Lecture 4: Charge in motor Dr. Anyuan Chen

Transcription

1 Lecture 4: Charge in motor Dr. Anyuan Chen Charges in motion (an electrical current) produce a magnetic field Magnetic field B generated by point charge q: B = μ 0 qv r 4π r 2 v r = v sin θ n n is a unit vector perpendicular to the plane containing v and r in the direction given by the right-hand rule μ 0 = 4π 10 7 H/m, permeability of free space θ

2 Force acting on point charge Force exerted by magnetic field B on a point charge Q is: F = qv B Magnetic force acting on a moving charge is always pependicular to it s moving directon, so magnetic force does no work, but changing the charge s moving direction

3 Example: Point charge s movement in constant uniform magnetic filed A constant uniform magnetic field B, a charge q with mass m is shot pependicularly into the magnetic field with speed V(0), what is the radius of charge? Centrifugal force F = mv2 = qvbsinθ (θ is the angle between v and B, here 90 degrees) r r = mv2 qvb

4 Magnetic field intensity Magnetic field intensity H= B, μ is called permeability and material μ dependent, the value of μ for free space is μ 0 = 4π 10 7 H/m. B = μ 0 4π qv r H = 1 qv r 4π r 2 The magnetic field intensity is independent of material property. r 2

5 Biot-savart s law : Magnetic field generated by current The produced magnetic field intensity by a current at certain space point Vector expression: H r = c I r dl R 4πR 3 B r = μ 0 4π න c I r dl R R 3 Idl R = I R dl sin θ n n is a unit vector perpendicular to the plane containing I and B in the direction given by the right-hand rule Scalar calculation H r = න c B r = μ 0 4π න c I r dl sinф 4πR 2 I r dl sinф R 2

6 example: Magnetic flux density db = μ 0 Idl R 4π R 2 B a μ B = 0 Idz a 4π R 2 a μ = Ƹ R 0 Idz a 4πR 2 = R a μ 0 I sin фdz a 4πR 2 sin ф = r r 2 + z 2 R 2 = r 2 + z 2 B = a a μ 0 I sin фdz 4πR 2 a = μ 0Ir 4π a dz (r 2+ z 2 ) 3 = μ 0I 2 2πr Τ 1 r a /2 Asumming a>>r, what is B?

7 Field on Axis of circular loop A ring with radius a and current I, calculate B at point on the z axis.

8 Magnetic flux and flux contiunity Magnetic flux ф isthe integral of the flux density accross surface ф = s B. ds, For en closed surface, the flux is zero B = 0 Theres is no magnetic monopole/ Continuous magnetic field.

9 Magnetic potential Magentic potential is difined based on B = 0 2 A = μj

10 Ampere s law The magnetic field in space around an electric current is proportional to the electric current which serves as its source. Stoke s theorem 1 μ 0 B = H = J In magnetostatic, for exapmle: constant DC current. D t = 0

11 Ampere s law: calculating the magentic field around a conductor B = a a μ 0 I sin фdz 4πR 2 a = μ 0Ir 4π a dz (a 2+ z 2 ) 3 = μ 0I 2 2πr Τ 1 r a /2 Stoke s theorem: න s න s B ds = රBdl = 2πrB(r) ൱ B ds = න s μ 0 J ds = μ 0 I 2πrB r = μ 0 I B r = μ 0I 2πr B = μ 0 J

12 Example: Solenoid න s න s H ds = රHdl = Hl H ds = න s J ds = NI H = NI l B = μh If the core is iron instead of air, the flux density B is much stronger.

13 Example: Magnetic field A coaxial line carrying curent I on the inner conductor and I on the outer. Calculate the magnetic field H at r distance, 1) 0<r<a, 2) a<r<b, 3) b<r<c 4) r>c

14 Magnetic field in material B = μ 0 H in vaccum Once there is magentic field applied to medium, the electronic spin motions in the atoms can be thougth of as corculating current that produce a field M, magnetization, which adds to magnetic field H. B = μ 0 (H + M) Magnetic susceptibillity χ m is used to quantify the addtional field M. B = μ χ m H = μ 0 μ r H = μh μ r is around 5000 for iron. Bad magnetic material relative permeability Silver:1 Copper 1 Gold: 1 Aluminium 1 Good magnetic materail relative permeability Iron alloy :

15 Field in magnetic material Magnetic material can be used to guide magnetic field path.

16 Permanent magnet Some materials after magnetization, the electron movement can remain when external magnetic field dispears. The electron movement can be distroted at high temperatures, or strong opposite external field.

17 Magnetic circuit: MMF: magnetomotive force: NI, HL R m = l μs The magnet (Hl) or current (NI) possesses a magneto motive force. The m.m.f generates a magnetic flux. The flux exists within the magnet and the air gap between the poles. The enclosed flux path is called a magnetic circuit. A stronger m.m.f. will produce more flux. The lower the reluctance of the magnetic circuit, the more flux will be produced

18 Magnetic circuit and electrical circuit

19 Magnetic material: B-H curve

20 Boundary condition for static magentic field 1 μ 0 B = H = J