Lecture 9 Magnetic Fields due to Currents Ch. 30• Cartoon - Shows magnetic field around a long current carrying wire and a loop of wire• Opening Demo - Iron filings showing B fields around wires with currents• Warm-up problem• Topics– Magnetic field produced by a moving charge– Magnetic fields produced by currents. Big Bite as an example.– Using Biot-Savart Law to calculate magnetic fields produced by currents.– Examples: Field at center of loop of wire, at center of circular arc of wire, at center ofsegment of wire.– Amperes’ Law : Analogous to Gauss’ L:aw in electrostatics, Useful in symmetric cases.– Infinitely long straight wire of radius a. Find B outside and inside wire.– Solenoid and Toroid Find B field.– Forces between current carrying wires or parallel moving charges• Demos– Torque on a current loop(galvanometer)– Iron filings showing B fields around wires with currents.– Compass needle near current carrying wire– Big Bite as an example of using a magnet as a research tool.– Force between parallel wires carrying identical currents.RemindersUngraded problem solutions are on website?UVa email activatedQuiz tomorrowClass in afternoon 1:00 PMGrades are on WebAssignMagnetic Fields due to CurrentsTorque on a coil in a magnetic field demo– left over from last time• So far we have used permanent magnets as our source of magneticfield. Historically this is how it started.• In early decades of the last century, it was learned that moving chargesand electric currents produced magnetic fields.• How do you find the Magnetic field due to a moving point charge?• How do you find the Magnetic field due to a current?– Biot-Savart Law – direct integration– Ampere’s Law – uses symmetry• Examples and DemosTorques on current loopsElectric motors operate by connecting a coil in a magnetic field to a current supply,which produces a torque on the coil causing it to rotate.PFiabFBBAbove is a rectangular loop of wire of sides a and b carrying current i.B is in the plane of the loop and ^ to a.Equal and opposite forces are exerted on the sides a.No forces exerted on b sinceSince net force is zero, we can evaluate t (torque) about any point. Evaluate itabout P (cm).iaBF=Bi!=NiabBsin"=NiABsin"i.A=ab=area of loop!= Fb sin"F = iaB!= iabB sin"For N loops we mult by NTorque on a current loop!= NiAB sin"µ= NiA = magnetic dipole moment!=µB sin"!=µ# Bˆn!B!ˆnMust reverse current usingcommutator to keep loopturning.Torque !!tends to rotate loop until plane is " to B( parallel to B).ˆnhttp://hyperphysics.ph y -a s tr.gsu.edu/hbase/magnetic/motdc .h tm l# c4Split ringcommutator+-Electron moving with speed v in a crossed electricand magnetic field in a cathode ray tube.  ! F = qE + q! v x! B yDiscovery of the electron by J.J. Thompson in 189722mv2qELy =1. E=0, B=0 Observe spot on screen2. Set E to some value and measure y the deflection3. Now turn on B until spot returns to the original positionBEvqvBqE==4 Solve foryE2LBqm22=This ratio was first measured by Thompson to be lighter than hydrogen by 1000Show demo of CRTTopic: Moving charges produce magnetic fieldsqr"rˆ20ˆ4 rrvqB!=!!"µ270 104AN!"=#µDetermined fomExperiment !v1. Magnitude of B is proportional to q, , and 1/r2.2. B is zero along the line of motion and proportional to sin at otherpoints.3. The direction is given by the RHR rotating into4. Magnetic permeabilityrˆ !v !vExample: A point charge q = 1 mC (1x10-3C) moves in the x direction withv = 108 m/s. It misses a mosquito by 1 mm. What is the B fieldexperienced by the mosquito?108 m/s90orˆ204 rvqB!µ=268327101101010mCBsmAN!!!"""=TB410=Recall the E field of a charge distribution“Coulombs Law”rˆrkdqEd2=!To find the field of a current distribution use:B!20rrˆsid4Bd!"µ=!!Biot-Savart LawTopic: A current produces a magnetic fieldˆrThis Law is found from experimentFind B field at center of loop of wire lying in a plane with radius Rand total current i flowing in it. B =µ04!"i d!l #ˆrr2Rirˆld !!is a vector coming out of the paper The anglebetween dl and r is constant and equal to 90degrees. !B = d!B!=µ04"ˆkidlR2!=µ04"ˆkiR2dl!=µ04"ˆkiR22"RkˆR2iB0µ=!Magnitude of B field at center ofloop. Direction is out of paper.Rkˆi d!l !ˆr = dl sin" ˆksin"= sin 90 = 1d!l !ˆr = dl ˆkrˆld!P"Magnitude is µ0i2RDirection is ˆk!B = d!B!=µ04"ˆkidlR2!=µ04"iR2ˆk dllilf!=µ04"iR2ˆk llilf=µ04"iR2ˆk(lf# li)=µ04"iR2ˆk(2"R # 0)=µ0i2Rˆkrˆld!P"Integration DetailsExample Loop of wire of radius R = 5 cm and current i = 10 A. What is B at thecenter? Magnitude and directionRiB20µ=)05(.21010427mABAN!"=#TB2610102.1 !"=#GaussTB 2.1102.14=!="Direction is out ofthe page.iWhat is the B field at the center of a segment or circulararc of wire?"0Rild!rˆTotal length of arc is S.Why is the contribution to the B field at P equal to zero from the straight section of wire?P!0= 100 deg=100 deg "6.28 rad360 deg= 100 " 0.0174 = 1.74 rad !B = d!B!=µ04"iR2ˆk dl!=µ04"iR2 ˆk S !B =µ04!iR "0ˆk where S is the arc length S = R"0"0 is in radians (not degrees) B =µ04!"i d!l #ˆrr2 d!l !!r = rdl sin" ˆksin"= sin 0 = 0 d!l "ˆr d!l "ˆrFind magnetic field at center of arc length004!"µRiB =Suppose you had the following loop.What is the magnitude and direction of B at the origin?SRiB204!µ=000002)2/(4!"µ!!"µRiRiRiB #=#=Next topic: Ampere’s LawAllows us to solve certain highly symmetric current problems for themagnetic field as Gauss’ Law did in electrostatics.Ampere’s Law iscIldB 0µ=!"!!Current enclosed by the path !E"!" d!A =qenc#0Gauss’s LawCharge enclosed by surfaceExample: Use Ampere’s Law to find B near a very long, straightwire. B is independent of position along the wire and only dependson the distance from the wire (symmetry).rld!iiBy symmetryldB!!irBdlBBdlldB 02µ!===="# ##!!riB!µ20=Suppose i = 10 AR = 10 cmT 1021010102517!!!"=""=BBShow Fe fillings around a straight wire with current, current loop, and solenoid.270 104AN!"=#µRules for finding direction of B field from a currentflowing in a wirer2iB0!µ=Force between two current carrying wiresFind the force due to the current element of the first wire and the magneticfield of the second wire. Integrate over the length of both wires. This will givethe force between the two wires.abbaBLiF!"! x =Force between two current carrying wiresFind the force due to the current element of the first wire and the magneticfield of the second wire.