Magnetic Forces Materials and devices INEL 4151 Dr

Magnetic Forces, Materials and devices INEL 4151 Dr. Sandra Cruz-Pol Electrical and Computer Engineering Dept. UPRM http: //www. treehugger. com/files/2008/10/spintronics-discover-could-lead-tomagnetic-batteries. php

Applications Motors Transformers MRI More… http: //videos. howstuffworks. com/hsw/1803 4 -electricity-and-magnetism-magneticlevitation-video. htm

Forces due to Magnetic fields Analogous to the electric force: The total force is given by: We have magnetic force: If the charge moving has a mass m, then:

Forces on a current element The current element can be expressed as: So we can write: Line current element surface current element volume current element

Force between two current elements • Each element produces a field B, which exerts a force on the other element. I 1 R 21 I 2

P. E. 8. 4 Find the force experienced by the loop if I 1=10 A, I 2=5 A, ro=20 cm, a=1 cm, b=30 cm z Divide loop into 4 segments. z I 1 F 2 I 2 ro a F 1 b F 3 r r F 4

For segment #1, Force #1 z I 1 F 1 ro I 2 b a r Since I 1 is infinite long wire:

For segment #2, z The B field at segment #2 due to current 1. F 2 I 1 I 2 ro b a r

For segment #3, Force #3 z I 1 I 2 ro b F 3 a r The field at segment 3:

For segment #4, The B field at segment #4 due to current 1. z I 1 I 2 b a ro F 4 r

The total force en the loop is I 1=10 A, I 2=5 A, ro=20 cm, a=1 cm, b=30 cm • The sum of all four: z F 2 F 1 F 3 r F 4 • Note that 2 terms cancel out:

Magnetic 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: z • Where m is the Magnetic Dipole moment: B • Where S is the area of the loop and an is its normal. This applies if B is uniform SIDE VIEW B

Torque on a Current Loop in a Magnetic Field CD Motor

Magnetic Dipoles, m Current loop Magnet Where the magnetic moment is:

Magnetic Torque and Moment The Magnetic torque can also be expressed as: The torque in [N m]is:

Magnetization (similar to Polarization for E) • Atoms have e- orbiting and spinning – Each have a magnetic dipole associated to it • Most materials have random orientation of their magnetic dipoles if NO external B-field is applied. • When a B field is applied, they try to align in the same direction. • The total magnetization [A/m]

Magnetization • The magnetization current density [A/m 2] • The total magnetic density is: • Magnetic susceptibility is: • The relative permeability is: • Permeability is in [H/m].

Classification of Materials according to magnetism Non-magnetic mr=1 Ex. air, free space, many materials in their natural state. Magnetic mr≠ 1 Diamagnetic mr≤ 1 Electronic motions of spin and orbit cancel out. lead, copper, Si, diamonds, superconductors. Are weakly affected by B Fields. Paramagnetic mr≥ 1 Temperature dependent. Not many uses except in masers Ferromagnetic mr>>1 Iron, Ni, Co, alloys Loose properties if heated above Curie T (770 C) Nonlinear: mr varies

B-H or Magnetization curve • When an H-field is applied to ferromagnetic material, it’s B increases until saturation. • curve. But when H is decreased, B doesn’t follow the same

Hysteresis Loop • Some 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.

Magnetic B. C. • We’ll use Gauss Law & • Ampere’s Circuit law

B 2 n B. C. : Two magnetic media • Consider the figure below: B 1 n B 1 DS B 1 t B 2 t m 1 m 2 Dh

B. C. : Two magnetic media • Consider the figure below: H 1 n H 1 q 1 m 1 H 2 n m 2 a b K H 1 t H 2 t d Dw Dh c

P. E. 8. 8 Find B 2 • Region 1 described by 3 x+4 y≥ 10, is free space • Region 2 described by 3 x+4 y≤ 10 is magnetric material with mr=10 • Assume boundary is current free.

P. E. 8. 7 B-field in a magnetic material. • In a region with mr=4. 6 • Find H and M and susceptability

How to make traffic light go Green when driving a bike or motorcycle • Stop directly on top of • http: //www. wikihow. com/ Trigger-Green-Traffic-Lights induction loop on the street • Attach neodymium magnets to the vehicle • Move on top of loop • http: //www. labreform. org • Push crossing button /education/loops. html • Video detectors

Inductors • If flux passes thru N turn, the total Flux Linkage is • This is proportional to the current • So we can define the inductance as: • The energy stored in the inductor is

When more than 1 inductor *Don’t confuse the Magnetization vector, M, with the mutual inductance!

Self -inductance • The total energy in the magnetic field is the sum of the energies: • The positive is taken if currents I 1 and I 2 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.

Magnetic Energy • The energy in a magnetostatic filed in a linear medium is: • Similar to E field

P. E. 8. 10 solenoid A long solenoid with 2 x 2 cm cross section has iron core (permeability is 1000 x ) and 4000 turns per meter. If carries current of. 5 A, find: • Self inductance per m • Energy per m stored in its field

Example: Calculate selfinductance of coaxial cable We’ll find it in two parts:

Example: (cont. ) coaxial We’ll find it in two parts:

Example: Find inductance for a 2 -wire transmission line • Lin is same as before: • We’ll find Lext from the definition of energy:

Magnetic Circuits • Manetomotive force • Ex: magnetic relays, motors, generators, transformers, toroids • Reluctance • Table 8. 4 presents analogy between magnetic and • Like V=IR electric circuits

8. 42 A cobalt ring (mr=600) has mean radius of 30 cm. • If a coil wound on the ring carries 12 A, calculate the N required to establish an average magnetic flux density of 1. 5 Teslas in the ring.

Force on Magnetic materials Relay • N turns, current I • B. C. B 1 n=B 2 n (ignore fringing) • Total energy change is used to displace bar a distance dl. • S=cross sectional area of core

P. E. 8. 16 U-shape electromagnet • Will lift 400 kg of mass(including keeper + weight) • Iron yoke has mr=3, 000 • Cross section =40 cm 2 • Air gap are 0. 1 mm long • Length of iron=50 cm • Current is 1 A Find number of turns, N Find force across one air gap Esto nos da el B en el air gap.