3Forces due to Magnetic fields B=mH= magnetic field density H=magnetic field intensity Force can be due to:B on moving charge, QB on current2 currentsB is defined as force per unit current element
4Forces on a Charge The total force is given by: Analogous to the electric force:We have magnetic force:The total force is given by:Note that Fm is perpendicular to both u and BIf the charge moving has a mass m, then:Usually Fm < Fe
5Forces on a current element The current element can be expressed as:So we can write:For closed path LineI=[A] current elementSurface K=[A/m]current elementVolume J=[A/m2]current element
6Force between two current elements Each element produces a field B, which exerts a force on the other element.I1R21I2
7P.E. 8.4 Find the force experienced by the loop if I1=10A, I2=5A, ro=20cm, a=1cm, b=30cm zDivide loop into 4 segments.F1F2rF3F4zI1I2brroa
8For segment #1, Force #1 z I1 I2 b F1 r a ro Since I1 is infinite long wire:zI1I2bF1rroa
9For segment #2, z F2 I1 I2 b r a ro The B field at segment #2 due to current 1.zF2I1I2brroa
10For segment #3, Force #3The field at segment 3:zI1I2bF3rroa
11For segment #4, z I1 I2 b r a ro F4 The B field at segment #4 due to current 1.zI1I2brroaF4
12The total force en the loop is I1=10A, I2=5A, ro=20cm, a=1cm, b=30cmF1F2rF3F4zThe sum of all four:Note that 2 terms cancel out:Attracts!
13Magnetic Torque and Moment Inside a motor/generator we have many loops with currents, and the Magnetic fields from a magnet exert a torque on them.The torque in [N m]is:Where m is the magnetic dipole moment:Where S is the area of the loop and an is its unit normal. This applies if B is uniformzBSIDE VIEWB
14Torque on a Current Loop in a Magnetic Field CD Motor
15Magnetic Torque and Moment The Magnetic torque can also be expressed as:The torque in [N m]is:
16Magnetization (similar to Polarization for E) Atoms have e- orbiting and spinningEach have a magnetic dipole associated to itMost materials have random orientation of their magnetic dipoles if NO external B-field is applied.When a B field is applied, they try to alignin the same direction.The total magnetization [A/m]
17Magnetization The magnetization current density [A/m2] The total magnetic density is:Magnetic susceptibility is:The relative permeability is:Permeability is in [H/m].
18Classification of Materials according to magnetism Non-magneticmr=1Ex. air, free space, most materials in their natural state.Magneticmr≠1Paramagneticmr≥1(air, platinum, tungsten, platinum )Temperature dependent.Not many uses except in masersFerromagneticmr>>1Iron, Ni,Co, steel, alloysLoose properties if heated above Curie T (770C)Nonlinear:mr variesDiamagneticmr≤1Electronic motions of spin and orbit cancel out.lead, copper, Si, diamonds, superconductors (mr=B=0).Are weakly affected by B Fields.
20B-H or Magnetization curve When an H-field is applied to ferromagnetic material,it’s B increases until saturation.But when H is decreased,B doesn’t follow the same curve.
21Hysteresis LoopSome ferrites, have almost rectangular B-H curves, ideal for digital computers for storing information.The area of the loop gives the energy loss per volume during one cycle in the form of heat.Tall-narrow loops are desirable for electric generators, motors, transformers to minimize the hysteresis losses.
22Magnetic B.C.We’ll use Gauss Law&Ampere’s Circuit law
23Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I B.C.: Two magnetic mediaConsider the figure below:B1nB1D SB1tm1DhB2nB2m2B2tElectrical Engineering, UPRM (please print on BOTH sides of paper)
24Dr. S. Cruz-Pol, INEL 4151-Electromagnetics I B.C.: Two magnetic mediaConsider the figure below:H1nH1q1abKm1H1tDhH2nm2H2cdH2tD wElectrical Engineering, UPRM (please print on BOTH sides of paper)
25P.E. 8.8 Find B2 Region 1 described by 3x+4y≥10, is free space Region 2 described by 3x+4y≤10 is magnetic material with mr=10Assume boundary is current free.OJO: Error en Solucionario!
26P.E. 8.7 B-field in a magnetic material. In a region with mr=4.6Find H and susceptabilitymA/m
27How to make traffic light go Green when driving a bike or motorcycle Stop directly on top of induction loop on the streetAttach neodymium magnets to the vehicleMove on top of loopPush crossing buttonVideo detectors
28Inductors So we can define the inductance as: then:If flux passes thru N turns, the total Flux Linkage isThis is proportional to the current I
30When more than 1 inductor *Don’t confuse the Magnetization vector, M, with the mutual inductance!
31Self -inductanceThe total energy in the magnetic field is the sum of the energies:The positive is taken if currents I1 and I2 flow such that the magnetic fields of the two circuits strengthen each other.See table 8.3 in textbook with formulas for inductance of common elements like coaxial cable, two-wire line, etc.
32P.E solenoidA long solenoid with 2 x 2 cm cross section has iron core (permeability is 1000x ) and 4000 turns per meter. If carries current of 0.5A, find:Self inductance per meterNote L is Independent of current
33Magnetic Circuits Magnetomotive force Reluctance Like V=IR Ex: magnetic relays, motors, generators, transformers, toroidsIn units of ampere-turnsTable 8.4 presents analogy between magnetic and electric circuits
36Ex. Find current in coil needed to produced flux density of 1 Ex. Find current in coil needed to produced flux density of 1.5T in air gap. Assume mr=50 and all cross sectional area is 10 cm2
378.42 A cobalt ring (mr=600) has mean radius of 30cm. If a coil wound on the ring carries 12A, calculate the N required to establish an average magnetic flux density of 1.5 Teslas in the ring.radius of 30cm
38Force on Magnetic materials RelayN turns, current IB.C. B1n=B2n (ignore fringing)Total energy change is used to displace bar a distance dl.S=cross sectional area of core
39P.E. 8.16 U-shape electromagnet Will lift 400kg of mass(including keeper + weight)Iron yoke has mr=3,000Cross section =40cm2Air gap are 0.1mm longLength of iron=50cmCurrent is 1AFind number of turns, NFind force across one air gapEsto nos da el B en el air gap.
40MagLev magnetic Levitation Floating one magnet over anotherUse diamagnetic materials, which repel and are repelled by strong H fields.Superconductors are diamagnetic.Regular train187 mphMaglev train312 mph