Phy2049: MagnetismhittMagnetic Field UnitsForce Between Two Parallel CurrentsForce Between Two Anti-Parallel CurrentsParallel Currents (cont.)B Field @ Center of Circular Current LoopCurrent Loop ExampleB Field of SolenoidField at Center of Partial LoopSlide 11Partial Loops (cont.)Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Phy2049: Magnetism•Last lecture: Biot-Savart’s and Ampere’s law:–Magnetic Field due to a straight wire–Current loops (whole or bits)and solenoids•Today: reminder and Faraday’s law.hittTwo long straight wires pierce the plane of the paper at vertices of an equilateral triangle as shown. They each carry 3A but in the opposite direction. The wire on the left has the current coming out of the paper while the wire on the right carries the current going into the paper. The magnetic field at the third vertex (P) has the magnitude and direction(North is up):(1) 20 μT, east (2) 17 μT, west (3) 15 μT, north (4) 26 μT, south (5) none of theseX4 cmMagnetic Field Units•From the expression for force on a current-carrying wire:–B = Fmax / I L–Units: Newtons/Am Tesla (SI unit)–Another unit: 1 gauss = 10-4 Tesla•Some sample magnetic field strengths:–Earth: B = 0.5 gauss = 0.5 x 10-4 T–Galaxy: B 10-6 gauss = 10-10 T–Bar magnet: B 100 – 200 gauss–Strong electromagnet: B = 2 T–Superconducting magnet: B = 20 – 40 T–Pulse magnet: B 100 T–Neutron star: B 108 – 109 T–Magnetar: B 1011 TForce Between Two Parallel Currents•Force on I2 from I1–RHR Force towards I1•Force on I1 from I2–RHR Force towards I2•Magnetic forces attract two parallel currentsI1I20 1 0 1 22 2 1 22 2I I IF I B L I L Lr rm mp p� �= = =� �� �I1I20 2 0 1 21 1 2 12 2I I IF I B L I L Lr rm mp p� �= = =� �� �Force Between Two Anti-Parallel Currents•Force on I2 from I1–RHR Force away from I1•Force on I1 from I2–RHR Force away from I2•Magnetic forces repel two antiparallel currentsI1I2I1I20 1 0 1 22 2 1 22 2I I IF I B L I L Lr rm mp p� �= = =� �� �0 2 0 1 21 1 2 12 2I I IF I B L I L Lr rm mp p� �= = =� �� �Parallel Currents (cont.)•Look at them edge on to see B fields more clearlyAntiparallel: repelFFParallel: attractFFBBBB21222111B Field @ Center of Circular Current Loop•Radius R and current i: find B field at center of loop–Direction: RHR #3 (see picture)•If N turns close together02iBRm=02N iBRm=Current Loop Example•i = 500 A, r = 5 cm, N=20( )( )7020 4 10 5001.26T2 2 0.05iB Nrpm-�= = =�B Field of Solenoid•Formula found from Ampere’s law–i = current–n = turns / meter–B ~ constant inside solenoid–B ~ zero outside solenoid–Most accurate when L>>R•Example: i = 100A, n = 10 turns/cm–n = 1000 turns / m0B inm=( )( )( )7 34 10 100 10 0.13TB p-= � =Field at Center of Partial Loop•Suppose loop covers angle •Use example where = (half circle)–Define direction into page as positive02 2iBRmfp� �=� �� �0 01 201 22 2 2 21 14i iBR RiBR Rm mp pp pm� � � �= -� � � �� � � �� �= -� �� �Partial Loops (cont.)•Note on problems when you have to evaluate a B field at a point from several partial loops–Only loop parts contribute, proportional to angle (previous slide)–Straight sections aimed at point contribute exactly nothing–Be careful about signs, e.g.in (b) fields partially cancel, whereas in (a) and (c) they addChapter 30 Induction and InductanceIn this chapter we will study the following topics: -Faraday’s law of induction -Lenz’s rule -Electric field induced by a changing magnetic field -Inductance and mutual inductance - RL circuits -Energy stored in a magnetic field(30 – 1)In a series of experiments Michael Faraday in England and Joseph Henry in the US were able to generate electric currents without the use of batteries Below we describe some of theFaraday's experimentsse experiments thathelped formulate whats is known as "Faraday's lawof induction"The circuit shown in the figure consists of a wire loop connected to a sensitiveammeter (known as a "galvanometer"). If we approach the loop with a permanent magnet we see a current being registered by the galvanometer. The results can be summarized as follows: A current appears only if there is relative motion between the magnet and the loop Faster motion results in a larger current If we1.2.3. reverse the direction of motion or the polarity of the magnet, the currentreverses sign and flows in the opposite direction. The current generated is known as " "; the emf that appears induced current is known as " "; the whole effect is called " "induced emf induction(30 – 2)In the figure we show a second type of experimentin which current is induced in loop 2 when the switch S in loop 1 is either closed or opened. Whenthe current in loop 1 is constant no induced current is observed in loop 2. The conclusion is that the magnetic field in an induction experiment can begenerated either by a permanent magnet or by anelectric current in a coil.loop 1loop 2Faraday summarized the results of his experiments in what is known as" "Faraday's law of inductionAn emf is induced in a loop when the number of magnetic field lines thatpass through the loop is changingFaraday's law is not an explanation of induction but merely a description of of what induction is. It is one of the four " of electromagnetism"all of which are statements of experimMaxwell's equationsental results. We have already encounteredGauss' law for the electric field, and Ampere's law (in its incomplete form)(30 – 3)BrˆnfdAThe magnetic flux through a surface that bordersa loop is determined as follows: BMagnetic Flux Φ1 we divide the surface that has the loop as its borderinto area elements of area . dA.For each element we calculate the magnetic flux
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