged paticle oing pependicula to >. Now we hae u 908, and sin u 1. The agnitude of the agnetic foce is thus F M q, and the foce is pependicula to the elocity. F M q q (a) (b) Figue 2 (a) When the elocity of a chaged paticle is paallel to the agnetic field >, the agnetic foce on the paticle is zeo. (b) When > akes a ight angle with > (u 908), the chaged paticle oes in a cicle that lies in a plane pependicula to >. Recall that when a paticle expeiences a foce of constant agnitude pependicula to its elocity, the esult is cicula otion, as shown in Figue 2(b). Hence, if a chaged paticle is oing pependicula to a unifo agnetic field, the paticle will oe in a cicle. This cicle lies in the plane pependicula to the field lines. The adius of the cicle can be deteined fo Newton s second law and centipetal acceleation. Recall that fo a paticle to oe in a cicle of adius, thee ust be a foce of agnitude 2 diected towad the cente of the cicle. Hee, the foce poducing cicula otion is the agnetic foce, so we hae F M 2 The agnetic foce is pependicula to the elocity and sin 908 1, so we can inset F M q: q 2 Soling fo gies q Now calculate the alue of fo an electon that has a speed of. 3 10 6 /s oing in a agnetic field of stength.0 3 10 4 T. Inseting these alues into the equation q and using 1 T 1 kg, we get C# s q 19.113 10 231 kg2 a. 3 10 6 s b kg 11.60 3 10 219 24 C2 a.0 3 10 C # b s 6.3 3 10 22 398 Chapte 8 Magnetic Fields NEL
This calculation shows that we can deteine the adius of a paticle s deflection if we know the ass of the paticle, its elocity, its chage, and the stength of the agnetic field though which it oes. Tutoial 1 Soling Pobles Related to Chaged Paticles in Cicula Motion in Magnetic Fields Saple Poble 1: An Electon in a Magnetic Field An electon stats fo est. A hoizontally diected electic field acceleates the electon though a potential diffeence of 37 V. The electon then leaes the electic field and oes into a agnetic field. The agnetic field stength is 0.26 T, diected into the page (Figue 3), and the ass of the electon is 9.11 3 10 231 kg. Figue 3 y (a) Deteine the speed of the electon at the oent it entes the agnetic field. (b) Deteine the agnitude and diection of the agnetic foce on the electon. (c) Deteine the adius of the electon s cicula path. Solution (a) Gien: DV 37 V; e 9.11 3 10 231 kg; q 1.60 3 10 219 C Requied: i Analysis: The decease in the electon s electic potential enegy equals the incease in its kinetic enegy, -DE E DE k, whee 2DE E qdv and E k 1 2 2. The speed of the electon befoe it entes the electic J field is zeo. Theefoe, DE k E kf. Use 1 V 1 C kg# and 2 1 J 1. s 2 Solution: 2DE E DE k qdv 1 2 i 2 i Å 2qDV Å 211.60 3 10 219 C2 137 V2 9.11 3 10 231 kg kg # 2 211.60 3 10 219 C2 a37 C # b s 2 ã 9.11 3 10 231 kg i 3.60 3 10 6 /s 1two exta digits caied2 Stateent: The initial speed of the electon at the oent it entes the agnetic field is 3.6 3 10 6 /s. (b) Gien: 0.26 T; q 1.60 3 10-19 C; u 908; i 3.60 3 10 6 /s Requied: F M and its diection Analysis: Use F M q sin u to deteine the agnitude of the foce. Then use the ight-hand ule to deteine the diection. The agnetic foce will be the opposite of this diection because the chage is negatie. Solution: F M q sin u 11.60 3 10 219 C2 a3.60 3 10 kg 6 b a0.26 s C # b 1 sin 9082 s F M 1. 3 10 213 N Apply the ight-hand ule fo an electic chage oing though a agnetic field: point the finges of you ight hand in the diection of the extenal agnetic field, into the page. Point you ight thub in the diection that the chage is oing, to the ight. You pal points in the diection of the agnetic foce fo a positie chage, up the page. The chage is negatie, so the agnetic foce is down the page. Stateent: The agnitude of the agnetic foce on the electon is 1. 3 10-13 N down the page. (c) Gien: e 9.11 3 10 231 kg; 0.26 T; q 1.60 3 10 219 C; 3.60 3 10 6 /s Requied: Analysis: The only foce acting on the electon is the agnetic foce. This foce is pependicula to the electon s elocity, causing it to oe in unifo cicula otion. The agnetic foce is the centipetal foce, F c 2. Solution: F M F c q 2 1because sin 908 12 q 19.11 3 10 231 kg2 a3.60 3 10 6 s b 11.60 3 10 219 C2 a0.26 kg C # b s 7.9 3 10 2 Stateent: The adius of the electon s cicula path is 7.9 3 10 -. NEL 8.4 Motion of Chaged Paticles in Magnetic Fields 399
Saple Poble 2: The Mass Spectoete: Identifying Paticles A eseache using a ass spectoete obsees a paticle taelling at 1.6 3 10 6 /s in a cicula path of adius 8.2 c. The spectoete s agnetic field is pependicula to the paticle s path and has a agnitude of 0.41 T. (a) Calculate the ass-to-chage atio of the paticle. (In 1910, Robet Millikan accuately deteined the chage caied by an electon. His finding allowed eseaches to calculate the ass of chaged paticles using the ass-to-chage atio.) (b) Identify the paticle using Table 1. Table 1 Isotope (kg) q (C) Solution (a) Gien: 1.6 3 10 6 /s; 8.2 c 0.082 ; u 908; 0.41 T q (kg/c) hydogen 1.67 3 10 227 1.60 3 10 219 1.04 3 10 28 deuteiu 3.3 3 10 227 1.60 3 10 219 2.09 3 10 28 titiu.01 3 10 227 1.60 3 10 219 3.13 3 10 28 Requied: q Analysis: The only foce acting on the electon is the agnetic foce, F M q sin u. This foce is pependicula to the electon s elocity, causing it to oe in unifo cicula otion. The agnetic foce is the centipetal foce, F c 2. Solution: F M F c q 2 q 1because sin 908 12 10.082 2 a0.41 kg C# s b 1.6 3 10 6 s q 2.1 3 1028 kg/c Stateent: The ass-to-chage atio of the paticle is 2.1 3 10 28 kg/c. (b) Accoding to Table 1, the paticle is the isotope deuteiu. Saple Poble 3: The Mass Spectoete: Sepaating Isotopes A eseache uses a ass spectoete in a cabon dating expeient (Figue 4). The incoing ions ae a ixtue of 12 C 1 and 14 C 1, and they hae speed 1.0 3 10 /s. The stength of the agnetic field is 0.10 T. The ass of the electon is 9.11 3 10 231 kg. The ass of the poton and the ass of the neuton ae both 1.67 3 10 227 kg. The eseache fist positions the ion detecto to deteine the alue of fo 12 C 1 and then oes it to deteine the alue of fo 14 C 1. How fa ust the detecto oe between detecting 12 C 1 and 14 C 1? incoing ions q ion detecto Figue 4 Gien: q 1.60 3 10 219 C; e 9.11 3 10 231 kg; p n 1.67 3 10 227 kg; 1.0 3 10 /s; 0.10 T Requied: Dd Analysis: Use the ass of the poton and the ass of the neuton to deteine the ass of each isotope. Then use the equation fo the adius of cuatue fo a paticle deflected in a agnetic field,. The detecto will hae to oe a q distance equal to twice the diffeence between the two adii. Solution: Deteine the ass of each isotope. C12 6 p 1 6 n 1 e 611.67310 227 kg21611.67310 227 kg2119.11310 231 kg2 C12 2.004 3 10 226 kg 1two exta digits caied2 C14 6 p 1 8 n 1 e 611.67310 227 kg21811.67310 227 kg2119.11310 231 kg2 C14 2.338 3 10 226 kg 1two exta digits caied2 Calculate the adius of cuatue of each paticle. C12 C12 q 12.004 3 10 226 kg2 a1.0 3 10 s b 11.60 3 10 219 C2 a0.10 kg C # b s C12 0.122 1two exta digits caied2 400 Chapte 8 Magnetic Fields NEL
C14 C14 q 12.338 3 10 226 kg2 a1.0 3 10 s b 11.60 3 10 219 C2 a0.10 kg C # s b C14 0.1461 1two exta digits caied2 Calculate how fa the detecto ust oe. Dd 21 C14 2 C12 2 210.1461 2 0.122 2 Dd 0.04 Stateent: The ion detecto ust oe a distance equal to the diffeence in the diaetes of the cicula tajectoies, so it ust oe a distance of 0.04. Pactice 1. A heliu 21 ion with chage 3.2 3 10-19 C and ass 6.7 3 10-27 kg entes a unifo 2.4 T agnetic fi eld at a elocity of 1. 3 10 7 /s, at ight angles to the fi eld. Calculate the adius of the ion s path. T/i [ans: 0.13 ] 2. A poton with ass 1.67 3 10 27 kg oes in a plane pependicula to a unifo 1. T agnetic fi eld in a cicle of adius 8.0 c. Calculate the poton s speed. T/i [ans: 1.1 3 10 7 /s] 3. Conside a ass spectoete used to sepaate the two isotopes hydogen and deuteiu. The isotope hydogen has a poton, and deuteiu has a poton and a neuton. Assue both ions hae a 11 chage and they ente the agnetic fi eld egion with a speed of 6.0 3 10 /s. Calculate the agnitude of the agnetic fi eld that is equied to gie a detecto placeent diffeence of 1. as easued fo the initial enty point into the spectoete copaed to when the ions leae the spectoete. T/i A [ans: 8.4 T] 4. The ainbidge-type ass spectoete uses a elocity selecto to select only those ions with the pope elocity. The selecto has two chaged paallel plates to ceate an electic fi eld pointing up, and coppe coils to ceate a agnetic fi eld. Positie ions pass though the selecto, with elocity diected to the ight (Figue ). T/i A (a) In which diection should the agnetic fi eld point in ode to balance the electic foce against the agnetic foce? (b) If the electic field has agnitude e, the agnetic field has agnitude, and the ion has chage q, deteine the pope elocity fo the ions to pass though the selecto without defl ection. (c) Pedict the paths of ions that hae too geat, and too sall, a elocity. Justify you answes. e Figue Mini inestigation Siulating a Mass Spectoete Mini inestigation Skills: Pefoing, Obseing, Analyzing, Counicating Equipent and Mateials: eye potection; 0 c wooden o plastic ap; ba agnet; 2 sall steel ball beaings of diffeent asses Wea closed-toed shoes fo this actiity. 1. Set up the ap at appoxiately a 48 angle. 2. Place the ba agnet on the leel suface at the ap s base. One pole of the agnet should be facing the botto of the ap. 3. Put on you eye potection. Roll a steel ball beaing down the ap, but not diectly at the agnet. SKILLS HANDOOK A2.1 4. Obsee the path of the ball.. Daw its tajectoy on a piece of pape. 6. Repeat Steps 3 to using a ball beaing with a diffeent ass. 7. Daw the new tajectoy next to the fi st one and note any diffeences. A. Copae this actiity to the function of a ass spectoete. How is it siila? K/U C. How is this actiity diffeent fo the function of a ass spectoete? K/U C NEL 8.4 Motion of Chaged Paticles in Magnetic Fields 401
Eath s Magnetic Field Chaged paticles taelling paallel to a agnetic field do not expeience a agnetic foce and continue oing along the field diection. Chaged paticles taelling pependicula to a agnetic field expeience a foce that keeps the oing in a cicula path. Chaged paticles with elocity coponents that ae both paallel and pependicula to a agnetic field expeience a cobination of these effects. The esult is a spial path that esebles the shape of a coil of wie. The paticle taels with a looping otion along the diection of the field (Figue 6). z q y x Figue 7 Eath s agnetic field deflects chaged paticles fo outside the atosphee. The paticles tael in spial paths along the field lines towad the agnetic poles. Figue 6 When the elocity of a chaged paticle has non-zeo coponents paallel and pependicula to the agnetic field, the paticle will oe along a spial path. Chaged paticles enteing Eath s agnetic field ae deflected in this way. Since they ae chaged paticles with a coponent of the elocity pependicula to the agnetic field, they will spial along the field lines towad the agnetic poles. This otion esults in a concentation of chaged paticles at Eath s noth and south agnetic poles (Figue 7). Collisions between the chaged paticles and atos in the atosphee elease light that causes the glow of the auoa boealis in the nothen heisphee and the auoa austalis in the southen heisphee (Figue 8). Eath s agnetic field Figue 8 The auoa austalis. The glow of the auoas occus when chaged paticles spial along Eath s agnetic field lines and collide with olecules in the atosphee aboe the pola egions. inne Van Allen belt oute Van Allen belt Figue 9 The Van Allen belts ae egions of chaged paticles and adiation tapped by Eath s agnetic field. At high altitudes in Eath s agnetic field ae zones of highly enegetic chaged paticles called the Van Allen adiation belts (Figue 9). Jaes A. Van Allen, an Aeican physicist, discoeed the tooidal (doughnut-shaped) zones of intense adiation while studying data fo a satellite he built in 198. Van Allen was able to show that chaged paticles fo cosic ays wee tapped in Eath s agnetic field. Most intense oe the equato, the Van Allen belts ae alost absent oe Eath s poles and consist of an inne egion and an oute egion. The oute Van Allen belt contains chaged paticles fo the atosphee and the Sun, ostly ions fo the sola wind. The inne Van Allen belt is a ing of highly enegetic potons. The concentation of chaged paticles and adiation can easily daage electonic equipent, so eseaches poga the paths and tajectoies of satellites and spacecaft to aoid the belts. 402 Chapte 8 Magnetic Fields NEL
Field Theoy We associate the te foce with a physical action of one object on anothe. When we talk about the foce of a bat against a baseball, ou inds use a concept of contact between the objects, which tansits the foce. To deelop a oe accuate concept of foce, we need to talk about it in tes of fields. We know that all objects ae ade of atos inteacting without actually touching each othe. Thee ae spatial gaps between the atos in a bat and a baseball, so the idea that the bat akes contact with the ball is deceptie. In eality, electoagnetic foces affect the inteacting atos in each object. How do we ceate an undestanding of the gaitational, electic, and agnetic foces? We need a scientific odel that descibes diffeent types of foces that exist at diffeent points in space, and field theoy does that. Field theoy is a scientific odel that descibes foces in tes of entities, called fields, that exist at eey point in space. The geneal idea of fields links diffeent kinds of foces once thought of as sepaate. Field theoy states that if an object expeiences a specific type of foce oe a continuous ange of positions in an aea, then a field exists in that aea. Field theoy can be applied in explaining the inute inteactions of subatoic paticles as well as descibing otions of galaxies thoughout the uniese. Studying gaitational, electic, and agnetic foces has eealed diffeences and siilaities between these foces and thei espectie fields. The electic and agnetic fields hae a stonge effect on the otion of subatoic paticles, such as potons and electons, but the gaitational field has a stonge effect on lage objects, such as planets, galaxies, and clustes of galaxies (Figue 10). field theoy a scientific odel that descibes foces in tes of entities that exist at eey point in space Figue 10 Gaity contols the collision of two clustes of galaxies, while electicity and agnetis affect the elease of adiation duing the collision. (Colous hae been added to the iage to enhance the isual epesentation.) The electic and gaitational foces eseble each othe in that the foce on an object depends on the location of the object. The agnetic foce, howee, depends on a chaged object s otion. The diection of electic and gaitational foces points fo the object towad the chage o ass souce. The diection of the agnetic foce depends on the otion of chaged paticles with espect to the agnetic field. Despite these siilaities and diffeences, field theoy states that electic and agnetic fields ae oe closely elated to one anothe than they ae to the gaitational field. In fact, the electic and agnetic fields ae thought to be diffeent aspects of a single field, the electoagnetic field. They ae used in conjunction with one anothe in a ultitude of innoatie technologies anging fo paticle acceleatos to atificial heats. NEL 8.4 Motion of Chaged Paticles in Magnetic Fields 403
8.4 Reiew Suay If a chaged paticle oes in a unifo agnetic field so that its initial elocity is paallel to the field, it will not expeience a agnetic foce. If it oes so that its initial elocity is pependicula to the field, it will oe in a cicula path in a plane pependicula to the agnetic field. If a chaged paticle oes in a unifo agnetic field with a elocity that is neithe paallel no pependicula to the field, it will oe in a spial path along the field lines. The adius of the cicula path a chaged paticle takes in a unifo agnetic field can be deteined fo Newton s second law and centipetal acceleation and is gien by, whee is the ass of the paticle, q q is its chage, is its speed, and is the agnitude of the agnetic field. Chaged paticles enteing Eath s agnetic field ae deflected and spial along the field lines towad the agnetic poles. This otion esults in a concentation of chaged paticles at Eath s noth and south agnetic poles. Field theoy states that if an object expeiences a specific type of foce oe a continuous ange of positions in an aea, then a field exists in that aea. Questions 1. Explain how a ass spectoete woks. Include a sketch as pat of you answe. K/U C 2. Conside a ass spectoete used to sepaate the two isotopes of uaniu, 238 U 31 (3.92 3 10-2 kg) and 23 U 31 (3.903 3 10-2 kg). Assue the ions ente the agnetic field egion of stength 9. T with identical speeds and leae the spectoete with a sepaation of 2.2 (as easued fo the enty point) afte copleting a half-cicle tun. Calculate the initial speed of the ions. T/I 3. An electon oes in a cicula path pependicula to a agnetic field of agnitude 0.424 T. The kinetic enegy of the electon is 2.203 3 10-19 J. Calculate the adius of the electon s path. Refe to Appendix fo the ass of the electon. T/I 4. A paticle caies a chage of 4 3 10-9 C. When it oes with elocity 3 3 10 3 /s [E 48 N], a unifo agnetic field exets a foce diectly upwad. When the paticle oes with a elocity of 2 3 10 4 /s diectly upwad, thee is a foce of 4 3 10 - N [W] exeted on it. What ae the agnitude and diection of the agnetic field? T/I. An electon, afte being acceleated though a potential diffeence of 100.0 V, entes a unifo agnetic field of 0.0400 T pependicula to its diection of otion. Calculate the adius of the path descibed by the electon. T/I 6. A elocity selecto is a deice that can choose the elocity of a chaged paticle oing though a egion in which the electic field is pependicula to the agnetic field, and with both fields pependicula to the initial elocity of the paticle (Figue 11). To ake the chaged paticle tael staight though the paallel plates, the downwad deflection due to the electic field ust equal the upwad deflection due to the agnetic field. Suppose you want to design a elocity selecto that will allow potons to pass though, undeflected, only if they hae a speed of.0 3 10 2 /s. T/I A e i P elocity selecto Figue 11 (a) The agnetic field is 0.00 T. Calculate the electic field you need. (b) What is the adius of the path the poton takes to get to point P? Refe to Appendix fo the ass of the poton. 404 Chapte 8 Magnetic Fields NEL