Chapter 19: Magnetic Forces and Fields

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1 Chapter 19: Magnetic Forces and Fields Magnetic Fields Magnetic Force on a Point Charge Motion of a Charged Particle in a Magnetic Field Crossed E and B fields Magnetic Forces on Current Carrying Wires Torque on a Current Loop Magnetic Field Due to a Current Ampère s Law Magnetic Materials 1

2 Magnetism (Just another Aspect of Electricity!) To pass on I will begin to discuss by what law of nature it comes about that iron can be attracted by that stone which the Greeks called magnet from the name of its home, because it is found within the national boundaries of the Magnetes. This astonishes men. In matters of this sort many principles have to be established Before you can give reason for the thing itself, you must approach by exceedingly long and roundabout ways LUCRETIUS THE ROMAN EPICUREAN SCIENTIST - Roman philosopher at beginning of the first century B.C. Magnetism has been known since antiquity: Iron oxide minerals like lodestone (Magnetite Fe 3 O 4 ) can act as bar magnets. What we will learn: Motion of electric charge creates a magnetic field If charges or charged particles are moving, there will be magnetism 2

3 Magnetic Fields Magnetic Dipole All magnets have at least one north pole and one south pole. Field lines emerge from north poles and enter through south poles. As for charges: there is a magnetic field, field lines (the pictorial representation of a magnetic field) and equipotential lines/planes. is always tangential to magnetic field lines. 3

4 Magnets exert forces on one another. Opposite magnetic poles attract and like magnetic poles repel. 4

5 Magnetic field lines are closed loops. There is no (known!) source of magnetic field lines. (No magnetic monopoles) If a magnet is broken in half you just end up with two magnets. 5

6 Nearly homogenous magnetic field between two large parallel magnets (horseshoemagnet) 6

7 Geomagnetism Near the surface of the Earth, the magnetic field is that of a dipole. Note the orientation of the magnetic poles! Magnetic south pole of the Earth is quite close to geographic North pole So magnetic north of compass needle points to geographic north of earth (Everything is only a convention) The magnetic field of the earth change with time 7

8 Away from the Earth, the magnetic field is distorted by the solar wind. The solar wind is a stream of charged particles (i.e., a plasma) which are ejected from the upper atmosphere of the sun. It consists mostly of high-energy electrons and protons (about 1 kev) Evidence for magnetic pole reversals has been found on the ocean floor. The iron bearing minerals in the rock contain a record of the Earth s magnetic field. 8

9 Magnetic Force on a Point Charge The magnetic force on a point charge is: ( v B) F = q B The unit of magnetic field (B) is the tesla (1T = 1 N/Am). A further unit for the magnetic field is gauss: 1G = 10-4 T (Magnetic field of earth around 0.5G) 9

10 ( ) The magnitude of F B is: F = qb vsinθ B where vsinθ is the component of the velocity perpendicular to the direction of the magnetic field. θ represents the angle between v and B. v θ B Draw the vectors tailto-tail to determine θ. 10

11 The direction of F B is found from the right-hand rule. For a general cross product: C = A B The right-hand rule is: using your right hand, point your fingers in the direction of A and curl them in the direction of B. Your thumb points in the direction of C. Note : C = A B B A 11

12 Example (text problem 19.15): An electron moves with speed m/s in a 1.2 Tesla uniform magnetic field. At one instant, the electron is moving due west and experiences an upward magnetic force of N. What is the direction of the magnetic field? F B = qbvsinθ sinθ = θ = 56 FB = qbv v (west) θ y θ F (up) x The angle can be either north of west OR north of east. 12

13 19.3 Charged Particle Moving Perpendicular to a Uniform B-field A positively charged particle has a velocity v (orange arrow) as shown. The magnetic field is into the page. The magnetic force, at this instant, is shown in blue. In this region of space this positive charge will move CCW in a circular path. 13

14 A positively charged particle experience a constant downward force (parabolic path) A positively charged particle entering a magnetic field experiences a horizontal force a right angles to its direction of motion. In this case the speed of the particle remains constant. A constant magnetic field cannot do work on a charged particle because the force is perpendicular to velocity. 14

15 Applying Newton s 2 nd Law to the charge: F = F B = ma r qvb = m v r 2 Radius of circular path r = mv/( q B) 15

16 Mass Spectrometer B A charged particle is shot into a region of known magnetic field. Detector Here, 2 v qvb = m r or qbr = mv Particles of different mass will travel different distances before striking the detector. (v, B, and q can be controlled.) V 16

17 Other devices that use magnetic fields to bend particle paths are cyclotrons and synchrotrons. Cyclotrons are used in the production of radioactive nuclei. For medical uses see the website of the Nuclear Energy Institute. Synchrotrons are being tested for use in treating tumors. 17

18 Motion of a Charged Particle in a Uniform B-field If a charged particle has a component of its velocity perpendicular to B, then its path will be a circle. If it also a component of v parallel to B, then it will move forward as well. This resulting path is a helix. 18

19 Crossed E and B Fields If a charged particle enters a region of space with both electric and magnetic fields present, the force on the particle will be F = = F e + qe + F B q( v B). 19

20 20 Consider a region of space with crossed electric and magnetic fields. B (into page) E Charge q>0 with velocity v

21 The value of the charge s speed can be adjusted so that Fnet = F e + FB = 0. The net force equal zero will occur when v=e/b. This region of space (with crossed E and B fields) is called a velocity selector. It can be used as part of a mass spectrometer. 21

22 Magnetic Force on a Current Carrying Wire The force on a current carrying wire in an external magnetic field is ( L B) F = I L is a vector that points in the direction of the current flow. Its magnitude is the length of the wire. 22

23 The magnitude of ( L B) F = I is F = ILBsinθ and its direction is given by the right-hand rule. 23

24 Example (text problem 19.43): A 20.0 cm by 30.0 cm loop of wire carries 1.0 A of current clockwise. (a) Find the magnetic force on each side of the loop if the magnetic field is 2.5 T to the left. I= 1.0 A B Left: F out of page Top: no force Right: F into page Bottom: no force 24

25 Example continued: The magnitudes of the nonzero forces are: F = ILBsinθ = ( 1.0 A)( 0.20 m)( 2.5 T) = 0.50 N sin 90 (b) What is the net force on the loop? F net = 0 25

26 Torque on a Current Loop Consider a current carrying loop in a magnetic field. The net force on this loop is zero, but the net torque is not. Force into page Axis B Force out of page L/2 L/2 26

27 The net torque on the current loop is: τ = NIAB sinθ N = number of turns of wire in the loop. I = the current carried by the loop. A = area of the loop. B = the magnetic field strength. θ = the angle between A and B. NIA magnetic moment 27

28 The direction of A is defined with a right-hand rule. Curl the fingers of your right hand in the direction of the current flow around a loop and your thumb will point in the direction of A. Because there is a torque on the current loop, it must have both a north and south pole. A current loop is a magnetic dipole. (Your thumb, using the above RHR, points from south to north.) 28

29 Magnetic Field due to a Current Moving charges (a current) create magnetic fields. The connection between current and magnetic field was discovered by Hans Christian Oersted. He discovered that a current creates a magnetic field (His compass deflected when it came close to the a current carrying wire). 29

30 The magnetic field at a distance r from a long, straight wire carrying current I is B = μ0i 2πr where μ 0 = 4π 10-7 Tm/A is the permeability of free space. The direction of the B-field lines is given by a right-hand rule. Point the thumb of your right hand in the direction of the current flow while wrapping your hand around the wire; your fingers will curl in the direction of the magnetic field lines. 30

31 A wire carries current I out of the page. The B-field lines of this wire are CCW. Note: The field (B) is tangent to the field lines. 31

32 Example (text problem 19.62): Two parallel wires in a horizontal plane carry currents I 1 and I 2 to the right. The wires each have a length L and are separated by a distance d. 1 I d 2 I (a) What are the magnitude and direction of the B-field of wire 1 at the location of wire 2? B μ0i 2πd 1 1 = Into the page 32

33 Example continued: (b) What are the magnitude and direction of the magnetic force on wire 2 due to wire 1? F 12 = = I I 2 2 LB LB 1 1 sinθ = μ I 0 1 I 2 2πd L F 12 toward top of page (toward wire 1) (c) What are the magnitude and direction of the B-field of wire 2 at the location of wire 1? B μ0i 2πd 2 2 = Out of the page 33

34 Example continued: (d) What are the magnitude and direction of the magnetic force on wire 1 due to wire 2? F 21 = = I I 1 1 LB LB 2 2 sinθ = μ I 0 1 I 2 2πd L F 21 toward bottom of page (toward wire 2) (e) Do parallel currents attract or repel? They attract. (f) Do antiparallel currents attract or repel? They repel. 34

35 The magnetic field of a current loop: The strength of the B-field at the center of the (single) wire loop is: B = μ 0 I 2R 35

36 The magnetic field of a solenoid: A solenoid is a coil of wire that is wrapped in a cylindrical shape. The field inside a solenoid is nearly uniform (if you stay away from the ends) and has a strength: B = μ0 ni Where n=n/l is the number of turns of wire (N) per unit length (L) and I is the current in the wire. 36

37 37

38 Ampère s Law Ampère s Law relates the magnetic field on a path to the net current cutting through the path. The tangential component of the magnetic field vector around any closed path is equal to the product of the permeability (of free space, μ 0 ) and the enclosed net current that pierces the loop 38

39 Example (text problem 19.65): A number of wires carry current into or out of the page as indicated. (a) What is the net current though the interior of loop 1? 39

40 Example continued: Assume currents into the page are negative and current out of the page are positive. Loop 1 encloses currents -3I, +14I, and -6I. The net current is +5I or 5I out of the page. (b) What is the net current though the interior of loop 2? Loop 2 encloses currents -16I and +14I. The net current is -2I or 2I into the page. 40

41 Define circulation: circulation = B Δl 41

42 Consider a wire carrying current into the page. Draw a closed path around the wire. Here the B-field is tangent to the path everywhere (hence the choice of a circular path). The circulation is B B( 2π ). Δl = r 42

43 Ampere s Law is B Δl = μ0i where I is the net current that cuts through the circular path. If the wire from the previous page carries a current I then the magnetic field at distance r from the wire is B = μ0i 2πr. 43

44 19.10 Magnetic Materials Ferromagnetic materials have domains, regions in which its atomic dipoles are aligned, giving the region a strong dipole moment. 44

45 When the domains are oriented randomly there will be no net magnetization of the object. When the domains are aligned, the material will have a net magnetization. 45

46 Summary Magnetic forces are felt only by moving charges Right-Hand Rules Magnetic Force on a Current Carrying Wire Torque on a Current Loop Magnetic Field of a Current Carrying Wire (straight wire, wire loop, solenoid) Ampère s Law 46

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