GENERAL PHYSICS II Electromagnetism & Thermal Physics 3/5/2008 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 §1 Magnetic interaction and magnetic field 1.1 Magnetic phenomena:  Some history:   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   Magnetic effects from natural magnets have been known for a long time Recorded observations from the Greeks more than 2500 years ago 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  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: the geographical pole ≠the magnetic pole N S South geographic pole North magnetic pole 3/5/2008 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 3/5/2008 N Electric Field Lines of an Electric Dipole Magnetic Field Lines of a bar magnet S 3/5/2008 N 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 ever been found ! 3/5/2008 no monopoles have 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 §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 = N / A.m B x x x x x x x x x x x x v x x x x x x F q 3/5/2008 B  v   q F B  v  q F=0 A Example 1:  Two protons each move at speed v (as shown in the diagram) in a region of space which contains a constant B field in the -z-direction Ignore the interaction between the two protons  What is the relation between the magnitudes of the forces on the two protons? (a) F1 < F B v B v z x (c) F1 > F2 – What is F2x, the x-component of the force on the second proton? (a) F2x < C (b) F1 = F y (b) F2x = (c) F 2x > – Inside the B field, the speed of each proton: (a) decreases 3/5/2008 (b) increases (c) stays the same 10 I R  Bz = dl +R ) 3/2  (z dl =  R      IR2 Bz = 2(z +R2 ) 3/2  IR2 Bz  2z Note the form the field takes for z >> R: • Expressed in terms of the magnetic moment: = I 3/5/2008 R 2    Bz  z3 note the typical dipole field behavior! 36 IR2 Bz  2 32 2   z R IR2 Bz   R z   2z Expressed in terms of the magnetic moment  I2  R  Bz   R z   2z Bz Note the typical 1/z3 dipole field behavior! 0 3/5/2008 R z 37 2.3 Force between two parallel current-carrying wires: • We know that a current-carrying wire can experience force from a B-field • We know that a a current-carrying wire produces a B-field • Therefore: We expect one current-carrying wire to exert a force on another current-carrying wire: d  F Ib  F Ia • Current goes together  wires come together • Current goes opposite  wires go opposite 3/5/2008 38 • Calculate force on length L of wire b due to field of wire a: The field at b due to a is given by:  F • Ib d Ia μI a Ba  2π d  Magnitude  of F on b =   I I L Fb I b L B a  a b 2 d Calculate force on length L of wire a due to field of wire b: The field at a due to b is given by:  F Ib d Ia ×  μI b Magnitude F I L  I a I b L 0 Bb Bb   = a a 2 d of F on a 2π d 3/5/2008 39 §4 Amper’s law and application: 4.1 Formulation of Amper’s law: For magnetic field there exists a law which plays the same role as the Gauss’s law for electric field The Amper’s law:   B d l I Line integral of magnetic field vector around a closed path Current “enclosed” by that path I 3/5/2008 40 Example: B-field of a straight wire: • Calculate field at distance R from wire using Ampere's Law:  Choose loop to be circle of radius R centered on the wire in a plane to wire  Why?  Magnitude of B is constant (function of R only)  Direction of B is parallel to the path  Line integral around the closed path:  Apply Ampere’s Law: I R   B d R  l B(2π ) Current enclosed by path = I  dl 2π μI RB  B  μI 2π R Ampere's Law simplifies the calculation thanks to symmetry of the current! (axial/cylindrical) 3/5/2008 41 4.2 Application of Amper’s law: a) B field inside a long wire:   xxxxx Suppose a total current I flows through the wire of radius a into the screen as shown xxxxxxxx xxxxxxxxx r xxxxxxxx Calculate B field as a function of r, the distance from the center of the wire a • B field is only a function of r  take path to be circle of radius r:  • Current passing through circle: • Ampere's Law: 3/5/2008 I enclosed   B d o  l μI enclosed xxxxx   B d r  l B (2 π ) r2  2I a  μI r B 2 πa 42 Therefore, • Inside the wire: (r < a) a I r B 2 a B • Outside the wire: ( r > a ) I B 2r 3/5/2008 r 43 b) B Field of an infinite current sheet  Consider an infinite sheet of current described by n wires/length each carrying current i into the screen as shown Calculate the B field • What is the direction of the field? • Symmetry  vertical direction • Calculate using Ampere's law for a square of side w: •   B d 0  l Bw Bw 2Bw • I nwi therefore, 3/5/2008   B d l μI  x x x w x x x x x x x x x constant μ 0ni B constant 44 c) B Field of a Solenoid:  A constant magnetic field can (in principle) be produced by an infinite sheet of current In practice, however, a constant magnetic field is often produced by a solenoid • A solenoid is defined by a current i flowing through a wire that is wrapped n turns per unit length on a cylinder of radius a and length L  L a To correctly calculate the B-field, we should use Biot-Savart, and add up the field from the different loops • If a