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Transcript
Lecture 13: June 19th 2009 Physics for Scientists and Engineers II Physics for Scientists and Engineers II , Summer Semester 2009 1 Ampere’s Law Magnetic field lines are tangentia l to a circle surroundin g the current. B ds I B a I Top View 0 I B d s B d s 2 r 2 r 0 I Physics for Scientists and Engineers II , Summer Semester 2009 2 Ampere’s Law Picking a different path. Top View r2 r1 I ds ds B ds Bds left semicircle Bds Bds straightsection Physics for Scientists and Engineers II , Summer Semester 2009 right semicircle Bds Bds straightsec tion 3 Ampere’s Law Bds Bds Bds left semicircle straightsec tion r1 0 I 0 I d s 0 2 r1 0 2 r2 I I 0 0 0 I 2 right semicircle Bds Bds straightsec tion r2 ds 0 0 2 The line integral B d s around any closed path equals I , where 0 I is the total steady current passing through any surface bounded by the closed path. Bds I 0 Physics for Scientists and Engineers II , Summer Semester 2009 4 Ampere’s Law Top View Picking a different path. r1 I r2 B ds ds B ds Bds inner semicircle Bds Bds straightsec tion Physics for Scientists and Engineers II , Summer Semester 2009 right semicircle Bds Bds straightsec tion 5 Ampere’s Law Bds inner semicircle Bds Bds straightsec tion r1 0 I 0 I d s 0 2 r1 0 2 r2 0 I 0 I 2 The line integral 2 right semicircle Bds Bds straightsec tion r2 ds 0 0 0 B d s around any closed path equals I , where 0 I is the total steady current passing through any surface bounded by the closed path. Bds I 0 Physics for Scientists and Engineers II , Summer Semester 2009 6 Example Application of Ampere’s Law I2 I1 Inner conductor : r r1 Insulator (air) : r1 r r2 Outer conductor : r2 r r3 Outside : r3 r Inner conductor : r r1 I1 I1 2 B d s 0 I inside B 2 r 0 r12 r B 0 2 r12 r Insulator : r1 r r2 B d s 0 I inside B 2 r 0 I1 B 0 Physics for Scientists and Engineers II , Summer Semester 2009 I1 2 r1 7 Example Application of Ampere’s Law Outer conductor : r2 r r3 2 r 2 r2 B d s 0 I inside B 2 r 0 I1 I 2 r32 r2 2 0 r 2 r2 2 B I1 I 2 2 r r3 2 r2 2 Outside : r3 r Bds I 0 inside B 2 r 0 I1 I 2 0 I1 I 2 B 2 r Physics for Scientists and Engineers II , Summer Semester 2009 8 Preventing Pitfalls ds Bds 0 Bds 0 I but not necessaril y that B 0 In fact, we know that B 0 along the blue path. Rather, B is perpendicu lar to the path (coming out of the page). Physics for Scientists and Engineers II , Summer Semester 2009 9 Preventing Pitfalls ds I Bds μ I 0 along the red path. Conclusion : You already need to know from symmetry etc. the direction of the magnetic field and that the magnitude of B constant along the path to draw the right conclusion s from Ampere' s laws. Physics for Scientists and Engineers II , Summer Semester 2009 10 A Long Solenoid (Wire wound in the form of a helix) Long solenoids produce reasonably uniform magnetic fields in their " interior". I Bds 0 Doesn' t mean that B 0 !!!! Bds μ I 0 B 0 outside the coil, but it is small. Physics for Scientists and Engineers II , Summer Semester 2009 11 A Long Solenoid (Wire wound in the form of a helix) Make the assumption that Bexternal Binternal Bds μ Binternal Bexternal Physics for Scientists and Engineers II , Summer Semester 2009 0 NI (N number of turns inside the blue loop) N B L μ 0 N I B μ0 I μ0 n I L N (n is the number of turns per unit length) L L 12 Problem 33 in the book J s (out of paper in y - direction) is a linear current density. Show that B is parallel to the sheet, J perpendicu lar to J s and B μ0 s . 2 z x L Physics for Scientists and Engineers II , Summer Semester 2009 13 Problem 33 in the book J s (out of paper in y - direction) is a linear current density. Symmetry shows that B is parallel to the sheet and perpendicu lar to J s . z x L Bds μ 0 I 2 B L μ0 I B μ0 Physics for Scientists and Engineers II , Summer Semester 2009 1 I 1 μ0 J s 2 L 2 14 Gauss’s Law in Magnetism Magnetic flux thoug h surface element d A : d B B d A Magnetic flux thoug h a surface : B B d A Units of magnetic flux : 1 Tm 2 1Wb (weber). Gauss' s law in magnetism : The magnetic flux throu gh any closed surface is always zero : B dA 0 Physics for Scientists and Engineers II , Summer Semester 2009 15 Gauss’s Law Comparison E dA qinside 0 Electric flux through closed surface is proportional to the amount of electric charge inside (electric monopoles). Physics for Scientists and Engineers II , Summer Semester 2009 B dA 0 Isolated magnetic monopoles have never been found. 16 Magnetism in Matter We now know how to build “electromagnets” (using electric current through a wire). We also found that a simple current loop produces a magnetic field / has a magnetic dipole moment. How about the “current” produced by an electron running around a nucleus? Let’s use a classical model (electron is a point charge orbiting around a positively charged nucleus. L r + I Orbital angular momentum of electron direction of motion of electron The tiny current loop produces a magnetic moment Physics for Scientists and Engineers II , Summer Semester 2009 17 Magnetism in Matter q e e ev I t T 2 2 r L r + I - ev 2 1 r e v r IA 2 2 r L L me v r v r me 1 L e L e 2 me 2me ( L) L = “orbital angular momentum” Physics for Scientists and Engineers II , Summer Semester 2009 18 Quantization Quantum Physics : Orbital angular momentum L is " quantized" h (means : it only occurs in multiples of 1.05 10 34 Js h Plank' s constant). 2π smallest nonzero value of : e e L 2 2me 2me (the factor 2 has to do with " total" orbital angular momentum, which is different from that only in one direction. ...this is beyond the scope of this class). Physics for Scientists and Engineers II , Summer Semester 2009 19 Spin Spin : An intrinsic property of electrons that contribute s to the total magnetic moment of an electron. Magnitude of spin angular momentum : S Magnetic moment due to spin : spin 3 2 e J 9.27 10 24 2 me T e B is called the Bohr magneton 2 me total orbital spin For atoms where all electrons are " paired" spin 0 For atoms with uneven number of electrons an unpaired electron must exist spin 0 The nucleus of an atom also has a magnetic moment due to it' s protons and electrons. nucleus electron Physics for Scientists and Engineers II , Summer Semester 2009 20