Solution Derivations for Capa #8

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1 Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A T field then bends the ion into a cicula path of adius What is the ass of the ion? V = Given q = Given = Given A ass spectoete will cause the paticle to follow a cicula patten. Thus, we know that F = a = v2 = qvb whee the last step follows as a esult of the agnetic field. To find the ass of the ion, we can solve fo. v2 = qvb = qvb v 2 = qb v Now the only poble is to find. It is elated to the kinetic enegy gained by acceleating though the electic field. Since it is initially at est, and exits the electic field befoe enteing the ass spectoete, the final kinetic enegy will be the potential enegy gained by taveling though the electic field. That is, 1 2 v2 = qv 2qV v = Plugging v into the above equation and squaing both sides, = qb 2qV 2 = q2 B 2 2 2qV = qb2 2 2V = qb2 2 2V 2) A chaged paticle is oving pependiculaly to a agnetic field B. Fill in the blank indicating the diection fo the quantity issing in the table. (If you thee answes fo top to botto ae +z, -x, -y, then ente: +z-x-y in the copute). Use the diaga on the uppe left fo the diections of the vaious axes. 1

2 Use the ight hand ule. Reebe that if the chage is negative, it will ove in the diection opposite that given by the ight hand ule. 3) The ectangula loop shown below is located in a unifo agnetic field of 0.50 T pointing in the positive x diection. What is the agnitude of the agnetic dipole oent of the loop? DATA: h = 6.5 c, w = 8.0 c, θ = 36.0, I = 5.0 A. h = Given w = Given θ = Given I = Given The agnetic dipole oent is defined by µ = IA = Ihw The diection is given by culing you finges in the diection of I. It is ipotant fo the next poble. 4) Calculate the agnitude of the toque about the z-axis on the loop. Toque is given by τ = µ B = µ B sin φ Iagine looking at the poble fo the top. In this case, B is pointing down (+x diection), and µ points to the uppe ight. The total angle between the two vectos is 180 θ (if µ is pointing oe into the page than out). You can see this because if the wie loop wee on the y-axis, µ would be antipaallel to B. Since the loop is tilted θ away fo the y-axis, so is µ. Thus, φ = 180 θ. 5) A poton, that is acceleated fo est though a potential of 7.0 kv entes the velocity filte consisting of a paallel-plate capacito and a agnetic field, shown below. The E-field between the paallel capacito plates is N/C. What B-field is equied so that the potons ae not deflected? (Ignoe elativistic effects fo high velocities.) V = Given E = Given q, = Given (poton o electon) B =? We can find the foce of the electic field and the foce due to the agnetic field. They ust be equal fo the paticle to tavel unaffected. F = qe = qvb B = E v 2

3 Fo poble 1, we know that v = 2qV. Thus, B = E 2qV 6) A ass spectoete, sketched below, is a device used to sepaate diffeent ions. Such ions with a well-defined velocity v 0 ente though a slit into a egion of unifo agnetic field B, whee they follow a seicicula path until they stike the detecto slit D. The distance between the enty and the detecto slits is d = Chloine ions of ass 35 au (1 au equals kg), caying a chage of +1e, ente the spectoete with initial speed of v 0 = /s. What value of B is equied fo these ions to hit the detecto? = Given q = Given v = Given B =? F = v2 B = v q = qvb Note that the ass will be soe constant ties one au. 7) A point chage oving in a agnetic field of 1.18 T esla expeiences a foce of N. The velocity of the chage is pependicula to the agnetic field. In this poble, we use the points of the copass and into and out of to indicate diections with espect to the page. If the agnetic field points west and the foce points out of the page, then give all possible coect answes fo the chage Q. (Give all coect stateents in alphabetical ode: ACD, BE, etc.) F = Given If the agnetic field points in one diection, the foce pependicula to it, the paticle ust ove in the othe pependicula diection. That is, it cannot ove paallel to one of the othe foces. That being said, use the ight hand ule to find the diection of the foce fo a positive paticle oving in the diection said. If the diection of the foce is opposite to that given, the chage ust be negative. 8) The speed of the chage in the poble above is /s. Calculate the agnitude of the chage. 3

4 v = Given q =? Using the equation fo agnetic foce, F = qvb q = F vb 9) In the poble above, the path of the chage in the agnetic field is a cicle. Assue that the chage is positive and has a ass of kg, the ass of a poton. What is the adius of the cicle? = Given =? Using the equation fo agnetic foce and the chage calculated in the pevious poble F = qvb = v2 = v2 qvb = v qb 10) In the poble above, what is the obital fequency, of the chage in the pevious poble? The book gives the equation (and the deivation) fo fequency on page (eqn 29-5). Using peviously calculated and given values, f = qb 2π 11) Five paticles follow the paths shown in the figue as they pass though a agnetic field B, which is indicated by the dots in the figue (B out of the page). What can you conclude about the chage of each paticle? (If you answes fo chages of paticles 1, 2, 3, 4, and 5, ae, espectively, +, 0, +,, 0, then ente into the copute.) Use the ight hand ule to find the diection of the foce. If the paticle oves in the diection of the foce, it is a positively chaged paticle. If it oves opposite of the foce, it is negatively chaged. If it is unaffected by the foce, then it is a neutal paticle. 12) A 7 kg paticle caying a chage of 65 µc entes a unifo 4 T agnetic field at a speed of 63 /s and with an angle of 63 with espect to the field lines, as shown in the figue. Answe the following questions: (Select T-Tue, F-False, If the fist is T and the est F, ente TFFFF). 4

5 QUESTION: A) The y-coponent of the paticle s velocity is unchanged as it passes though the B-Field. B) The Foce on the paticle is in the -z diection. C) The paticle s speed vaies as it passes though the B-Field. D) The wok done by the field on the paticle is zeo as the foce is noal to the displaceent. E) The paticle follows a helical path. ANSWER: A) False. Only the diection paallel to the agnetic field will be unchanged. Anothe way to see this is that the foce into the page will cause the paticle to ove in a cicula diection aound the x-axis. This ust change the y-velocity (and the z-velocity). B) Tue. Use the ight hand ule. C) False. The agnetic field cannot change the paticles speed, only the diection of the velocity. D) Tue. The foce of the agnetic field is always pependicula to the oveent of the paticle. Thus, the dot poduct involved in wok equates to zeo. E) Tue. As descibed in (A) and page 732 (fig ) of the book, a paticle oving in a agnetic field will tavel a helical path. 5

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