27_InstructorSolutions - MAGNETIC FIELD AND MAGNETIC FORCES...

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27-1 M AGNETIC F IELD AND M AGNETIC F ORCES 27.1. I DENTIFY and S ET U P : Apply Eq.(27.2) to calculate . F ! Use the cross products of unit vectors from Section 1.10. E XECUTE : ( ) ( ) 4 4 ° ° 4.19 10 m/s 3.85 10 m/s = + × + − × v i j ! (a) ( ) ° 1.40 T = B i ! ( ) ( ) ( ) ( ) 8 4 4 ° ° ° ° 1.24 10 C 1.40 T 4.19 10 m/s 3.85 10 m/s q = × = − × × × × × F v B i i j i ! ! ! ° ° ° ° ° 0, × = × = − i i j i k ( ) ( ) ( ) ( ) ( ) 8 4 4 ° ° 1.24 10 C 1.40 T 3.85 10 m/s 6.68 10 N = − × × = − × F k k ! E VALUATE : The directions of and v B ! ! are shown in Figure 27.1a. The right-hand rule gives that × v B ! ! is directed out of the paper (+ z -direction). The charge is negative so F ! is opposite to ; × v B ! ! Figure 27.1a F ! is in the - z direction. This agrees with the direction calculated with unit vectors. (b) E XECUTE : ( ) ° 1.40 T = B k ! ( ) ( ) ( ) ( ) 8 4 4 ° ° ° ° 1.24 10 C 1.40 T 4.19 10 m/s 3.85 10 m/s q = × = − × + × × × × F v B i k j k ! ! ! ° ° ° ° ° ° , × = − × = i k j j k i ( ) ( ) ( ) ( ) ( ) 4 4 4 4 ° ° ° ° 7.27 10 N 6.68 10 N 6.68 10 N 7.27 10 N = − × + × = × + × F j i i j ! E VALUATE : The directions of and v B ! ! are shown in Figure 27.1b. The direction of F ! is opposite to × v B ! ! since q is negative. The direction of F ! computed from the right-hand rule agrees qualitatively with the direction calculated with unit vectors. Figure 27.1b 27.2. I DENTIFY : The net force must be zero, so the magnetic and gravity forces must be equal in magnitude and opposite in direction. S ET U P : The gravity force is downward so the force from the magnetic field must be upward. The charge±s velocity and the forces are shown in Figure 27.2. Since the charge is negative, the magnetic force is opposite to the right-hand rule direction. The minimum magnetic field is when the field is perpendicular to v ! . The force is also perpendicular to B ! , so B ! is either eastward or westward. E XECUTE : If B ! is eastward, the right-hand rule direction is into the page and B F ! is out of the page, as required. Therefore, B ! is eastward. sin mg q vB φ = . 90 φ = ° and 3 2 4 8 (0.195 10 kg)(9.80 m/s ) 1.91 T (4.00 10 m/s)(2.50 10 C) mg B v q × = = = × × . 27
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27-2 Chapter 27 E VALUATE : The magnetic field could also have a component along the north-south direction, that would not contribute to the force, but then the field wouldn±t have minimum magnitude. Figure 27.2 27.3. I DENTIFY : The force F ! on the particle is in the direction of the deflection of the particle. Apply the right-hand rule to the directions of v ! and B ! . See if your thumb is in the direction of F ! , or opposite to that direction. Use sin F q vB φ = with 90 φ = ° to calculate F . S ET U P : The directions of v ! , B ! and F ! are shown in Figure 27.3. E XECUTE : (a) When you apply the right-hand rule to v ! and B ! , your thumb points east. F ! is in this direction, so the charge is positive. (b) 6 3 sin (8.50 10 C)(4.75 10 m/s)(1.25 T)sin90 0.0505 N F q vB φ = = × × = ° E VALUATE : If the particle had negative charge and v !
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