Magniticfield 1
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  • 3. Magnetic Field • A magnetic field is a region in which a body with magnetic properties experiences a force.
  • 4. Sources of Magnetic Field • Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscope currents associated with electrons in atomic orbits.
  • 5. Magnetic Field Lines • A magnetic field is visualised using magnetic lines of force which are imaginary lines such that the tangent at any point gives the direction of the magnetic field at that point.
  • 6. Magnetic Flux Pattern
  • 7. The Earth’s Magnetic Field • The Earth's magnetic field appears to come from a giant bar magnet, but with its south pole located up near the Earth's north pole.
  • 8. Magnetic flux pattern due to current in a straight wire at right angles to a uniform field Net flux is lesser on this side of the wire Net flux is greater on this side of the wire I
  • 9. Fleming’s Left Hand Rule • If you point your left forefinger in the direction of the magnetic field, and your second finger in the direction of the current flow, then your thumb will point naturally in the direction of the resulting force!
  • 10. Magnetic Force θ θ sin sin qv F B qvBF ≡ = • A magnetic force acts on a charge moving in a magnetic field. • Use to define magnetic field B. Direction of Force (Conductor in Field) B I F • The right-hand rule can be used to obtain the direction of the force on a conductor placed in a magnetic field.
  • 11. Helical Path of Charged Particle qB mv r = • Charged particle can undergo circular motion perpendicular to magnetic field. • Velocity component parallel to field results in helical motion. • Example: earth’s radiation belts. Field of a Long Straight Wire r I B π µ 2 0 = + B I • Magnetic field of a long straight wire depends on current and radial distance.
  • 12. Magnetic Field of a Point Charge It is found that the magnetic field is perpendicular to both the velocity of the charge and the unit vector from the charge to the point in question The magnitude of the field is given by 2 0 sin 4 r vq B φ π µ = where φ is the angle between the velocity and unit vector The magnetic field lines form concentric circles about the velocity vector
  • 13. Force on a current-carrying conductor • The direction of magnetic force always perpendicular to the direction of the magnetic field and the direction of current passing through the conductor. →→→ ×= BIF  θsinIF = magnetic field: FB = qv × B = |q|vB sin θ
  • 14. Magnetic Flux Density • The magnetic flux density is defined as the force per unit length per unit current acting on a current-carrying conductor at right angle to the field lines. I F B = Unit : tesla (T) or gauss (G), 1 G = 10-4 T or weber/m2
  • 15. Example: Helical particle motion Proton (m=1.67*10-27 kg) moves in a uniform magnetic field B=0.5 T directed along x-axis at t=0 the proton has velocity components 5 5 1.5 10 / 0 2 10 / x y z v m s v v m s = × = = × a) At t=0 find the force on the proton and its acceleration. b) find the radius of the helical path and the pitch of the helix zv v⊥ = z zF qv B qv B j ∧ = × = 12 2 9.58 10 / F a m s m = = × 7 7 4.18 4.79 10 / 2 1.31 10 s 19.7 z x mv R mm q B q B rad s m T pitch v T mm ω π ω − = = = = × = = × = =
  • 16. The Earth’s Magnetosphere Earth can be viewed as a gigantic bar magnet spinning in space. • Its toroidal magnetic field encases the planet like a huge inner tube. • This field shields Earth from the solar wind—a continuous stream of charged particles cast off by the sun.  Produces a bullet-shaped cavity called the magnetosphere.  It also traps charged particles – leads to the radiation or Van Allen belts.
  • 17. The Earth’s Radiation Belts Particles are trapped by the non-uniform geomagnetic field – much like a magnetic bottle – they bounce back and forth from one hemisphere to another. The trapped particles tend to congregate in distinct bands based on their charge, energy, and origin. Two primary bands of trapped particles exist: the one closer to Earth is predominantly made up of protons, while the one farther away is mostly electrons.
  • 18. Magnetic Field Measurements • Using a current balance (d.c.) • Using a search coil (a.c.) • Using a Hall probe (d.c.)
  • 19. Magnetic flux density due to a straight wire • Experiments show that the magnetic flux density at a point near a long straight wire is r I B ∝ r P וTThis relationship is valid as long as r, the perpendicular distance to the wire, is much less than the distance to the ends of the wire.
  • 20. Calculation of B near a wire r I B o π µ 2 = )m(HAmT104 -1-17− ×= πµo Where µo is called the permeability of free space. Permeability is a measure of the effect of a material on the magnetic field by the material.
  • 21. Magnetic Field due to a Solenoid • The magnetic field is strongest at the centre of the solenoid and becomes weaker outside.
  • 22. Crossed magnetic and electric fields r F=qE+q r vx r B y
  • 23. Magnetic Flux Density due to a Solenoid • Experiments show that the magnetic flux density inside a solenoid is IB ∝  N B ∝and So we have  NI B oµ = or nIB oµ= where  N n =
  • 24. Variation of magnetic flux density along the axis of a solenoid • B is independent of the shape or area of the cross-section of the solenoid. • At a point at the end of the solenoid, nB oµ 2 1 '= B Distance from the centre of the solenoid0 nIB oµ= nB oµ 2 1 '= 2 1 2 1 −
  • 25. Magnetic Flux Density due to Some Current-carrying conductors(1) • Circular coil r NI B o 2 µ = • Helmholtz coils r NI r NI B oo µµ 72.0 125 8 ≈=
  • 26. Magnetic Flux density due to Some Current-carrying Conductors (2)
  • 27. Force on a moving charge in a magnetic field • The force on a moving charge is proportional to the component of the magnetic field perpendicular to the direction of the velocity of the charge and is in a direction perpendicular to both the velocity and the field. θsinqvBF = BvqvBF ⊥= formax BvF //for0=
  • 28. Right Hand Rule • Direction of force on a positive charge given by the right hand rule. →→→ ×= BvqF
  • 29. Free Charging Moving in a Uniform Magnetic Field • If the motion is exactly at right angles to a uniform field, the path is turned into a circle. • In general, with the motion inclined to the field, the path is helix round the lines of force.
  • 30. Mass Spectrometer • The mass spectrometer is used to measure the masses of atoms. • Ions will follow a straight line path in this region. • Ions follow a circular path in this region. qvBqE = r mv qvB 2 '=
  • 31. Aurora Borealis (Northern Lights) • Charged ions approach the Earth from the Sun (the “solar wind” and are drawn toward the poles, sometimes causing a phenomenon called the aurora borealis.
  • 32. Causes of Aurora Borealis • The charged particles from the sun approaching the Earth are captured by the magnetic field of the Earth. • Such particles follow the field lines toward the poles. • The high concentration of charged particles ionizes the air and recombining of electrons with atoms emits light.
  • 33. Hall Effect • When a current carrying conductor is held firmly in a magnetic field, the field exerts a sideways force on the charges moving in the conductor. • A buildup of charge at the sides of the conductor produces a measurable voltage between the two sides of the conductor. • The presence of this measurable transverse voltage is called the Hall effect.
  • 34. Hall Voltage • The transverse voltage builds up until the electric field it produces exerts an electric force on the moving charges that equal and opposite to the magnetic force. • The transverse voltage produced is called the Hall voltage.
  • 35. Charge Carriers in the Hall Effect • The Hall voltage has a different polarity for positive and negative charge carriers. • That is, the Hall voltage can reveal the sign of the charge carriers.
  • 36. Hall Probe • Basically the Hall probe is a small piece of semiconductor layer. • When control current IC is flowing through the semiconductor and magnetic field B is applied, the resultant Hall voltage VH can be measured on the sides • Four leads are connected to the midpoints of opposite sides.
  • 37. Force between two parallel current-carrying straight wires (1) 1. Parallel wires with current flowing in the same direction, attract each other. 2. Parallel wires with current flowing in the opposite direction, repel each other.
  • 38. Force between two parallel current-carrying straight wires (2) • Note that the force exerted on I2 by I1 is equal but opposite to the force exerted on I1 by I2. a II F o π µ 2 21  =
  • 39. Definition of the ampere • The ampere is the constant current which, if maintained in two parallel conductors of infinite length, of negligible cross-section, and placed one metre apart in a vacuum, would produce between these conductors force of 2 x 10- 7 N per metre of length.
  • 40. Torque on a Rectangular Current- carrying Coil in a Uniform Magnetic Field • Let the normal to the coil plane make an angle θ with the magnetic field. • The torque τ is given by θτ sinNBAI=
  • 41. Moving Coil Galvanometer • A moving coil galvanometer consists of a coil of copper wire which is able to rotate in a magnetic field. • The magnetic field is produced in the narrow air gap between concave pole pieces of a permanent magnet and a fixed soft-iron cylinder. • The coil is pivoted on jewelled bearings and its rotation is resisted by a pair of spiral hair springs.
  • 42. Radial Magnetic Field • In order to have a meter with a linear scale, the field lines in the gap should be always parallel to the plane of the coil as it rotates. • This could be achieved if we have a radial magnetic field. The soft iron cylinder gives us this field shape.
  • 43. The Principle of a Moving Coil Galvanometer • The torque due to the current in the coil is given by NBAI=τ • The coil rotates until • The resisting couple due to the hair springs is ατ k=' 'ττ = • Then we have k NBAI =α Where α is the angle of deflection and k is the torsion constant. Iei ∝α..
  • 44. Current Sensitivity • The current sensitivity of a galvanometer is defined as the deflection per unit current. k NBA I = α • High current sensitivity can be achieved by • A coil of large area, • A coil of large number of turns, • Large value of B which could be achieved by using strong magnet and narrow air gap. • Hair springs with small torsion constant k. Unit : Rad A-1 or mm A-1
  • 45. Limitation of High Current Sensitivity • If the coil is too large, the moment of inertia is also large and hence the coil would swing about its deflection position before a reading can be taken. • If the coil has a large number of turns, the air gap needs to be wide. • If the hair springs have small torsion constant, the restoring torque would become weak and the coil would swing before coming to rest.
  • 46. Voltage Sensitivity • The voltage sensitivity of a galvanometer is defined as the deflection per unit voltage across the galvanometer. kR NBA V = α Unit : rad V-1 or mm V-1 Where R is the resistance of the galvanometer. • High voltage sensitivity is desirable in circuits of relatively low resistance.
  • 47. Example: A point charge q = 1 mC moves in the x direction with v = 108 m/s. It misses a mosquito by 1 mm. What is the B field experienced by the mosquito? 108 m/s 90o rˆ 2 0 4 r v qB π µ = 26 83 2 7 10 1 101010 m CB s m A N − −− ×××= TB 4 10=
  • 48. To find the E field of a charge distribution use: Use: 2 ˆ r rkdq Ed =  To find the field of a current distribution use:B  Note vdq dt ds dqds dt dq ids  === 2 0 ˆ 4 r rsid Bd × =  π µ
  • 49. Topic: Biot – Savart Law Use to find B field at the center of a loop of wire. 2 0 ˆ 4 r rlid B × = ∫  π µ R i Loop of wire lying in a plane. It has radius R and total current i flowing in it. First find rld ˆ×  rld ˆ×  is a vector coming out of the paper at the same angle anywhere on the circle. The angle is constant.rˆ rˆ ld  ld  R R i dl R i R idl dBB π π µ π µ π µ 2 444 2 0 2 0 2 0 ==== ∫∫∫ R i B 2 0µ = k R i B ˆ 2 0µ =  Magnitude of B field at center of loop. Direction is out of paper. R kˆ i
  • 50. Example: Find magnetic field inside a long, thick wire of radius a Cross-sectional view Path of integral dl a r rBdlBBdlldB r π π 2 2 0 ===⋅ ∫ ∫∫  CIurB 02 =π IC = current enclosed by the path i a r i a r IC 2 2 2 2 == π π r i a r B π µ 2 1 2 2 0= 2 0 2 a ir B π µ = Example: Find field inside a solenoid. See next slide.
  • 51. Toroid ∫ =⋅ CIldB 0µ  ∫ = NiBdl 0µ NirB 02 µπ = r Ni B π µ 2 0 = a < r < b a b ld  BdlldB =⋅  10cos =o Tokamak Toroid at Princeton I = 73,000 Amps for 3 secs N is the total number of turns
  • 52. Torque on a Current Loop It is straightforwardto see how this force is turned into a torque about the axis of a motor. Consider a rectangularloop of wire which lies in a uniform magnetic field created by a magnet: FB = i L x B There is no force on the ends of the rectangularwire, sincethe current is eitherparallel or anti-parallelto the field so the cross product between L and B is zero. The segments along the length L of the rectangular loop are perpendicularto the field. Relative to the magnetic field, the current runs in opposite directions through the top and the bottom sides. Thus, the force on these segmentsis equal and opposite - there is no net force on the loop. There is a torque, however, as clearlyillustrated in the end view on the right above. In a motor, this torque is transmitted to the shaft and a commutator (not shown) reverses the direction of current every ½ revolution so the torque acts in the same direction. N N S S Copyright © 2009 Pearson Education, Inc. Chapter 27 Magnetism Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley PowerPoint® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman Lectures by James Pazun Chapter 28 Sources of Magnetic Field Chapter 29. Magnetic Field Due to Currents 29.1. What is Physics? 29.2. Calculating the Magnetic Field Due to a Current 29.3. Force Between Two Parallel Currents 29.4. Ampere's Law 29.5. Solenoids and Toroids 29.6. A Current-Carrying Coil as a Magnetic Dipole Dr. Jie Zou PHY1361 1 Chapter 29 Magnetic Fields Physics 1304: Lecture 12, Pg 1 The Laws of Biot-Savart & Ampere × dl I Ch. 32 – The Magnetic Field Adobe Acrobat Document Adobe Acrobat Document Module 12 – The Magnetic Field Serway & Faughn Chapters 19 & 20 Fall 2008Physics 231 Lecture 8-1 Sources of the Magnetic Field WHICH YOU WANT PRESS L_CLICK
  • 53. Power in Electric Circuits A battery sets up a current in a circuit containing an unspecified conducting device The battery maintains a voltage V across its own terminals, and thus across the terminals of the unspecified device. The amount of charge dq that moves between those terminals in time interval dt is equal to i dt. This charge moves through a potential decrease in magnitude by the amount The moving charge loses potential at rate of dU/dt = i V The power associated with this transfer of energy is the rate of transfer, which is The unit of power that follows Other forms