mv = ev ebr Application: circular motion of moving ions In a uniform magnetic field: The mass spectrometer KE=PE magnitude of electron charge

Transcription

1 1.4 The Mass Spectrometer Application: circular motion of moving ions In a uniform magnetic field: The mass spectrometer mv r qb mv eb magnitude of electron charge 1 mv ev KEPE v 1 mv ebr m v e r m B m e r m ev er V B B The mass spectrum of naturally occurring neon, showing three isotopes. 1

2 Example: A singly charged positive ion has a mass of.5 x 10-6 kg. After being accelerated through a potential difference of 50 V, the ion enters a magnetic field of 0.5 T, in a direction perpendicular to the field. Calculate the radius of the path of the ion in the field.

3 Example: A singly charged positive ion has a mass of.5 x 10-6 kg. After being accelerated through a potential difference of 50 V, the ion enters a magnetic field of 0.5 T, in a direction perpendicular to the field. Calculate the radius of the path of the ion in the field. q 1.6x10 19 m.5x10 6 V 50 V B 0.5 T r? C kg mv FB Fc qvb r r V v W q K q 1 mv q mv qb We need to solve for the velocity! 19 Vq (50)(1.6x10 ) m.5x10 m/s r (.5x10 (1.6x10 6 )(56,568) 19 )(0.5) m 3

4 1.3 The Motion of a Charged Particle in a Magnetic Field Example: Velocity Selector A velocity selector is a device for measuring the velocity of a charged particle. The device operates by applying electric and magnetic forces to the particle in such a way that these forces balance. Given B and q, wow should an electric field be applied so that the force it applies to the particle can balance the magnetic force? Solution: by RHR-1: the velocity is to the right (thumb) and the magnetic field (finger) is into the page ( x marks the tail of an arrow), so the magnetic force on a positive charge is upward (palm up). Here θ 90 F B qvb We therefore need the electric force F E that points down that has the same magnitude: for a positive charge the electric field then needs to point in the same direction as the desired force, and F E qe. We want F E F B qe qvb E vb (and pointing downward) 4

5 1.5 The Force on a Current in a Magnetic Field Magnetic force on a current The magnetic force on the moving charges pushes the wire to the right. Since a current consists of moving charges then a current is subject to the Lorentz force Magnetic force on a straight current segment of length L F qvbsinθ In Class Demo F q t L I ( v t ) Bsinθ F ILBsinθ Here θ is the angle between the direction of the current and the magnetic field 5

6 1.5 The Force on a Current in a Magnetic Field Example: A wire carries a current of.0 A from west to east. Assume that at this location the magnetic field of Earth is horizontal and directed from south to north and that it has a magnitude of B T. (a) Find the magnitude and direction of the magnetic force on a 36.0 m length of wire. (b) Calculate the gravitational force on the same length of wire if it s made of copper and has a cross-sectional area of m. 6

7 1.5 The Force on a Current in a Magnetic Field Example: A wire carries a current of.0 A from west to east. Assume that at this location the magnetic field of Earth is horizontal and directed from south to north and that it has a magnitude of B T. (a) Find the magnitude and direction of the magnetic force on a 36.0 m length of wire. (b) Calculate the gravitational force on the same length of wire if it s made of copper and has a cross-sectional area of m. Solution : (a) By RHR -1 the force is pointing upward F ILB sinθ B whereθ is the angle between the direction of the current segment and the magneticfield Here θ 90 F (b) Density of copper is ρ Mass of wire : M ρ( volume) ρal M F F G G ( Mg / F B 3 ( kg)(9.81m/s 199 B kg/m ILB 3 (.0 A)(36.0 m)( )( m kg/m ) 7.89 N 3-5 )(36.0 m) kg T) N (up) 7

8 1.7 Magnetic Fields Produced by Currents In Class Demo A LONG ( ), STRAIGHT WIRE carrying current produces a magnetic field µ B oi π r Result to remember: derivation requires integral calculus µ o 4π 10 7 T m A Magnitude permeability of free space Direction Right-Hand Rule No.. Curl the fingers of the right hand into the shape of a half-circle. Point the thumb in the direction of the conventional current, and the tips of the fingers will point in the direction of the magnetic field. 8

9 1.7 Magnetic Fields Produced by Currents Current carrying wires can exert forces on each other. Example: What is the attraction/repulsion force per meter of wire between two infinite straight wires carrying 1.00A each, separated by 1.00 m? 9

10 1.7 Magnetic Fields Produced by Currents Current carrying wires can exert forces on each other. Example: What is the attraction/repulsion force per meter of wire between two infinite straight wires carrying 1.00A each, separated by 1.00 m? B 1 F F L µ 0I1 (4π 10 T m/a)(1.00 A) π r π (1.00 m) I LB1 sinθ, θ 90 µ 0I1I I B1 π r 7 (1.00 A)( T) 7 N s C C m s N m T 1 m This is in fact the original SI definition of an ampere (A). 10

11 1.7 Magnetic Fields Produced by Currents SOLENOID number of turns per unit length n N / L Magnetic field in the interior of a solenoid Is approximately uniform In the limit of infinite length, but finite n B µ ni o Magnitude µ oni L Result to remember: derivation requires integral calculus 11

12 1.7 Magnetic Fields Produced by Currents A CIRCULAR LOOP OF WIRE R B R.H. Two different applications of RHR- for the same field configuration At the center of the circular loop only Magnitude µ B oi R Result to remember: derivation requires integral calculus 1

13 1.7 Magnetic Fields Produced by Currents Example: Net Magnetic Field A long straight wire carries a current of 8.0 A and a circular loop of wire carries a current of.0 A and has a radius of m. Find the magnitude and direction of the magnetic field at the center of the loop. 13

14 1.7 Magnetic Fields Produced by Currents Example: Net Magnetic Field A long straight wire carries a current of 8.0 A and a circular loop of wire carries a current of.0 A and has a radius of m. Find the magnitude and direction of the magnetic field at the center of the loop. B µ I 1 µ oi µ o I1 π r R π r I R o B ( 7 π 10 T m A) A.0 A T π ( m) m 14

15 1.7 Magnetic Fields Produced by Currents The field lines around a current loop resemble those around the bar magnet. Attraction Repulsion 15

16 1.6 The Torque on a Current-Carrying Coil Put a rectangular loop of current I and length (height) L, and width w in a uniform magnetic field B. The loop is mounted such that it is free to rotate about a vertical axis through its center. We will consider the forces on each segment and the resulting torque from each. Using RHR-1: The force on segment 3 points down, and that on segment 4 points up. F 3 and F 4 are also equal in magnitude and cancel one another. The magnitudes F 3 F 4 IwBsin(90 -φ) IwBcosφ also change with the rotation angle φ But both F 3 and F 4 are directed parallel to the axis, and results in no torque. Top view φ is the angle between the normal to the loop and the magnetic field 16

17 1.6 The Torque on a Current-Carrying Coil Looking at segments 1 and which have the current running vertically. By RHR-1, force F 1 on segment 1 (current up) points into the page, for all values of φ. Also by RHR-1, force F on segment 1 (current down) points out of the page. They cancel each other to yield no net force on the loop. However, F 1 and F both tend to turn the loop in the clockwise sense (as seen in the top view). The torques from the two forces are each w τ 1, ( F1, ) sinφ F ILB sinθ sinceθ 90 τ τ 1 + τ Fwsinφ ILB Top view 1 φ is the angle between the normal to the loop and the magnetic field 17

18 1.6 The Torque on a Current-Carrying Coil τ τ 1 + τ Fwsinφ The torque τ is maximum when the normal of the loop is perpendicular to the magnetic field, and zero when the normal is parallel to the field. F ILB sinθ sinceθ 90 ILB The torque tends to cause the loop normal to become aligned to the field, just like on a bar magnet. Current loop magnetic dipole ( wsinφ) IAB sinφ Net torque τ ILB Top view A Lw area of loop τ magnetic moment m NIA B sinφ number of turns of wire 18

19 The Torque on a Current-Carrying Coil Example The Torque Exerted on a Current-Carrying Coil A coil of wire has an area of.0x10-4 m, consists of 100 loops or turns, and contains a current of A. The coil is placed in a uniform magnetic field of magnitude 0.15 T. (a) Determine the magnetic moment of the coil. (b) Find the maximum torque that the magnetic field can exert on the coil.

20 0 1.6 The Torque on a Current-Carrying Coil Example The Torque Exerted on a Current-Carrying Coil A coil of wire has an area of.0x10-4 m, consists of 100 loops or turns, and contains a current of A. The coil is placed in a uniform magnetic field of magnitude 0.15 T. (a) Determine the magnetic moment of the coil. (b) Find the maximum torque that the magnetic field can exert on the coil. (a) m magnetic moment ( )( )( 4 ) 4 NIA A.0 10 m A m (b) τ magnetic moment NIA Bsinφ ( A m )( 0.15 T) sin N m

21 1.6 The Torque on a Current-Carrying Coil Application: The basic components of a dc motor. The brushes switches the direction of the current so that the torque is always in the same direction continuous rotation 1

22 1.8 Ampere s Law AMPERE S LAW FOR STATIC MAGNETIC FIELDS For any current geometry that produces a magnetic field that does not change in time, B µ oi net current passing through surface bounded by path positive sense for the current bounded is related to the direction of the path traveled by RHR- Positive direction for bound currents

23 1.8 Ampere s Law Example An Infinitely Long, Straight, Current-Carrying Wire Use Ampere s law to obtain the magnetic field. 3

24 1.8 Ampere s Law Example An Infinitely Long, Straight, Current-Carrying Wire Use Ampere s law to obtain the magnetic field. 1. By symmetry, the magnetic field must have cylindrical symmetry: depends only on distance r to the wire.. The magnetic field should be generally perpendicular to the current (by Right Hand Rule). Choose a circular path (at constant r) to match the symmetry The magnetic field lines circulate around the line by RHR-: traverse path in the CCW direction as seen from the top The field is parallel to the path and has constant magnitude on the path. B B µ I o ( ) µ I o B(π r) µ I µ o B I π r o 4