Essential Physics II. Lecture 8:

Transcription

1 Essential Physics II E II Lecture 8:

2 News Schedule change: Monday 7th December ( ) NO CLASS! Thursday 26th November ( ) 18:15-19:45 This week s homework: 11/26 (next lecture) Next week s homework: 12/14 (class 11/26) Class 11/30 homework: 12/14

3 What is magnetism?

4 What is magnetism? Quiz Magnetism is created by...? (a) electric charge q (b) electric currents (c) magnetic charge B (d) magnetic currents do not exist

5 What is magnetism? Magnetic force is seen between... magnets S N S N magnet and certain materials, e.g. iron electric currents

6 What is magnetism? Like with electric field, Ē, define the magnetic field, B. B

7 What is magnetism? The magnetic field, B, exerts a force on moving electric charges. x x x x x x x x B into page x x x x x q x x x x x x x x x x x x x x x x x x x x x x x x x x x if v =0, FB =0 magnetic force

8 What is magnetism? The magnetic field, B, exerts a force on moving electric charges. F x x x x B x x x x B into page x x x x x q x x v x x x x x x x x x x x x x x x x x if v 6= 0, FB 6=0 x x x x x x x x

9 What is magnetism? Magnetic force depends on: The charge, q q The magnetic field, B B The charge velocity, v q v The angle,, between and. v B v B

10 What is magnetism? The angle,, between and. v B B v Magnetic force is max: = 90 v perpendicular to B Magnetic force is 0: =0 v parallel to B max min Therefore: F B / sin

11 What is magnetism? Magnetic force magnitude: F B = q vb sin NOT Scalar product! vector 1 vector 2 cos scalar product = vector 1 vector 2 vector 1 vector 2 sin vector product 0 apple apple 180 = vector 1 x vector 2 F B = q v B magnetic force unit: Tesla (T)

12 What is magnetism? 1 T is a strong magnetic field. Often use Gauss : Earth s magnetic field Fridge magnets MRI Magnetar

13 What is magnetism? Magnetic force direction: right-angles (perpendicular) to v and B. F B = q v B Found using right hand rule v B

14 What is magnetism? Quiz A proton moves in a magnetic field. What will the direction of the magnetic force on the proton be in cases (a)? (A) parallel to B (B) out of the page v (C) into the page (D) zero v F B = q v B B

15 What is magnetism? Quiz A proton moves in a magnetic field. What will the direction of the magnetic force on the proton be in cases (b)? (A) parallel to B (B) out of the page v (C) into the page (D) zero v F B = q v B B but force is weaker

16 What is magnetism? Quiz Put the magnetic forces from the 3 velocity directions in order of size, largest -> smallest (A) a, b, c (B) c, b, a (C) b, a, c (c) (D) a, c, b

17 What is magnetism? Quiz A positive charge enters a uniform magnetic field. What is the direction of the magnetic force? (A) down (B) out of the page (C) into the page (D) to the right v x x x x x x x x x x x x v x x x x x x x x x x x x x x x x q x x x x F B x= q v x x B x x x x x (E) to the left B into page B

18 What is magnetism? Quiz A positive charge enters a uniform magnetic field. What is the direction of the magnetic force? B (A) zero (B) out of the page sin 0 = 0 v (C) into the page q (D) to the right (E) to the left

19 What is magnetism? Quiz 3 protons enter a 0.10 T magnetic field. v =2.0Mm/s Find the magnitude of the force on (1) B (A) 0 N (B) 200, 000N (1) q v v q (3) (C) 200, 000N v (D) (E) N N q (2) e = C

20 What is magnetism? Quiz 3 protons enter a 0.10 T magnetic field. v =2.0Mm/s Find the magnitude of the force on (1) (A) 0 N F B = q v B sin (B) (C) (D) (E) 200, 000N 200, 000N N N =( )( )(0.1) sin(90) = N

21 What is magnetism? Quiz 3 protons enter a 0.10 T magnetic field. v =2.0Mm/s Find the magnitude of the force on (2) B (A) 0 N sin 0 = 0 (B) 200, 000N (1) q v v q (3) (C) 200, 000N v (D) (E) N N q (2) e = C

22 What is magnetism? Quiz 3 protons enter a 0.10 T magnetic field. v =2.0Mm/s Find the magnitude of the force on (3) B (A) 0 N (B) 200, 000N (1) q v v q (3) (C) 200, 000N 0 apple apple 180 v (D) (E) N N But! Direction is opposite q (2) e = C

23 What is magnetism? The magnetic force is separate from the electric force. Combined: F = qē + q v B electric force magnetic force

24 What is magnetism? The magnetic force is separate from the electric force. Combined: F = qē + q v B electromagnetic force

25 Charged particles in a magnetic field The magnetic force is always perpendicular to the velocity. The force changes the direction, but not the speed and it does no work. (no component in direction of motion) If charge moves perpendicular to the field q F B v q uniform circular motion F = qvb sin 90 = qvb = m v2 r r = mv qb circular motion radius of path magnetic B q out of page

26 Charged particles in a magnetic field r = mv qb radius of path For constant charge (q) and momentum (p = mv) stronger magnetic field increases the force and decreases the radius. Charge sign reverses direction.

27 Charged particles in a magnetic field r = mv qb radius of path Period, T, for the circular orbit: q T = 2 r v = 2 v mv qb = 2 m qb r F B v Does not depend on v or r: r = mv qb the higher v, the larger r. T does not change. B out of page

28 Charged particles in a magnetic field r = mv qb radius of path Period, T, for the circular orbit: q T = 2 r v = 2 v mv qb = 2 m qb r F B v and frequency: f = 1 T = qb 2 m cyclotron frequency B out of page

29 Charged particles in a magnetic field Quiz 2 particles of the same mass enter a magnetic field with the same speed and follow the paths shown. Which particle has the bigger charge? (A) A (B) B r = mv qb (C) both are equal bigger q = smaller r x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x (D) cannot tell from the picture A B B into page

30 Charged particles in a magnetic field If v is not perpendicular to the field: v components perpendicular to B Gives circular motion in plane perpendicular to B parallel to B Unaffected by field Easy to move charge along field line. But cannot move perpendicular ( circular motion) Charges are frozen to the field lines like beads on a wire.

31 Magnetic force on a current An electric current is moving charges v d It feels a force in a magnetic field. Each charge: Fq = q v d B Force per length l: F = nalq v d B A l volume charges / volume n

32 Magnetic force on a current An electric current is moving charges v d It feels a force in a magnetic field. Each charge: F q = q v d B Force per length l: F = nalq v d B define current: I = Q t = nalq l/v d = naqv d F = I l B magnitude: wire length direction: along wire

33 Magnetic force on a current An electric current is moving charges v d It feels a force in a magnetic field. Each charge: F q = q v d B Force per length l: F = nalq v d B define current: I = Q t = nalq l/v d = naqv d F = I l B magnetic force on a current

34 Magnetic force on a current F = I l B magnetic force on a current This force moves electrons to one side of the wire. Creates electric force across wire.

35 Magnetic force on a current Quiz A flexible conducting wire passes through a magnetic field that points out of the page. The wire is deflected upward. Which direction is current flowing in the wire? (A) to the left (B) to the right F B = q v B v B B out of page

36 Magnetic force on a current Quiz A wire carrying 15A current makes a 25 degree angle with a uniform magnetic field. The magnetic force / unit length is 0.31 N/m. What is the magnetic field strength? (A) -156 mt F = I l B (B) 23 mt (C) 49 mt B = F Ilsin = F l 1 I sin (D) 21 mt = 0.31N/m (15A) sin(25 ) = 49mT

37 Magnetic force on a current Quiz A wire carrying 15A current makes a 25 degree angle with a uniform magnetic field. Magnetic field strength is 48.9 mt What is the maximum force / unit length possible by turning the wire? (A) 0.73 N/m Max when (B) N/m (C) 0.52 N/m F l = IB = (15A)(48.9mT) =0.73N/m (D) 0 N/m

38 Origin of magnetic field A charge feels a force in an electric field Ē And a charge creates an electric field. A moving charge feels a force in a magnetic field A moving charge creates a magnetic field

39 Origin of magnetic field Biot-Savart Law Beo-savaar d B = µ 0 4 Id l r r 2 Gives magnetic field, d B, at point P from current element, Id l (Similar to Coulomb s law for electric field, dē, from charge element, dq ) µ 0 = N/A 2 permeability constant B = Z d B = µ 0 4 Z Id l ˆr r 2

40 Origin of magnetic field Electric field lines begin and end on charges. But the magnetic field does not begin and end on moving charges The magnetic field encircles the moving charge or current. Magnetic field lines do not begin or end. Direction from right-hand grip rule

41 Using the Biot-Savart Law A current loop Find B at point P on axis of a circular loop carrying current I. Direction of B given by right hand grip rule. Perpendicular field components cancel. Parallel add.

42 Using the Biot-Savart Law A current loop Z B = db x = Z db cos = µ 0I 4 Z loop dl x 2 + a 2 r 2 a p x2 + a 2 cos

43 Using the Biot-Savart Law A current loop Z B = db x = Z db cos = µ 0I 4 Z loop dl x 2 + a 2 a p x2 + a 2 B = µ 0 Ia 4 (x 2 + a 2 ) 3/2 Z loop distance, x, to P is same for everywhere on loop (move outside integral) B = µ 0 Ia 2 2(x 2 + a 2 ) 3/2 dl loop circumference: 2 a

44 Using the Biot-Savart Law Field of a straight wire Find B at point P from an infinitely long straight wire carrying current I. Direction of B given by right hand grip rule. B out of page d B = µ 0 4 Id l r r 2 = µ 0I 4 dl sin r 2 sin =sin( + 90) = sin 90 cos + cos 90 sin sin = y r = y p x2 + y 2 = cos = y r

45 Using the Biot-Savart Law Field of a straight wire Find B at point P from an infinitely long straight wire carrying current I. Direction of B given by right hand grip rule. B out of page d B = µ 0 4 Id l r r 2 = µ 0I 4 dl sin r 2 = µ 0I 4 ydl (x 2 + y 2 ) 3/2 on axis: dl = dx

46 Using the Biot-Savart Law Field of a straight wire Find B at point P from an infinitely long straight wire carrying current I. Direction of B given by right hand grip rule. B out of page d B = µ 0 4 Id l r r 2 = µ 0I 4 dl sin r 2 = µ 0I 4 ydl (x 2 + y 2 ) 3/2 B = Z db = µ 0Iy 4 Z 1 1 dx (x 2 + y 2 ) 3/2 = µ 0I 2 y

47 Using the Biot-Savart Law Since magnetic fields produce a force on current-carrying wires: F = I l B and current-carrying wires produce a magnetic field... Current-carrying wires exert a force on each other. d Field from wire 1 at wire 2: B 1 = µ 0I 1 2 d This field is perpendicular to wire 2: F 2 = I 2 l B 1 = µ 0I 1 I 2 l 2 d magnetic force between 2 wires

48 Using the Biot-Savart Law F = µ 0I 1 I 2 l 2 d Force on each wire Parallel currents attract. If one current is reversed, the sign will change since: I = naqv d F = µ 0I 1 ( I 2 )l 2 d = µ 0I 1 I 2 l 2 d Anti-parallel currents repel

49 Using the Biot-Savart Law Quiz A flexible wire is wound into a flat spiral. If a current flows in the direction shown, will the coil tighten or become looser? (A) Tighten (B) Loosen current is parallel

50 Magnetic dipoles Field from a loop: B = µ 0 Ia 2 2(x 2 + a 2 ) 3/2 For x >> a: B ' µ 0Ia 2 2x 3 = 2µ 0IA 4 x 3 A = loop area Compare with on-axis electric dipole: Ē = 2kp x 3 î magnetic dipole moment: µ = IA B = µ 0 2 µ x 3 The current loop is a magnetic dipole.

51 Magnetic dipoles The magnetic dipole moment is a vector: µ = IĀ Direction given by right hand grip rule. Current loops normally have many turns: µ = NIĀ magnetic dipole moment for N-turn loop True for any shape closed loop Any current loop is an magnetic dipole

52 Magnetic dipoles Far from the dipole, electric dipoles and magnetic dipoles look the same. Near the dipole, they look different. Electric dipoles can be split into positive and negative charges: But magnetic charges or monopoles has never been found. Electric field lines start and stop on charges. Magnetic field lines do not begin or end, but circle moving charge.

53 Magnetic dipoles Because magnetic field lines do not begin or end the magnetic flux is always zero! I Never for magnetic fields! X X B dā =0 Gauss s law for magnetism X If monopoles are discovered... Gauss s law will break! X

54 Magnetic dipoles Quiz Which field lines could be a magnetic field? (A) a I B dā =0 (B) b Net field lines through a closed surface = 0

55 Torque on a current loop Like the electric dipole, the magnetic dipole feels a torque (turning force) in a magnetic field. Opposite sides of loop have current in opposite directions In uniform field, net force = 0 I I But component perpendicular to B creates a torque F = I l B = IaB For each perpendicular side a I I side = rf sin = 1 2 bf sin = 1 2 biab sin b net =2 side = IabB sin = IAB sin = µb sin = µ B loop area

56 Torque on a current loop The magnetic dipole turns to align with magnetic field. If the current is reversed, the dipole will keep turning. In an DC electric motor, the current loop spins between magnetic poles Just before the dipole aligns with the field, the current direction is changed. Loop keeps spinning in same direction.

57 Torque on a current loop

58 Torque on a current loop Quiz A rectangular current loop is in a uniform magnetic field. What is the direction of the net force on the loop? a) + x Forces all point out. No torque, no net force. b) + y I c) zero F = I l B z B F B = q v B d) - x I x y e) - y B

59 Torque on a current loop Quiz If there is a current in the loop in the direction shown, the loop will: a) move up b) move down c) rotate clockwise F N S F d) rotate counterclockwise e) both rotate and move S N

60 End~ (First half of C26)

61 Water delivery The Earth formed dry so where did our water come from? A likely option is after the Earth formed ( ) icy meteorites hit our planet and delivered the water

62 Hayabusa2 Hayabusa2 launched December 2014 It will visit asteroid Ryugu Take 3 samples And return to Earth in 2020 It may bring with it the beginnings of our water