Chapter 29. Magnetic Fields

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1 Chapter 29 Magnetic Fields

2 A Partial History of Magnetism 13 th century BC Chinese used a compass 800 BC Uses a magnetic needle Probably an invention of Arabic or Indian origin Greeks Discovered magnetite (Fe 3 O 4 ) attracts iron

3 A Partial History of Magnetism Hans Christian Oersted Danish Physicist Discovered the relationship between electricity and magnetism in 1819 during a lecture demonstration! An electric current in a wire deflected a nearby compass needle

4 Demo Ec4: Oersted s Experiment When a current flows through the conductor the compass needle deflects due to the magnetic field set up by the current. Reversing the direction of the current reverses the direction of the magnetic field. Oersted in 1819 revolutionised the study of electricity & magnetism, sparking scientists to investigate the subject. He himself couldn t set up experiments and had to rely on assistants. He got left behind!

5 Demo Ec1: Lines of force in a magnetic field Iron filings sprinkled onto perspex sheet on top of bar magnet

6 Magnetic Poles Every magnet, regardless of its shape, has two poles Called north and south poles Poles exert forces on one another Similar to the way electric charges exert forces on each other Like poles repel N-N or S-S Unlike poles attract N-S

7 Magnetic Poles, cont. The poles received their names due to the way a magnet behaves in the Earth s magnetic field If a bar magnet is suspended so that it can move freely, it will rotate Magnetic north pole points toward the Earth s north geographic pole i.e. Earth s north geographic pole is a magnetic south pole Similarly, the Earth s south geographic pole is a magnetic north pole

8 Magnetic Poles, final The force between two poles varies as the inverse square of the distance between them A single magnetic pole has never been isolated i.e. magnetic poles always found in pairs There is some theoretical basis for the existence of monopoles single poles

9 Magnetic Fields (magnetic induction) A vector quantity Symbol B Direction given by that which the north pole of a compass needle points Magnetic field lines trace this direction in space

10 MFA02VD1: Compass needle shows direction of magnetic field around a bar magnet

11 Magnetic Field Lines for a Bar Magnet Compass can be used to trace the field lines The lines outside the magnet point from the North pole to the South pole

12 MFM02VD1: Magnetic Field lines around bar magnets

13 Magnetic Field Lines, Bar Magnet Iron filings are used to show the pattern of the magnetic field lines The direction of the field is the direction a north pole would point

14 Magnetic Field Lines, Unlike Poles Iron filings are used to show the pattern of the electric field lines The direction of the field is the direction a north pole would point c.f. electric field produced by an electric dipole

15 Magnetic Field Lines, Like Poles Iron filings are used to show the pattern of the electric field lines The direction of the field is the direction a north pole would point c.f. electric field produced by like charges

16 Magnetic Fields and Forces The magnetic field, B, at some point in space can be defined in terms of the magnetic force, F B Note: B is sometimes known as the Magnetic Induction A magnetic force will be exerted on a charged particle moving in a magnetic field

17 F B on a Charge Moving in a Magnetic Field Magnitude proportional to charge and speed of the particle Direction depends on the velocity of the particle and the direction of the magnetic field It is perpendicular to both F B = q v x B F B is the magnetic force q is the charge v is the velocity of the moving charge B is the magnetic field

18 Direction F B perpendicular to plane formed by v & B Oppositely directed forces are exerted on charges of different signs cause the particles to move in opposite directions

19 Direction given by Right-Hand Rule Fingers point in the direction of v (for positive charge; opposite direction if negative) Curl fingers in the direction of B Then thumb points in the direction of v x B; i.e. the direction of F B

20 The Magnitude of F The magnitude of the magnetic force on a charged particle is F B = q vb sin is the angle between v and B F B is zero when v and B are parallel or antiparallel = 0 or 180 F B is a maximum when v and B are perpendicular = 90

21 Example Proton moves at v = 8x10 6 m/s in x-direction. Enters region where B = 2.5 T directed at 60 to x-axis in xy-plane. What is the initial force on the proton? Right Hand rule puts F in the z-direction. F qvb sin( ) ( C)( m/s)(2.5 T)sin(60 ) N. Tiny!

22 Differences Between Electric and Magnetic Fields Direction of force Electric force acts along direction of electric field Magnetic force acts perpendicular to magnetic field Motion Electric force acts on a charged particle regardless of whether the particle is moving Magnetic force acts on a charged particle only when the particle is in motion

23 Differences Between Electric and Magnetic Fields Work Electric force does work in displacing a charged particle Magnetic force associated with a steady magnetic field does no work when a particle is displaced Force is perpendicular to the displacement, so F.ds=0

24 Work in Magnetic Fields The kinetic energy of a charged particle moving through a magnetic field cannot be altered by the magnetic field alone The field can only alter the direction of the particle, not its speed

25 Units of Magnetic Field The SI unit of magnetic field is the tesla (T) Since F B = q vb sin 1 T 1 The cgs unit is a gauss (G) 1 T = 10 4 G N C m/s 1 N A m

26 Typical Magnetic Field Values

27 Quick Quiz 29.1 The north-pole end of a bar magnet is held near a positively charged piece of plastic. The plastic is (a) attracted (b) repelled (c) unaffected by the magnet

28 Quick Quiz 29.1 Answer: (c). The magnetic force exerted by a magnetic field on a charge is proportional to the charge s velocity relative to the field. If the charge is stationary, as in this situation, there is no magnetic force.

29 Quick Quiz 29.2 A charged particle moves with velocity v in a magnetic field B. The magnetic force on the particle is a maximum when v is: (a) parallel to B (b) perpendicular to B (c) zero

30 Quick Quiz 29.2 Answer: (b). The maximum value of sin θ occurs for θ = 90.

31 Quick Quiz 29.3 An electron moves in the plane of this paper toward the top of the page. A magnetic field is also in the plane of the page and directed toward the right. The direction of the magnetic force on the electron is (a) toward the top of the page (b) toward the bottom of the page (c) toward the left edge of the page (d) toward the right edge of the page (e) out of the page (f) into the page

32 Quick Quiz 29.3 Answer: (e). Out of the page. The right-hand rule gives the direction. Be sure to account for the negative charge on the electron.

33 Magnetic Force on a Current Carrying Conductor A force is exerted on a current-carrying wire placed in a magnetic field The current is a collection of many charged particles in motion The direction of the force is given by the right-hand rule

34 Ec9: Magnetic force on current carrying conductor Flexible wire placed between poles of a strong magnet. When current is passed through it, the wire dramatically jumps out. Can demonstrate F is perpendicular to L and B by suspending wire in different orientations. See also corridor display.

35 Note on Notation The dots indicate the direction is out of the page The dots represent the tips of the arrows coming toward you The crosses indicate the direction is into the page The crosses represent the feathered tails of the arrows

36 MFA03AN2: Force on a current-carrying conductor

37 Force on a Wire When no current flows there is no force Therefore, the wire remains vertical

38 Force on a Wire (2) With B into the page, and Current up the page, then Force is to the left Apply the right hand rule

39 Force on a Wire, equation Magnetic force is exerted on each moving charge in the wire F = q v d x B Total force is the product of force on one charge times the number of charges F = (q v d x B)nAL n is the number of charges per unit volume

40 Force on a Wire, (4) In terms of the current, this becomes F = I L x B L is a vector that points in the direction of the current (i.e. of v D ) Magnitude is the length L of the segment I is the current = nqav D (think about the units to see this) B is the magnetic field

41 Force on a Wire of Arbitrary Shape Consider a small segment of the wire, ds The force exerted on this segment is F = I ds x B The total force is a b F I d s B

42 Force on a Wire, Case 1 Suppose that B is uniform. b Then a F I d s B becomes F=I L xb L is the vector sum of all the length elements from a to b Thus, the magnetic force on a curved currentcarrying wire in a uniform field is equal to that on a straight wire connecting the end points and carrying the same current

43 Force on a Wire, Case 2 An arbitrary shaped closed loop carrying current I in a uniform magnetic field F B I d s B 0 since d s 0 The length elements form a closed loop, so the vector sum is zero Thus, the net magnetic force acting on any closed current loop in a uniform magnetic field is zero

44 Quick Quiz 29.4 The four wires shown below all carry the same current from point A to point B through the same magnetic field. In all four parts of the figure, the points A and B are 10 cm apart. Which of the following choices ranks the wires according to the magnitude of the magnetic force exerted on them, from greatest to least? (a) a, b, c (b) a, c, b (c) a, c, d (d) a, d, c (d) b, c, d (e) c, b, d

45 Quick Quiz 29.4 Answer: (c). The order is (a), (b) = (c), (d). The magnitude of the force depends on the value of sin θ. The maximum force occurs when the wire is perpendicular to the field (a), and there is zero force when the wire is parallel (d). Choices (b) and (c) represent the same force because a straight wire between A and B will have the same force on it as the curved wire for a uniform magnetic field.

46 Quick Quiz 29.5 A wire carries current in the plane of this paper towards the top of the page. The wire experiences a magnetic force toward the right edge of the page. The direction of the magnetic field causing this force is (a) in the plane of the page and toward the left edge (b) in the plane of the page and toward the bottom edge (c) out of the page (d) into the page

47 Quick Quiz 29.5 Answer: (c). Use the right-hand rule to determine the direction of the magnetic field.

48 Charged Particle in a Magnetic Field Consider a particle moving in an external magnetic field with its velocity perpendicular to the field The force is always directed toward the centre of the circular path The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle

49 Force on a Charged Particle Equating the magnetic and centripetal forces: Solving gives r = mv/qb F qvb mv 2 r r is proportional to the momentum of the particle and inversely proportional to the charge and to the magnetic field

50 More About Motion of Charged Particle The angular speed of the particle is v r qb m The angular speed, is also referred to as the cyclotron frequency The period of the motion is T 2 r v 2 2 m qb

51 Van Allen Radiation Belts The Van Allen radiation belts consist of charged particles surrounding the Earth in doughnut-shaped regions Cosmic ray particles from the Sun are trapped by the Earth s magnetic field The particles spiral from pole to pole Can result in Auroras

52 The Earth s Magnetosphere Charged particles trapped in the Van Allen belts and create aurorae

53 Aurorae: MFM11VD1 Typically km above the Earth

54 Ec5: Deflection of electron beam by magnetic & electric fields Use either a cathode ray tube or electron beam deflection tube. Possible to determine e/m by size of the deflection. Mark North pole of magnet to show that F = q v x B.

55 MFM03AN1: Lorentz force in electric & magnetic fields

56 Charged Particles Moving in Electric and Magnetic Fields In many applications, charged particles will move in the presence of both magnetic and electric fields In that case, the total force is the sum of the forces due to the individual fields In general: F = qe + qv x B

57 Hall Effect (not examinable) When a current carrying conductor is placed in a magnetic field, a potential difference is generated in a direction perpendicular to both the current and the magnetic field This phenomena is known as the Hall effect It arises from the deflection of charge carriers to one side of the conductor as a result of the magnetic forces they experience

58 Hall Voltage Observing the Hall effect The Hall voltage is measured between points a and c

59 Hall Effect When the charge carriers are negative, the upper edge of the conductor becomes negatively charged When the charge carriers are positive, the upper edge becomes positively charged Sign of Hall voltage, V H, gives the sign of the charges

60 Hall Voltage In equilibrium qe H = qv d B V H = E H d = v d B d d is the width of the conductor v d is the drift velocity So can be found if B and d are known and V H measured Since v d I nqa, then V H IBd nqa Where R H = 1 / nq is called the Hall coefficient and y is the thickness of the conductor, with area A = yd A properly calibrated conductor can be used to measure the magnitude of an unknown magnetic field IB nqy R H IB y

61 Quick Quiz 29.8 A charged particle is moving perpendicular to a magnetic field in a circle with a radius r. An identical particle enters the field, with v perpendicular to B, but with a higher speed v than the first particle. Compared to the radius of the circle for the first particle, the radius of the circle for the second particle is (a) smaller (b) larger (c) equal in size

62 Quick Quiz 29.8 Answer: (b). The magnetic force on the particle increases in proportion to v, but the centripetal acceleration increases according to the square of v. The result is a larger radius, as we can see from r = mv/qb.

63 Quick Quiz 29.9 A charged particle is moving perpendicular to a magnetic field in a circle with a radius r. The magnitude of the magnetic field is increased. Compared to the initial radius of the circular path, the radius of the new path is (a) smaller (b) larger (c) equal in size

64 Quick Quiz 29.9 Answer: (a). The magnetic force on the particle increases in proportion to B. The result is a smaller radius, as we also see from r = mv/qb.

65 End of Chapter

66 Quick Quiz 29.6 Rank the magnitudes of the torques acting on the rectangular loops shown in the figure below, from highest to lowest. (All the loops are identical and carry the same current.) (a) a, b, c (b) b, c, a (c) c, b, a (d) a, c, b. (e) All loops experience zero torque.

67 Quick Quiz 29.6 Answer: (c). Because all loops enclose the same area and carry the same current, the magnitude of μ is the same for all. For part (c) in the image, μ points upward and is perpendicular to the magnetic field and τ = μb, the maximum torque possible. For the loop in (a), μ points along the direction of B and the torque is zero. For (b), the torque is intermediate between zero and the maximum value.

68 Quick Quiz 29.7 Rank the magnitudes of the net forces acting on the rectangular loops shown in this figure, from highest to lowest. (All the loops are identical and carry the same current.) (a) a, b, c (b) b, c, a (c) c, b, a (d) b, a, c (e) All loops experience zero net force.

69 Quick Quiz 29.7 Answer: (e). Because the magnetic field is uniform, there is zero net force on all three loops.

70 Quick Quiz 29.10a Three types of particles enter a mass spectrometer like the one shown in your book as Figure The figure below shows where the particles strike the detector array. Rank the particles that arrive at a, b, and c by speed. (a) a, b, c (c) c, b, a (b) b, c, a (d) All their speeds are equal.

71 Quick Quiz 29.10a Answer: (d). The velocity selector ensures that all three types of particles have the same speed.

72 Quick Quiz 29.10b Rank the particles that arrive at a, b, and c by m/q ratio. (a) a, b, c (c) c, b, a (b) b, c, a (d) All their m/q ratios are equal.

73 Quick Quiz 29.10b Answer: (c). We cannot determine individual masses or charges, but we can rank the particles by m/q ratio. Equation indicates that those particles traveling through the circle of greatest radius have the greatest m/q ratio.

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