Physics 1B Part II: Magnetism

Transcription

1 Physics 1 Part : Magnetism colors reversed (color code varies) 6/5/2012 1

2 We start with the macroscopic What did historical people observe? How do magnets behave? s electricity related to magnetism? f so, how? Then we proceed to the microscopic How do particles behave? Lorentz magnetic force Michael Faraday Hans Christian Oersted 6/5/2012 2

3 The nature of research ut Mr. Faraday, of what use is all this? - unknown woman Madam, of what use is a newborn baby? - Michael Faraday With electromagnetism, as with babies, it s all a matter of potential. - ill Nye, the Science Guy 6/5/2012 3

4 Compass Two thousand years ago: Hang lodestone from string: it point north Magic! ut useful c. 4 th century CE 6/5/2012 4

5 Magnets reinforce each other Magnets align to create a stronger field Magnets move to increase -field This is opposite of electric dipoles Charges move to reduce E-field -field points out of North end -field points in to South end 6/5/2012 5

6 Where is the Earth s magnet? A C N S N S S N D E S N N S 6/5/2012 6

7 The north face ulk materials are neither north or south Only faces are faces have magnetic lines of force piercing them north faces attract south ends of compasses (-field comes out) south faces attract north ends of compasses (-field goes in) The bulk volume cannot be called north or south S N south face north face efore break it, the face looking left is already south; the face looking right is already north 6/5/2012 7

8 Dipolar disorder reak an electric dipole in two... You get one + and one -- Two monopoles Only possible because lines terminate reak a magnet in two... You get two dipole magnets Magnetic lines never terminate There are no magnetic charges N N N S N S 6/5/ S S

9 Electric charge doesn t interact with magnets ut electric current does! There is a connection between electricity and magnetism 6/5/2012 9

10 Electrical connection A connection between electricity and magnetism Compass points perpendicular to radius Tangent to circle around wire wire creates -field from slightly above creates current out of page top view 6/5/ N S

11 A current carrying wire lies in the plane of the compass. How does the needle respond? A nothing N points left C N points down D N points right E the compass explodes 6/5/

12 Force on a length of current and l must have consistent signs external -field l external -field l F m = l F m = l from slightly above top view 6/5/

13 Units of F m = l => N = A-m[] => [] = N/(A-m) or tesla, T Fundamentally, [] = kg-m/s 2 /(C/s-m) = kg/(c-s) ut we don t care about fundamental units here 6/5/

14 What is the net effect of the -field A net force up C on the current loop? net force down net torque clockwise D net torque counter-clockwise E nothing external -field from slightly above 6/5/

15 What is the net effect of the -field on the current loop? A net force up C net force down net torque clockwise D net torque counter-clockwise E nothing external -field top wire from slightly above 6/5/

16 Magnets from electricity: iot-savart Current generates -field Voltage has no effect iot-savart is the Coulomb s Law of electromagnets ds rˆ dsrˆ dssin d(at P) km, d k 2 m k 2 m 2 r r r k m 7 10 T-m/A ds r P d rˆ 6/5/ ds θ ˆr r P d

17 What is the -field from the given current element at P? A zero into the page C out of the page D up E down ds P 6/5/

18 Creating a magnetic dipole: a current loop Current flows in loops: creates a magnetic field Magnetic flux always forms closed loops -field inside the loop follows the right-hand rule Outside, in the equatorial plane, points opposite to inside axis N outside inside 6/5/ S

19 Current loop in a -field: redux How are the given -field and that produced by the loop related? external -field Magnetic forces pull & twist to increase the magnetic field top wire external -field F F -field from loop net torque 6/5/

20 Our 2.5 right-hand rules (RHRs) RHR #1: from current element creates from #1 we can derive #3 RHR #3: inside from a current loop axis N from slightly above RHR #2: cross product z y x x y z 6/5/ inside S

21 Particles Force of -field on a conductor depends on current only, independent of the conductor ut different conductors have different mobile charge densities and average speeds: this tells us something Consider 1-second s worth of mobile charge in a conductor t s v d m long Total mobile charge in the volume is Current through conductor of area A is: Magnetic force on any current is: one second s worth of volume v d (1 s) F l qnv Al m d external -field Q = qn(v d A)(1 s) Q Q qnvd A t 1s A A 6/5/ l

22 Particles (2) Magnetic force on a wire is due to magnetic force on mobile charges (individual particles) in it: F l qnv Al F qv nal qv N m d m d d Lorentz magnetic force on a single particle: Fm qv Fm qv The macroscopic magnetic force tells us about the microscopic magnetic force Total electromagnetic (EM) force Force is a vector: vectors add: F E v E v total q q q v d (1 s) external -field A A 6/5/ l

23 Newton s 3 rd law? And thirdly, the code is more what you d call guidelines than actual rules. Magnetic forces do not obey Newton s 3 rd guideline ut golly, professor, what of conservation of momentum? Electromagnetic waves carry off the remaining momentum, and total momentum is conserved etween isolated particles, Coulomb forces dominate Magnetic forces are only significant at relativistic speeds Walt Disney Pictures. Used without permission. So sue me. 2 (@ q 1 ) q 1 q 2 1 (@ q 2 ) = 0 => F 21 = 0 6/5/ F 12

24 Ampere s Law Symmetry simplifies the -field from a current For any 2D surface: s around Follows from iot-savart law d 0 through Similar to Gauss Law for any volume: E da surface q in t φˆ r wire ds Amperian loop t φˆ Example: from a wire d s t2 r 0 t 6/5/ r through Confusion over ds (should use dr in Ampere s): n iot-savart, ds is length of current element. n Ampere s Law, ds is displacement in space.

25 Solenoid: A better electromagnet Multiple turns increase -field Permeable (e.g. iron) core increases -field Ampere s Law in action Make 3 contributions zero Solves for the core -field N soft magnetic core ds ds = 0 very far Amperian loop d s 0 through N N h h l l ds very small, ds = /5/ S

26 The magnetic facts of life: where do magnets come from? They come from currents ut where do permanent magnets come from? The stork brings them? From microscopic currents in the magnet? A teeny bit From the intrinsic magnetic dipole moment of unpaired electrons 6/5/

27 nduced magnetic fields are not necessarily induced to reinforce The induced field opposes the change in the primary field Then, the resulting magnetic fields push and pull to reinforce as best they can Or at least, to minimize cancellation induced current -field from induced current primary Snapshot when the primary current is first turned on (increasing) 6/5/

28 nduced motional voltage (EMF) and current We quantify the induced voltage from our existing knowledge The conducting bar moves to the right with velocity, v We will return to -fields and work later primary -field moving conductor l - F - - charges accumulate + charges accumulate + + v ΔV F qv, q 0 6/5/ U q v e U e V v q The book calls this electro-motive force, or emf: E

29 efore equilibrium, which way is the current? A up F qv, q 0 down C all around D zero l - v U e V q v U q e v 6/5/

30 f the mobile charges were positive, then before equilibrium, which way would the current be? A up down C all around D zero l F + v F qv, q 0 U e V qv U q e v 6/5/

31 Faraday s Law, part 1 R Complete the circuit Write the voltage in terms of flux da recall: E da A primary -field area l - F + v ΔV We da V v q dt 6/5/ E area d dt d A dt Lenz Law: nduced current,, creates secondary -field which opposes the change in primary flux, Φ

32 R Faraday s Law, the sequel f multiple edges move, voltages add V Vi segments d dt An arbitrary shape is a sum of short segments: primary -field l v + - F v v Faraday's Law: V loop, RHR t Lenz Law: current opposes change in flux, Φ 6/5/ R A v v

33 Faraday s Law, part trois t still holds for stationary wires and changing -field: Cannot be derived from moving wires induced current primary increasing Faraday's Law: top view V loop, RHR R A t primary increasing Snapshot when the primary current is first turned on (increasing) Lenz Law: current opposes change in flux, Φ 6/5/

34 Work and magnetic fields There s a subtlety: Strictly speaking, magnetic fields do no work ecause the force is always perpendicular to the motion ut motors are magnetic, and they certainly do work Strictly speaking, magnetic fields create electric fields, which do work The net effect is that magnetic fields indirectly do work have ignored this indirection, and taken the results as work done by the magnetic field 6/5/