Chapter5 SteadyMagneticFieldsMagnetic Flux Density,Field EquationsBoundary Conditions1. Magnetic Flux Density, Flux, and Field Lines2.Eguations for Steady Magnetic Fields in Free Space3.Vector&ScalarMagnetic Potentials4.MagnetizationofMedia5.EguationsforSteadyMagneticFieldsinAMedium6.Boundary Conditionsfor Steady Magnetic FieldsV
Chapter 5 Steady Magnetic Fields Magnetic Flux Density, Field Equations Boundary Conditions 1. Magnetic Flux Density, Flux, and Field Lines 2. Equations for Steady Magnetic Fields in Free Space 3. Vector & Scalar Magnetic Potentials 4. Magnetization of Media 5. Equations for Steady Magnetic Fields in A Medium 6. Boundary Conditions for Steady Magnetic Fields
1.MagneticFluxDensity,Flux,andFieldLinesA magnetic field exerts a force on a moving charge. Hence, theforce acting on the moving charges, the current element, or thetorque acting on a small current loop can be used to quantify themagneticfields.Experiments show that the magnetic force acting on a movingcharge is related not only to the magnitude and the speed of thecharge, but also to the direction of motion
1. Magnetic Flux Density, Flux, and Field Lines A magnetic field exerts a force on a moving charge. Hence, the force acting on the moving charges, the current element, or the torque acting on a small current loop can be used to quantify the magnetic fields. Experiments show that the magnetic force acting on a moving charge is related not only to the magnitude and the speed of the charge, but also to the direction of motion
The magnetic force will be maximum when the charge is movingalong a certain direction, and will be zero when the motionisperpendicular to it. We define the direction in which the force is zeroas the in-line direction, as shown in the following figure.Assuming the maximum forceis Fm,ifthe angle between the direction of chargeIn-linemotion and the in-line directionis α, theBDirectionforcewill beFFm sin αThe magnitude of the force F is proportional to the product ofthe magnitude of the charge g and the magnitude of the velocity yThis forceis called LorentzforceV
F B v In-line Direction The magnetic force will be maximum when the charge is moving along a certain direction, and will be zero when the motion is perpendicular to it. We define the direction in which the force is zero as the in-line direction, as shown in the following figure. Assuming the maximum force is Fm , if the angle between the direction of charge motion and the in-line direction is , the force will be Fm sin The magnitude of the force F is proportional to the product of the magnitude of the charge q and the magnitude of the velocity v. This force is called Lorentz force
We define a vector B whose magnitude is Fm with the directionbeing the in-line direction.The relationship between the vector Bthe charge g, the velocity v, and the force F isF = qv× B0In-lineWherevectorBis calledmagnetic fluxBDirectiondensity, and the unit is tesla ( T ).FLorentz force is always perpendicularto the direction of chargemotion. Consequently, the Lorenzforce can only change the directionof the charge in motion and there is no work done in this actionU7
We define a vector B whose magnitude is with the direction being the in-line direction. The relationship between the vectorB, the charge q, the velocity v, and the force F is qv Fm F = qv B Where vector B is called magnetic flux density, and the unit is tesla ( T ). Lorentz force is always perpendicularto the direction of charge motion. Consequently, the Lorenz force can only change the direction of the charge in motion and there is no work done in this action. In-line Direction F B v
The current elementis a segment of current-carrying wire.Themagnitude of the line element vector dl stands for the length of thecurrent element I, and the direction is that of the current IIfthe currentflowingin the currentelementIdlBis I,thendldq dlIdl =dg = vdqdtdtAnd the force F acting on the current element in a magnetic fieldwith magnetic fluxdensityBisF = Idl× Bif the currentis parallelto the magnetic flux density B, the force willbe zero. If it is perpendicularto B, the force is maximumThe direction of the magnetic force on a currentis always perpen-diculartothe direction ofthecurrentflowUV
The current element is a segment of current-carrying wire. The magnitude of the line element vector dl stands for the length of the current element I , and the direction is that of the current I. F B Idl If the current flowing in the current element is I,then q q t t q I d d d d d d d d v l l = l = = And the force F acting on the current element in a magnetic field with magnetic flux density B is F = Idl B if the current is parallel to the magnetic flux density B, the force will be zero. If it is perpendicularto B, the force is maximum. The direction of the magnetic force on a current is always perpendicular to the direction of the current flow
The torque on a small current loopThe small current loopis a plane square frame with four sides oflength I each, and the direction of flow currentis shown in figure.When viewedfrom a large distance, thecurrent loop may be considered a magneticdHdipole.BSThe magnetic field in the plane of theframe current can be takento be a uniformIf the magnetic flux density B is parallel to the plane of the frame,no force will act on the sides ab and cd, while thedirections of theforces on the sides ad and bc are opposite. The magnitude of thetorque T on the frame currentisT = FI = IIBl = II- B = ISBwhere Sis the area ofthe frameU
The torque on a small current loop. d c a b F F B S When viewed from a large distance, the current loop may be considered a magnetic dipole. T = Fl = IlBl = Il B = ISB 2 where S is the area of the frame. The magnetic field in the plane of the frame current can be taken to be a uniform. The small current loop is a plane square frame with four sides of length l each, and the direction of flow current is shown in figure. If the magnetic flux density B is parallel to the plane of the frame, no force will act on the sides ab and cd, while the directions of the forces on the sides ad and bc are opposite. The magnitude of the torque T on the frame current is
If B is perpendicularto theplane of theB1Fframe, the forces on the four sides aredirected outsideand will cancel each otherFSThe torgue acting on the frame current iszeroIf the angle between the vector B and theB.Rnormal to theplane of theframe is O, thevector B maybe resolvedintotwoBScomponents B, and B,. Then, the magnitudeof the torque T on the current loopisT = ISB, = ISBsin @U7
F d c a b F F F B S d c a b F F B B n Bt F F S If B is perpendicular to the plane of the frame, the forces on the four sides are directed outside and will cancel each other. The torque acting on the frame current is zero. If the angle between the vector B and the normal to the plane of the frame is , the vector B may be resolved into two components Bn and Bt . Then, the magnitude of the torque T on the current loop is T = ISBt = ISBsin
Requiring the direction ofthe directed surface S and the directionof the current to obey the right hand rule, the above equation can bewritteninthefollowingvectorformasT =I(S×B)Itis validforanysmall currentloop.In general, theproductIsiscalledthe magnetic moment of the currentloop, andit is denoted asm, so thatm= ISThe aboveequation can be written asT=mxBwhich states thatif the magnetic moment mis parallelto the magneticflux density B, the torque acting on the frame is zero. If they areperpendicular to each other, the torqueis maximumUV
Requiring the direction of the directed surface S and the direction of the current to obey the right hand rule, the above equation can be written in the following vector form as T = I(S B) It is valid for any small current loop. In general, the product IS is called the magnetic moment of the current loop, and it is denoted as m, so that m = IS The above equation can be written as T = mB which states that if the magnetic moment m is parallel to the magnetic flux density B, the torque acting on the frame is zero. If they are perpendicular to each other, the torque is maximum
The flux of themagneticflux density B througha directed surfaceSis called magnetic flux, andit is denoted as , given byD=JBdsThe unit ofmagneticflux is weber (Wb)The magnetic flux density can also be described using a set of curves.The tangentialdirection at a point on the curve stands for the directionof magneticflux density, and these curves are called magnetic field linesThe vector equation forthe magnetic field line isBxdl = 0The magneticfield lines cannotalso beintersected.As the electric field lines,the density of the magnetic field lines candescribe the intensity of the magnetic field.A larger density of magneticfield linesstandsfor strongermagneticfieldintensityu
The magnetic flux density can also be described using a set of curves. The tangential direction at a point on the curve stands for the direction of magnetic flux density, and these curves are called magnetic field lines. The vector equation for the magnetic field line is Bdl = 0 The magnetic field lines cannot also be intersected. The flux of the magnetic flux density B through a directed surface S is called magnetic flux, and it is denoted as , given by B dS = S The unit of magnetic flux is weber (Wb). As the electric field lines, the density of the magnetic field lines can describe the intensity of the magnetic field. A larger density of magnetic field linesstandsfor stronger magnetic field intensity
2.EquationsforSteadyMagneticFieldsinFreeSpaceThe magnetic flux density B of a steady magnetic fieldin vacuumsatisfiesthefollowingequationsB·dS = 0f, B dl = μo ILeft equation is called Ampere's circuitallaw, where μo is thepermeability of vacuum, μ。 = 4π ×10-7 H/m , and I is the currentenclosed bytheclosed curve.Ampere's circuitallaw:The circulation of the magnetic flux densityin vacuum around a closed curveis egual to the current enclosed bythecurve multiplied by the permeability ofvacuumRight equation shows thatthe totalmagnetic flux through a closedsurfaceis equal to zeroThe magnetic field lines are closed everywhere, with no beginningor end.This may be called the principle of magnetic flux continuityV
2. Equations for Steady Magnetic Fields in Free Space The magnetic flux density B of a steady magnetic field in vacuum satisfies the following equations I l d = 0 B l = S B dS 0 Left equation is called Ampere’s circuital law, where 0 is the permeability of vacuum, H/m , and I is the current enclosed by the closed curve. 7 0 4π 10− = Ampere’s circuital law: The circulationof the magnetic flux density in vacuum around a closed curve is equal to the current enclosed by the curve multiplied by the permeability of vacuum. The magnetic field lines are closed everywhere, with no beginning or end. This may be called the principle of magnetic flux continuity. Right equation shows that the total magnetic flux through a closed surface is equal to zero