Last Time. Magnetic Torque Magnetic Dipole in a B-Field: Potential Energy

Transcription

1 Last Time Magnetic Torque Magnetic Dipole in a B-Field: Potential Energy 1

2 Today Magnetic Force Motors and Generators Gauss' Law

3 Force on a Magnetic Dipole df df Idl B IBdl df IBdl sin df df If θ =, i.e. uniform field, then F =. F net IB sin dl F net RIBsin F net B sin R 3

4 U m Force on a Magnetic Dipole B x F F by _ usx Um B by _ us U x m F x B x db < dx db dx F by _ us F r mag U F x U m, 1 m, du dx d(μ B) F = dx Force exerted on dipole by magnetic field 4

5 Force on a Magnetic Dipole F x db dx F x d bar 3 dx 4 x B x 4 x bar 3 F x 6 4 x bar 4 Two magnets F x x 5

6 Exercise What is the distance at which one magnet can pick up another? F x x =mg.1 N 1. Am x F x. m 6

7 F m How to Make an Electric Motor Il B B Run the current one way Then reverse the current B Torque is in same direction 7

8 Electric Generators Tangential speed: v h / Force on charges: F m qv B F m qvbsin F qe e F m E vbsin emf left V we wvbsin emf total wvbsin emf total whb sin emf t ABsint 8

9 Electric Generators emf I t t emf R ABsint t AC generator AB R sin t 9

10 Power Required to Turn a Generator emf I t t ABsint AB R sin t W IwBsin h P dw dt d IwhB sin dt P IAB sint P I emf I R P ~ sin t 1

11 Demos: 6B-9 Magneto (Electric Generator) 3E-1 Steam Engine 11

12 Building up the Solid Sphere Total Charge: Total Volume: Charge Volume What Uniform we stop charge here at means r? Assume: these Charge are the distributed SAME uniformly Charge so far: r Volume so far: R Charge Volume 1

13 Where's the Source? Follow the Flow! We only see water flowing out. We see the water flow in. We see the same amount flow out. The source of this fountain must be in the bowl. The source of this fountain is not in the picture. 13

14 What is the Source? The Source of Water is a Water Faucet The Source of E-Field is Charge Water flowing OUT of bowl there is a Source inside E-field "flowing" out of sphere there is a Source inside 14

15 Electric Flux: Surface Area Flux through small area: flux ~ E nˆ A The n points OUTWARD from the surface. Out is +, In is - Definition of electric flux on a surface: surface E nˆ A 15

16 Electric Flux: Perpendicular Field or Area Perpendicular field E nˆ A AE cos E nˆ A Perpendicular area EAcos Exy cos x y q E nˆ A AE E nˆ A EA 16

17 Adding up the Flux surface E nˆ A E nˆ da E da da electric flux on a closed surface E da 17

18 Gauss s Law surface E nˆ A q inside E nda ˆ q inside Features: 1. Proportionality constant. Size and shape independence 3. Independence on number of charges inside 4. Charges outside contribute zero 18

19 1. Gauss s Law: Proportionality Constant surface surface Q qinside E E nˆ A 4 r 1 4 Q r Q r Q r A surface 4r r ˆ n ˆ A Q What if charge is negative? Works at least for one charge and spherical surface 19

20 . Gauss s Law: The Size of the Surface surface 1 Q qinside E E nˆ A 4 r 1 E ~ r A ~ r E 1 ~ r universe would be much different if exponent was not exactly!

21 Gauss' Law Size of Surface Point Charge: s Integrated over any concentric sphere: 1

22 3. Gauss s Law: The Shape of the Surface surface E nˆ A q inside E ˆnA EA surface surface A R (r tan) r All elements of the outer surface can be projected onto corresponding areas on the inner sphere with the same flux A / A 1 r / r 1 E A / E 1 A 1 1

23 4. Gauss s Law: Outside Charges surface E nˆ A q inside surface E nˆ A A ~ r 1 E ~ r surface EA A 1E1 A E Outside charges contribute to total flux 3

24 5. Gauss s Law: Superposition E surface E surface surface E 1 nˆ nˆ A A nˆ A 3 Q 1 Q surface E nˆ A q inside 4

25 iclicker Question One point charge Q is placed inside a closed surface with complicated shape. Which of the following statements is correct on the flux through the surface? a) It s not possible to calculate the flux since the details of the surface shape are not given. b) The flux is Q/ε c) The location of the charge Q inside the surface is need in order to calculate the flux. d) The flux is not zero and must be negative. +Q

26 iclicker Question Two identical point charges are placed, at the center of a large sphere in one case, and outside an identical sphere in the other case. Which statement about the net electric flux through the surface of the sphere is true? a) The flux is larger when the charge is inside. b) The flux is larger when the charge is outside. c) The flux is the same (and not zero). d) Not enough information to tell. e) The flux is zero in both cases. Q> Q>

27 Again: Continuous Charge Distribution 1: Charged Line At a point P on perpendicular axis: dx L / E E y k L / x y cos L / y k dx 3/ x y L / ( x y ) k k y y sec ( y tan y ) 3/ d k cos d (sin ) y tan x L / and E k / y 1

28 Infinitely long uniformly charged line E ( end caps) ( side) ( side) E rh Gauss s Law: E E r h k r Same result but much less work! r E h

29 Charged Disk At a point P on axis: E E k kr x kq x ( x ( x R) R)

30 Uniformly charged thin, infinite sheet Gauss s Law! E ( end caps) ( side) ( end caps) A EA Qenc A E h

31 Long Cylindrical Capacitor 1. Put charges +q on inner cylinder of radius a, -q on outer cylindrical shell of inner radius b.. Calculate E by Gauss Law q E rl E 3. Calculate V q rl a b q q V E dl dr ln b / a rl L b a 4. Divide q by V q q L C C L V q ln ln b / a b / a C dep. log. on a, b L L b q a +q -q V

32 . Calculate E by Gauss s Law q E 4 r q Spherical Capacitor 1. Put charges +q on inner sphere of radius a, -q on outer shell of inner radius b. 3. Calculate V from E q dr q 1 1 V Va Vb 4 r 4 b a b 4. Divide q by V q is proportional to V ab C 4 V b a a C C only depends on a,b 4 a as b (isolated sphere)

33 Today Gauss' Law 33