AP Physics Electromagnetic Wrap Up


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1 AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle is the angle between the field and the elocity, q epesents the chage of the paticle. If the elocity is F Ilsin pependicula to the field, then you hae a maximum foce and it is simply equal to F q. This is the equation fo the magnitude of the foce exeted on a staight length of wie by a magnetic field. Whee F is the magnetic foce, I is the cuent, l is the length of the wie, and is the angle between the wie and the magnetic field. 0I This is the magnetic field stength aound a staight section of a cuent caying conducto. is the magnetic field stength, 0 is the pemeability of fee space, I is the cuent, and is the distance to the cente of the conducto. The alue fo the pemeability of fee space is: x10 T m A m A Acos This is the equation fo magnetic flux. m is the magnetic flux, is the magnetic field stength, A is the aea of the loop, and is the angle between and a nomal to the plane of the loop. ag m t This is the equation fo induced emf. ag is the induced emf, m is the change in magnetic flux, and t is the change in time. l This is the equation fo the induced emf in a loop of width l that is moing into o out of a magnetic field at a elocity. 430
2 Hee s what you gots to be able to do: Magnetostatics 1. Foces on Moing Chages in Magnetic Field a. You should undestand the foce expeienced by a chaged paticle in a magnetic field so you can: (1) Calculate the magnitude and diection of the foce in tems of q,, and, and explain why the magnetic foce can pefom no wok. Okay, this is simply using the old F qsin equation. The magnetic foce can pefom no wok because the diection of the magnetic foce is always pependicula to the motion of the paticle, so the wok is always zeo. () Deduce the diection of a magnetic field fom infomation about the foces expeienced by chaged paticles moing though that field. This is simply the old ight hand ule. (3) State and apply the fomula fo the adius of the cicula path of a chage that moes pependicula to a unifom magnetic field, and deie this fomula fom Newton s Second Law and the magnetic foce law. To do this, you simply deie the fomula fo the centipetal foce and set it equal to the magnetic foce. Sole fo the adius. The handout is quite happy to show you how to do this. (4) Descibe the most geneal path possible fo a chaged paticle moing in a unifom magnetic field, and descibe the motion of a paticle that entes a unifom magnetic field moing with specified initial elocity. The most geneal path is a helix spial kind of deal. If the paticle s elocity is pependicula to the field, then the paticle will follow a cicula path. (5) Descibe quantitatiely unde what conditions paticles will moe with constant elocity though cossed electic and magnetic fields. The deal hee is that the two foces, the electic and magnetic foces, must somehow add up to zeo if the paticle is to hae a constant elocity. So the foces hae to be the same magnitude and be in opposite diections. Since the electic foce is in the diection of the electic field and the magnetic foce is pependicula to the magnetic field, then the two fields hae to be pependicula to each othe.. Foces on cuentcaying Wies in Magnetic Fields a. You should undestand the foce expeienced by a cuent in a magnetic field so you can: 431
3 (1) Calculate the magnitude and diection of the foce on a staight segment of cuentcaying wie in a unifomed magnetic field Just use the equation; F Ilsin () Indicate the diection of magnetic foces on a cuentcaying loop of wie in a magnetic field, and detemine how the loop will tend to otate as a consequence of these foces. Use the good old ight hand ule. 3. Fields of Long Cuentcaying Wies a. You should undestand the magnitude field poduced by a long staight cuentcaying wie so you can: (1) Calculate the magnitude and diection of the field at a point in the icinity of such a wie. Use the equation. 0I () Use supeposition to detemine the magnetic field poduced by two long wies. Supeposition means ecto addition. Each of the wies poduces its own field. You just add them up as ectos. (3) Calculate the foce of attaction o epulsion between two long cuent caying wies. I Use the F Ilsin. Electomagnetism 1. Electomagnetic Induction 0 equation and the equation fo the foce exeted by a wie, a. You should undestand the concept of magnetic flux so you can: (1) Calculate the flux of a unifom magnetic field though a loop of abitay oientation. Simply use the equation fo magnetic flux. m A Acos b. You should undestand Faaday s Law and Lenz s Law so you can: 43
4 (1) ecognize situations in which changing flux though a loop will cause an induced emf o cuent in the loop. Faaday s law is simply that a cuent is induced when the magnetic flux though the conducto is changing. Hae to be alet fo the idea that it is only when the flux is changing that an emf will be induced. Typical situations whee this happens would be whee a loop taels into o out of a magnetic field. Please note that once it is in a field, een though it is moing, no emf is induced because the flux is not changing. The othe situation is a loop that is otating within a magnetic field. () Calculate the magnitude and diection of the induced emf and cuent in: (a) A squae loop of wie pulled at a constant elocity into o out of a unifom magnetic field. Use the equation ule. l. To figue out the cuent diection, use the ight hand (b) A loop of wie placed in a spatially unifom magnetic field whose magnitude is changing at a constant ate. Use the equation ag m. t (c) A loop of wie that otates at a constant ate about an axis pependicula to a unifom magnetic field. This is a bit moe complicated. Thee ae some examples of how to do this in the last electomagnetic handout. Check it out. (d) A conducting ba moing pependicula to a unifom magnetic field. Just use the good old l equation. Fom 000: 7. A paticle with unknown mass and chage moes with constant speed V = 1.9 x 10 6 m/s as it passes undeflected though a pai of paallel plates, as shown aboe. The plates ae sepaated by a distance d = 6.0 x 103 m, and a constant potential diffeence V is maintained between them. A unifom magnetic field of magnitude = 0.0 T diected into the page exists both between the plates and in a egion to the ight of them as shown. Afte the paticle passes into the egion to the ight of the plates whee only the A V magnetic field exists, its tajectoy is cicula with adius = 0.10 m. 433
5 (a) What is the sign of the chage of the paticle? Check the appopiate space below. Positie _xx_ Negatie Neutal It cannot be detemined fom this infomation. If the paticle was positie, the magnetic foce would be up (ight hand ule) and the paticle would cue aboe the plates, it s motion is the opposite, so the paticle must hae a negatie chage. etween the plates, the negatie paticle is deflected downwad. Theefoe the electic field must foce the negatie paticle up, but diection of the field is the diection a positie test chage would go so the field must be down. (b) On the diagam aboe, clealy indicate the diection of the electic field between the plates. (c) Detemine the magnitude of the potential diffeence V between the plates. V The magnetic field stength is gien by: E d Magnetic foce: F qsin q F Electic Foce: E F qe q Set the electic foce and the magnetic foce equal to one anothe: V q qe q q V d d 6 m 3 3 V 1.9 x T6.0 x10 m.8 x10 V s (d) Detemine the atio of the chage to the mass ( q/m) of the paticle. This sounds bad. Okay, the paticle follows a cued path in the magnetic field. So we can set the centipetal foce = magnetic foce. m q Soling fo q/m q q m m q m x 95 x10 C 9.5 x10 C m s m T kg kg
6 Fom 1999: A ectangula conducting loop of width w, height h, and esistance is mounted etically on a nonconducting cat as shown aboe. The cat is placed on the inclined potion of a tack and eleased fom est at position P 1 at a height y o aboe the hoizontal potion of the tack. It olls with negligible fiction down the incline and though a unifom magnetic field in the egion aboe the hoizontal potion of the tack. The conducting loop is in the plane of the page, and the magnetic field is diected into the page. The loop passes completely though the field with a negligible change in speed. Expess you answes in tems of the gien quantities and fundamental constants. (a) Detemine the speed of the cat when it eaches the hoizontal potion of the tack. 1 KE ME m mgh gh gyo (b) Detemine the following fo the time at which the cat is at position P with onethid of the loop in the magnetic field. i. The magnitude of the emf induced in the conducting loop A 1 wh wh w x x 3 w t t gyo 3 gy wh 3 h gy t w 3 gy o o o ii. The magnitude of the cuent induced in the conducting loop 435
7 V I I V h gyo (c) On the following diagam of the conducting loop, indicate the diection of the cuent when it is at position P. (d) i. Using the axes below, sketch a gaph of the magnitude of the magnetic flux though the loop as a function of the hoizontal distance x taeled by the cat, letting x = 0 be the position at which the font edge of the loop just entes the field. Label appopiate alues on the etical axis. wh ii. Using the axes below, sketch a gaph of the cuent induced in the loop as a function of the hoizontal distance x taeled by the cat, letting x = 0 be the position at which the font edge of the loop just entes the field. Let counteclockwise cuent be positie and label appopiate alues on the etical axis. h gy o h gyo 436
8 Fom 1996: A igid od of mass m and length is suspended fom two identical spings of negligible mass as shown in the diagam below. The uppe ends of the spings ae fixed in place and the spings stetch a distance d unde the weight of the suspended od. (a) Detemine the sping constant k of each sping in tems of the othe gien quantities and fundamental constants. F F mg 0 F mg s s but Fs kx so kx mg ut each sping holds ½ the total mass 1 mg mg kx mg k x d As shown below, the uppe end of the spings ae connected by a cicuit banch containing a battey of emf and a switch S, so that a complete cicuit is fomed with the metal od and spings. The cicuit has a total esistance, epesented by the esisto in the diagam below. The od is in a unifom magnetic field, diected pependicula to the page. The uppe ends of the spings emain fixed in place and the switch S is closed. When the system comes to equilibium, the od is loweed an additional distance d. (b) What is the diection of the magnetic field elatie to the coodinate axes shown on the ight in the peious diagam? Using the ight hand ule, the field is out of the page, along the positie z axis. The cuent is to the ight thumb points that way and the ba is pushed down, so the palm points down. The finges end up pointing out of the page. (c) Detemine the magnitude of the magnetic field in tems of m,, d, d,,, and fundamental constants. Gaity initially displaced the ba a distance d. Then the magnetic field displaced the ba an exta distance, d. Set the sping foce equal to the magnetic foce: 437
9 F S F kx I ut, thee ae two spings kx I x is d, so plug that in: kd I Sole fo : Now we need to get in tems of the items listed in (c): kd I If the intenal esistance is not mentioned, we can assume that it is zeo. So V V V I I so I Plug this into the equation we deeloped fo I: kd kd kd I Also, we aleady found a alue fo k mg d Plug this into the equation fo k: mg d kd d mgd d (d) When the switch is suddenly opened, the od oscillates. Fo these oscillations, detemine the following quantities in tems of d, d, and fundamental constants: i. The peiod. T m ut thee ae two spings. k T m Plug in the alue fo k. k T m mg d d g 438
10 ii. The maximum speed of the od. Use conseation of enegy: 1 1 US K kx m The in font of U S because thee ae two spings kd k kx kx m d and m m m k mg d mg d mg g d d d m m d d Fom 1995: In a linea acceleato, potons ae acceleated fom est though a potential diffeence to a speed of appoximately 3.1 x 10 6 metes pe second. The esulting poton beam poduces a cuent of.0 x 106 ampee. a. Detemine the potential diffeence though which the potons wee acceleated. The electic potential enegy is equal to the wok done on the paticle. This must equal its kinetic enegy. 1 qq 1 m U qv W qv qv m V 4 q 1 E 0 Note, we can look up the mass of a poton (it is 1.67 x 107 kg) kg m s 19 C m 4 V V q
11 b. If the beam is stopped in a taget, detemine the amount of themal enegy that is poduced in the taget in one minute. ecause enegy is conseed, the themal enegy poduced in the taget is simply equal to the powe of the beam of paticles. W P and P IV W Pt W IVt t 6 4 W IVt.0 x10 A 5.0 x10 V 60 s 600 x10 A 6.0 J E W 6.0 J Themal The poton beam entes a egion of unifom magnetic field, as shown aboe, that causes the beam to follow a semicicula path. c. Detemine the magnitude of the field that is equied to cause an ac of adius 0.10 mete. ac F FNet ma so FC m o F FC F qsin so qsin m sin x10 kg3.1 x10 m s m q m 0.3 T q x C m d. What is the diection of the magnetic field elatie to the axes shown aboe on the ight? Positie Z  out of the pape.. Use the ight hand ule. Thumb points in diection of elocity, palm points in diection of magnetic foce, and the finges point in the diection of the field. At the dot on the cued path of the poton, the palm points to the left towads the cente of the cicle. The thumb points down (the elocity at that point is a tangent to the cicle and is down). You finges end up pointing out of the page. Fom 1994: 440
12 A foce F is applied to a conducting od so that the od slides with constant speed oe a fictionless pai of paallel conducting ails that ae sepaated by a distance. The od and ails hae negligible esistance, but the ails ae connected by a esistance, as shown below. Thee is a unifom magnetic field pependicula to and diected out of the plane of the pape. a. On the following diagam, indicate the diection of the induced cuent in the esisto. Using the ight hand ule, the palm must point to the left (the diection of the foce which opposes the motion of the ba), the finges point in the diection of the field out of the page. This thumb then points in the diection of the cuent though the ba, which is down. Theefoe the cuent is clockwise and the cuent though the esisto is up. Detemine expessions fo the following in tems of,,, and. b. The induced emf in the od. l c. The electic field in the od. F q E F qsin F q so E E q q d. The magnitude of the induced cuent in the esisto. We know that the emf induced is: l We also know that, if the esistance in the od and ails is negligible, then the emf is equal to the oltage. So: V Now we can apply Ohm s law. 441
13 V V I I I e. The powe dissipated in the esisto as the od moes in the magnetic field. P IV P P f. The magnitude of the extenal foce F applied to the od to keep it moing with constant speed. At constant speed the net foce must equal zeo. F 0 F F 0 F F F I F F Fom 1998: The long, staight wie shown in Figue 1 below is in the plane of the page and caies a cuent I. Point P is also in the plane of the page and is a pependicula distance d fom the wie. Gaitational effects ae negligible. a. With efeence to the coodinate system in Figue 1, what is the diection of the magnetic field at point P due to the cuent in the wie? Use the ight hand ule. Into the page, in the minus z diection. A paticle of mass m and positie chage q is initially moing paallel to the wie with a speed o when it is at point P, as shown in Figue below. 44
14 b. With efeence to the coodinate system in Figue, what is the diection of the magnetic foce acting on the paticle at point P? ight hand ule. Foce is down, in the minus y diection. Thumb points in diection of 0, finges point into page (diection of field), so the palm points down, the diection of the foce. c. Detemine the magnitude of the magnetic foce acting on the paticle at point P in tems of the gien quantities and fundamental constants. o I o I sin o I o o I F q q q d d d d. An electic field is applied that causes the net foce on the paticle to be zeo at point P. i. With efeence to the coodinate system in Figue, what is the diection of the electic field at point P that could accomplish this? The electic field diection is defined as the diection a positie paticle moes. The positie paticle must moe up to counte the downwad magnetic field. ii. Detemine the magnitude of the electic field in tems of the gien quantities and fundamental constants. F E F E F E q E F q oi qo d E q o oi d 443
15 Fom 199: The figue shows a coss section of a cathode ay tube. An electon in the tube initially moes hoizontally in the plane of the coss section at a speed of.0 x 10 7 metes pe second. The electon is deflected upwad by a magnetic field that has a field stength of 6.0 x 104 Tesla. a. What is the diection of the magnetic field? Out of the page. b. Detemine the magnitude of the magnetic foce acting on the electon. F q m F 1.6x10 C x10 6 x10 T 1.9 x10 N s c. Detemine the adius of cuatue of the path followed by the electon while it is in the magnetic field. ac F ma Fc m 7 m x10 31 s m 9.11 x10 kg Fc 15 kg m 1.9 x10 s 0.19 m An electic field is late established in the same egion as the magnetic field such that the electon now passes though the magnetic and electic fields without deflection. d. Detemine the magnitude of the electic field. 15 F 1.9 x10 N 444
16 F E Q x10 N N E x10 C C e. What is the diection of the electic field? The foce needs to be downwad on the electon to cancel the upwad magnetic field foce. Unlike the magnetic field which is pependicula, the electic field is paallel to its foce. The electic field is the diection a positie chage moes. Electons moe opposite the field. So if the electon needs to moe down the electic field is up. Fom 1993: A paticle of mass m and chage q is acceleated fom est in the plane of the page though a potential diffeence V between two paallel plates as shown. The paticle is injected though a hole in the ighthand plate into a egion of space containing a unifom magnetic field of magnitude oiented pependicula to the plane of the page. The paticle cues in a semicicula path and stikes a detecto. Neglect elatiistic effects thoughout this poblem. a. Detemine the following i. State whethe the sign of the chage on the paticle is positie o negatie. Positie. The ight plate is negatie and the left is positie. The paticle is attacted to the negatie plate and epelled by the positie plate so it has to be positie. ii. State whethe the diection of the magnetic field is into the page o out of the page. Out of the page. Look at the paticle at the mid point of the cue (whee the little aowhead is located). Its elocity is down, so point you thumb down. It is being foced towads the cente by the magnetic foce, so point you palm to the left. This leaes you finges pointing out of the page, which is the diection of the magnetic field. b. Detemine each of the following in tems of m, q, V, and. i. The speed of the chaged paticle as it entes the egion of the magnetic field. 1 W qv Wok Incease in KE k m 1 qv qv m m ii. The foce exeted on the chaged paticle by the magnetic field. 445
17 F qsin sin F q iii. The distance fom the point of injection to the detecto. F Fc q m o qv m m m m qv d d q q q m i. The wok done by the magnetic field on the chaged paticle duing the semicicula tip. No wok is done since the magnetic field is pependicula to the displacement. 446
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