From Stokes'theoremwehavef B d/ = I,(V×B) dsAnd considering I-{J dsThen, from Ampere's circuitallaw we have[.(V×B- HoJ)-dS = 0Since the above equation holds for any surface, the integrandshould bezero, leadingtoV×B=μoJwhich states that the curl of the magnetic flux density of a steadymagnetic field at a pointin vacuumis equal to the product of thecurrent density at the point and the permeability of vacuum.U7
From Stokes’ theorem we have = l S B dl ( B) dS And considering = S I J dS ( − 0 )d = 0 S B J S Then, from Ampere’s circuital law we have Since the above equation holds for any surface, the integrand should be zero, leading to B = 0 J which states that the curl of the magnetic flux density of a steady magnetic field at a point in vacuum is equal to the product of the current density at the point and the permeability of vacuum
From the divergence theorem we havefB ds =J, V.BdVConsidering B.ds = O, we obtain[,V·BdV =0Since the equation holds everywhere, the integrand should be zero,i.e.V.B=0which states that the divergence of the magnetic flux density of a steadymagneticfieldis equal to zero everywhereConsequently, we find the differential form of the equations for thesteady magnetic field in vacuum asVxB=μoJV-B=0The steady magnetic field in vacuumis a solenoidal field.u7
From the divergence theorem we have = S V B dS BdV = V BdV 0 B = 0 Since the equation holds everywhere, the integrand should be zero, i.e. which states that the divergence of the magnetic flux density of a steady magnetic field is equal to zero everywhere. Consequently, we find the differential form of the equations for the steady magnetic field in vacuum as B = 0 J B = 0 The steady magnetic field in vacuum is a solenoidal field. Considering , we obtain d = 0 S B S
Based on Helmholtz's theorem, the magnetic flux density BshouldbeB(r) = -V@(r)+ V× A(r)WhereV'xB(r)".B(rav"dyA(r)=^D(r) =r1Considering VxB = μ.J , V.B=O we haveJroA(r)=@(r) = 04元andB(r) = V× A(r)which shows that the magnetic flux density of a steady magnetic fieldat a pointin vacuum is equal to the curl of the vectorfunctionA at thepoint.U
Based on Helmholtz’s theorem, the magnetic flux density B should be B(r) = −(r) + A(r) V V − = d ( ) 4π 1 ( ) r r B r r V V − = d ( ) 4π 1 ( ) r r B r A r Where (r) = 0 V V − = d ( ) 4π ( ) 0 r r J r A r which shows that the magnetic flux density of a steady magnetic field at a point in vacuum is equal to the curl of the vector function A at the point. Considering , we have B = 0 J B = 0 and B(r) = A(r)
If the distribution of the currentis known, the vector magneticpotentialA at a point can be found, and we can calculate the magneticflux density at the point.The relationship between the magnetic flux density and thecurrentisJ(r)x(r-r)B(r)= Ho dvr-r4元which is called the Biot-Savart'slaw.The current can be distributedin a volume,on a surface, orin aline,and they are called volume current, surface current, and linecurrent,respectivelyAn equivalentrelation among these currents is asJdV = JdS = IdlWhere J, is the surface current density (A/m), and the direction of dlis theflow directionofthe linecurrentU7
If the distribution of the current is known, the vector magnetic potential A at a point can be found, and we can calculate the magnetic flux density at the point. V V − − = d ( ) ( ) 4π ( ) 3 0 r r J r r r B r which is called the Biot-Savart’s law. The current can be distributed in a volume, on a surface, or in a line, and they are called volume current, surface current, and line current, respectively. JdV J dS Idl = S = The relationship between the magnetic flux density and the current is An equivalent relation among these currents is as Where JS is the surface current density (A/m), and the direction of dl is the flow direction of the line current
The vector magnetic potentials and the magnetic flux densitiescaused by a surface current and a line current are, respectivelyJs(r)x(r-r)40-禁道B(r)= 4o[dsdsr-r4元JsIdl'x(r-r)PorB(r) = 44元-For some steady magnetic fields, it will be simple to calculate themagnetic flux density based onAmperes circuitallaw.f, B. dl = μo IFor this, we need to find a closed curve along which the magnitude ofthe magnetic flux densityis constanteverywhere,and the directioncoincideswiththe tangentialdirectionof the curve.Thenthevectorintegral becomes a scalarintegral, B can be taken out of the integral,anditcanbedetermined凹√拉
The vector magnetic potentials and the magnetic flux densities caused by a surface current and a line current are, respectively S S S − = d ( ) 4π ( ) 0 r r J r A r S S S − − = d ( ) ( ) 4π ( ) 3 0 r r J r r r B r − = l r r l A r d 4π ( ) 0 I − − = l r r l r r B r 3 0 d ( ) 4π ( ) I For some steady magnetic fields, it will be simple to calculate the magnetic flux density based on Ampere’s circuital law. For this, we need to find a closed curve along which the magnitude of the magnetic flux density is constant everywhere, and the direction coincideswith the tangential direction of the curve. Then the vector integral becomes a scalar integral, B can be taken out of the integral, and it can be determined. I l d = 0 B l