HW7 Fri Emf & Induction Mon.

Transcription

1 Wed Ohm s Law & mf Thurs. HW7 Fri mf & nduction Mon. Wed. Fri. xam (Ch 3 & 5) nductance and nergy of Maxwell s uations Mon Potential Formulation HW8

2 Ohm s Law charged particles moing in a wire The classical Drude model lectron gas thermal kt Periodic collisions with lattice impurities and ibrations t collisions l collisions thermal Get s nowhere on aerage With lectric Field While the instantaneous elocity is hary change drift thermal thermal The random thermal motion aerages out while the slow drift forward remains etween collisions ma F J ae thermal f thermal collision ae t collision m m thermal n carriers drift n molecules thermal f l a t collision carriers/ molecule thermal m tcollision drift n molecules Ohm s Law the current density is proportional to the field f carriers/ molecule l collisions m thermal J uctiity

3 Ohm s Law charged particles moing in a wire The classical Drude model J ncarriers drift nmolecules fcarriers/ molecule J uctiity Ohm s Law the current density is proportional to the field l m collisions thermal Or, if we integrate oer the cross-section perpendicular to the current / field, J R ntegrate along path current follows. is uniformly perpendicular to, and constant oer this area (otherwise, we d hae a curl) 1 Resistance For steady current, it s d 1 constant oer the path R d (sign is usually neglected, but means current flows down hill. )

4 d nergy Dissipation nergy transferred to differential bit of charges when accelerated through a potential difference: W field d ut in steady-state, charges moing in a resistie material hae no aerage gain in energy because they repeatey collide with impurities and ibrations and transfer the energy to the atoms of the wire. W wire d The wire warms up and, unless it melts first (thus stopping the current and the heating), it must shed the energy by heating the enironment uction and radiation (as in an incandescent lamp s filament.) W Rate at which energy is transferred : P wire W enironment d d For ohmic materials (for which ohm s law applies) so P R R R

5 xample: Pr. 7.4 Two long, coaxial metal cylinders separated by a material with uctiity (s) = k/s. What is the resistance, R? ẑ a b a b 1 R a b a ( s) R 1 k L b L / a k s 0 L / b a k L 1 sd dz b a 0 1 kd L

6 xample: Pr. 7.3 Two metal objects are embedded in a weakly ucting material of constant uctance find the relationship between this, R, and C. +Q -Q A key point is that current is free to flow any direction out of one object to arrie at the other, so when it comes to integrating J oer an area, it s oer a closed area surrounding one of the objects. R So, now, RC J while, / / Q C Q Q Q o o So, And by Gauss s Law,

7 mf (lectro-motie force ) Some process inside a battery causes charge separation across terminals. The field of those charges drie the current through a circuit. n euilibrium, F drie Of course, battery battery. term dp battery. term 0 or F. term. term drie F drie battery W drie mf

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9 Real Chemical attery - + SO HSO 4 H 4 Pb HSO PbSO H e 4 4 PbO HSO 4 3H e PbSO4 H O Net ffect Pb PbO H SO4 PbSO4 H O

10 Real Chemical attery - e - + Pb PbO H SO4 PbSO4 H O W chem W chem "mf" W chem G reaction Difference in Gibbs Free nergy for assembling products and reactants (Phys 344)

11 Motional mf Moe ucting bar across magnetic field F mag e Mobile electrons moe in response to magnetic force F lectron surplus accumulates at one end, deficiency at other elect Resulting electric field and force F + + F mag elect e grows until Felect Fmag 0 e e 0 n terms of oltage and mf: F mag mf mag d yˆ dy L L d dy

12 Motional mf mf L L mag so L R Of course, now that we hae established another component of R charge motion, a current flowing up, there s another component of magnetic force, Lxˆ F mag xample: f the bar has mass m and initial speed o, what will it be at time t? dp F mag Show that eentually, all the initial kinetic energy of the bar gets radiated away by the resistor d L m Lxˆ P R t L t R mr L d L oe R R 0 d L R L t mr L 1 mr t m o e 1 mr L d e L t o R mr t o e R t m o 1

13 Motional mf xample 7.4 : Faray Disk A metal disk of radius a rotates with an angular freuency (counterclockwise iewed from aboe) about an axis parallel to a uniform magnetic field. A circuit is made by a sliding contact. What is the current through the resistor R? R R a R mf mf mf a 0 F mag F mag s d ds a 0 sds s sˆ a

14 Motional mf and Magnetic Flux L y mf mag L dx da d a d y Will proe generality next time xercise: A suare loop is cut out of a thick sheet of aluminum. t is placed so that the top portion is in a uniform, horizontal magnetic field of 1 T into the page (as shown below) and allowed to fall under graity. The shading indicates the field region. What is the terminal elocity of the loop? How long does it take to reach 90% of the terminal elocity? F mag F gra

15 Motional mf and Magnetic Flux L y mf mag L dx da d a d y Will proe generality next time xercise: A suare loop is cut out of a thick sheet of aluminum. t is placed so that the top portion is in a uniform, horizontal magnetic field of 1 T into the page (as shown below) and allowed to fall under graity. The shading indicates the field region. What is the terminal elocity of the loop? How long does it take to reach 90% of the terminal elocity? F mag F gra

16 Generalization of Flux Rule S(t) S(t+) Using ector identity (1) A C Thus change in magnetic flux through the loop d rate of change in magnetic flux through the loop d f mag mfmag d mfmag Warning: our deriation used that the changing, /, corresponded to moing charge,. Not applicable when that s not the case. thar be paradoxes A C (We will later extend this reasoning to discuss stationary charges but changing fields)

17 Wed Ohm s Law & mf Thurs. HW7 Fri mf & nduction Mon. Wed. Fri. xam (Ch 3 & 5) nductance and nergy of Maxwell s uations Mon Potential Formulation HW8 Where we e been Stationary Charges producing and interacting ia lectric Fields Steady Currents producing and interacting ia Magnetic Fields Where we re going arying currents and charge distributions producing and interacting with arying lectric and Magnetic Fields A step closer to

18 where = Force between moing charges a = r Q and = or = lectric Depends on obserer s perception of source charge s elocity and acceleration Magnetic Also depends on obserer s perception of recipient charge s elocity