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