# Chapter X Magnetic Field

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## Chapter X Magnetic Field

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 Some history:  Magnetic effects from natural magnets have been known for a long time. Recorded observations from the Greeks more than 2500 years ago.  The word magnetism comes from the Greek word for a certain type of stone (lodestone) containing iron oxide found in Magnesia, a district in northern Greece.  Properties of lodestones: could exert forces on similar stones and could impart this property (magnetize) to a piece of iron it touched.

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## Nội dung Text: Chapter X Magnetic Field

1. GENERAL PHYSICS II Electromagnetism & Thermal Physics 3/5/2008 1
2. Chapter X Magnetic Field §1. Magnetic interaction and magnetic field §2. Magnetic forces on a moving charged particle and on a current-carrying conductor §3. Magnetic field of a current – magnetic field calculations §4. Amper’s law and application 3/5/2008 2
3. §1. Magnetic interaction and magnetic field 1.1 Magnetic phenomena:  Some history:  Magnetic effects from natural magnets have been known for a long time. Recorded observations from the Greeks more than 2500 years ago.  The word magnetism comes from the Greek word for a certain type of stone (lodestone) containing iron oxide found in Magnesia, a district in northern Greece.  Properties of lodestones: could exert forces on similar stones and could impart this property (magnetize) to a piece of iron it touched.  Bar magnet: a bar-shaped permanent magnet. It has two poles: N and S Like poles repel; Unlike poles attract. We say that the magnets can interact each with other. This kind of interaction differs from electric interactions, and is called magnetic interaction 3/5/2008 3
4.  We have known that the means of transfering interactions between electric charges is electric field. By analogy to electric interaction we introduce for magnetic interaction the concept of magnetic field which is the means of transfering magnetic interactions: A magnet sets up a magnetic field in the space around it and the second magnet responds to that field.  The direction of the magnetic field at any point is defined as the direction of the force that the field would exert on a magnetic north pole of compass needle North geographic pole South magnetic N S pole The earth itself is a magnet. Note that for the earth magnet: N S the geographical pole ≠the magnetic pole South geographic pole North magnetic pole 3/5/2008 4
5. 1.2 Magnetic field vector and magnetic field lines: By analogy to electric field vector E we can introduce magnetic field vector B : + The direction of magnetic field vector at each point in the space can be defined experimentally by a compass + The mathematical expression for magnetic field vector (magnitude and direction) will be defined below (the law of Biot and Savart) Magnetic field lines can be drawn in the same manner as electric field lines (direction and density) S N 3/5/2008 5
6. Electric Field Lines of an Electric Dipole Magnetic Field Lines of a bar magnet S N 3/5/2008 6
7. Magnetic Monopoles ?  Perhaps there exist magnetic charges, just like electric charges. Such an entity would be called a magnetic monopole (having + or - magnetic charge).  How can you isolate this magnetic charge? Try cutting a bar magnet in half: S N S N S N Even an individual electron has a magnetic “dipole”! • Many searches for magnetic monopoles no monopoles have ever been found ! 3/5/2008 7
8. Source of Magnetic Fields?  What is the source of magnetic fields, if not magnetic charge?  Answer: electric charge in motion!  e.g., current in wire surrounding cylinder (solenoid) produces very similar field to that of bar magnet.  Therefore, understanding source of field generated by bar magnet lies in understanding currents at atomic level within bulk matter Motions of electrons on orbits and intrinsic motions produce magnetic field. Orbits of electrons about nuclei Intrinsic “spin” of electrons (more important effect) 3/5/2008 8
9. §2. Magnetic forces on a moving charged particle and on a current-carrying conductor: 2.1 Magnetic force on a moving charge: • The force F on a charge q moving with velocity v through a region of space with magnetic field B is given by:   Magnetic Force: F  v  q B (Lorentz force) • In the formula B is measured in Tesla (T): 1T = 1 N / A.m B B B x x x x x x   x x x x x x v v v x x x x x x    q q F q F F=0 3/5/2008 9
10. Example 1: y  Two protons each move at speed v (as v A 1 shown in the diagram) in a region of space which contains a constant B field B in the -z-direction. Ignore the interaction 2 v between the two protons.  What is the relation between the magnitudes of the forces on the two z x protons? (a) F1 < F 2 (b) F1 = F 2 (c) F1 > F2 B – What is F2x, the x-component of the force on the second proton? (a) F2x < 0 (b) F2x = 0 (c) F 2x > 0 C – Inside the B field, the speed of each proton: (a) decreases (b) increases (c) stays the same 3/5/2008 10
11.  Two independent protons each move at speed v (as shown in the diagram) y A in a region of space which contains a v 1 constant B field in the -z-direction. Ignore the interaction between the two B protons. v 2  What is the relation between the magnitudes of the forces on the two protons? z x (a) F1 < F2 (b) F1 = F2 (c) F1 > F2 • The magnetic force is given by:    F q v   F qvB sin θ B • In both cases the angle between v and B is 90 !! Therefore F1 = F2. 3/5/2008 11
12. F1 y  Two independent protons each move v at speed v (as shown in the diagram) F2 1 B in a region of space which contains a B constant B field in the -z-direction. v Ignore the interaction between the two 2 protons. z x  What is F2x, the x-component of the force on the second proton? (a) F2x < 0 (b) F2x = 0 (c) F2x > 0 • To determine the direction of the force, we use the corkscrew rule (or right-hand rule).    F q v B • As shown in the diagram, F2x < 0. 3/5/2008 12
13. C  Two protons each move at speed v (as y shown in the diagram) in a region of space which contains a constant B 1 v field in the -z-direction. Ignore the B interaction between the two protons. v 2  Inside the B field, the speed of each proton: z x (a) decreases (b) increases (c) stays the same Although the proton does experience a force (which deflects it), this force always ⊥ to v. Therefore, there is no possibility to do work, so kinetic energy is constant and | v | is constant. 3/5/2008 13
14. Example 2: Determine the ratio of charge to mass for an electron ? e- 1) Turn on electron ‘gun’ 1 mv 2  qV R 2 2) Turn on magnetic field B mv V R  qB ‘gun’ 3) Rearrange in terms of measured values, V, R and B 2 q q  v 2 2V and v 2  RB  m m   q 2V  2 2 m R B Experiment gave this ratio = 1.76  11 C/kg 10 3/5/2008 14
15. • The scheme for the “gun” • The principle of a selector: 3/5/2008 15
16. 2.2 Magnetic force on a current-carrying conductor:  Consider a current-carrying wire in the N S presence of a magnetic field B.  There will be a force on each of the charges moving in the wire. What will be the total force dF on a length dl of the wire?  Suppose current is made up of n charges/volume each carrying charge q and moving with velocity v through a wire of cross-section A.    Force on each charge = qv  B     Total force = dF  ( dl ) qv  nA B dq nAv ( dt ) q     Current = I  nAvq  d F Id l B dt dt The case for a straight length of wire L carrying    F  I L B a current I, the force on it is: 3/5/2008 16
17. 2.3 Magnetic force and torque on a current loop: F B x x x x x x x  Consider loop in magnetic field as on right: If field is to plane of loop, the x x x x x x x net force on loop is 0! Fx x x x x x x x x x x x x x F x x x x x x x  Force on top path cancels force on bottom path (F = IBL) F I  Force on right path cancels force on left path. (F = IBL) B • If plane of loop is not to field, there will x be a non-zero torque on the loop! F F . 3/5/2008 17
18. Example: A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. • What is the force on section a-b of the loop? a) zero b) out of the page c) into the page ab: Fab = 0 = Fcd since the wire is parallel to B. • What is the force on section b-c of the loop? a) zero b) out of the page c) into the page bc: Fbc = ILB RHR: I is up, B is to the right, so F points into the screen. 4) What is the net force on the loop?   a) zero b) out of the page c) into the page      By symmetry: Fda  Fbc  F  n e t Fa b Fb c Fc d Fd a 0 3/5/2008 18
19. Calculation of Torque: • Suppose the loop has width w (the side we see) and length L (into the screen). B The torque is given by: x    w F τ r  F  w  F .  τ 2 F sin   2   F = IBL    sin AIB r  r ×F where A = wL = area of loop   Note: if loop B, sin 0   0 = = F maximum  occurs when loop parallel to B 3/5/2008 19
20. • We can define the magnetic dipole moment of a current loop as follows: B + magnitude:  AI x   F + direction: to plane of the loop in  the direction the thumb of right hand F  . points if fingers curl in the direction of  current.  • Torque on loop can then be rewritten as:      sin  AIB τμ B  • Note: if loop consists of N turns, = NAI Concerning to magnetic dipole μ N moment we have an analogue: (N loops  N bar magnets) = 3/5/2008 20