Chapter6ElectromagneticInductionLaw ofElectromagnetic InductionInductancesEnergy and Force1.Law of ElectromagneticInduction2.Inductances3. Energy in Steady Magnetic Fields4. Magnetic Forces
Chapter 6 Electromagnetic Induction Law of Electromagnetic Induction Inductances Energy and Force 1. Law of Electromagnetic Induction 2. Inductances 3. Energy in Steady Magnetic Fields 4. Magnetic Forces
1.LawofElectromagneticInductionFrom physics we know that when the magnetic flux @through aclosed coil is changing, an induced electromotiveforce e willbegenerated in the coil, with the relationdddtwhere the positive direction of the electromotive force e and that of themagnetic flux comply with theleft hand rule.If the magnetic flux is increased with time, the direction of theinduced electromotiveforceand that of the magnetic flux obeythe lefthand rule ifthe magnetic flux is decreased with time, they will obey theright handrule
1. Law of Electromagnetic Induction From physics we know that when the magnetic flux through a closed coil is changing, an induced electromotive force e will be generated in the coil, with the relation t e d d = − where the positive direction of the electromotive force e and that of the magnetic flux comply with the left hand rule. If the magnetic flux is increased with time, the direction of the induced electromotive force and that of the magnetic flux obey the left hand rule. if the magnetic flux is decreased with time, they will obey the right hand rule
The induced magnetic flux caused by theinduced current in the coil always resists thechange of the original magnetic flux. Theinduced magnetic flux is called the reactionmagnetic flux, and the induced electromotiveforceis called the backelectromotiveforce.When the induced electromotiveforceis generated in the coil.there is an electric field to push the charges to move in the coil, andthis induced electricfieldis denoted as EThe lineintegral of theinduced electricfield intensityaroundthe closed coil is equal to the induced electromotiveforcein the coile, i.e.d@fE-dl=e -dtUV
e I The induced magnetic flux caused by the induced current in the coil always resists the change of the original magnetic flux. The induced magnetic flux is called the reaction magnetic flux, and the induced electromotive force is called the back electromotive force. When the induced electromotive force is generated in the coil, there is an electric field to push the charges to move in the coil, and this induced electric field is denoted as E. t e l d d d = = − E l The line integral of the induced electric field intensity around the closed coil is equal to the induced electromotive force in the coil e, i.e
Considering @= [B.ds, we havefedl--%[B.dsatWhichis called the lawof electromagneticinduction,and it shows thatwhen the magnetic field through a closed coilis changing, an inducedelectric field will be generated in the coilThe law of electromagnetic induction shows that a time-varyingmagnetic field can produce a time-variableelectric fieldBased on Stokes'theorem, fromthe above eguation we haveaB(V×E)+dS = 0atSince the equation holds for any area S, the integrand must bezero, so thataBVxE=atUEV
Considering , we have = S B dS = − l S t E dl B dS Which is called the law of electromagnetic induction, and it shows that when the magnetic field through a closed coil is changing, an induced electric field will be generated in the coil. The law of electromagnetic induction shows that a time-varying magnetic field can produce a time-variable electric field. Based on Stokes’ theorem, from the above equation we have ( ) d 0 = + S B E S t Since the equation holds for any area S, the integrand must be zero, so that t = − B E
aBVxE-atwhich is called the differential form of law of electromagneticinduction, and it means that the negative time rate of change ofthe magnetic flux density at a point is equal to the curl of the timevariableelectricfieldintensityatthatpointThe law of electromagnetic induction is one of basic laws fortime-varying electromagnetic fields, and it is also one of Maxwell'sequations.U7
which is called the differential form of law of electromagnetic induction, and it means that the negative time rate of change of the magnetic flux density at a point is equal to the curl of the timevariable electric field intensity at that point. The law of electromagnetic induction is one of basic laws for time-varying electromagnetic fields, and it is also one of Maxwell’s equations. t = − B E
2.InductancesIn a linear medium, the magnetic flux @through the closed circuitis alsoproportionaltothe currentI.The magnetic flux linked with the current I is called the magneticflux linkage with the current I, and it is denoted as Y. The ratio of ytoIis denoted byL,henceyL.1It is called the inductance of the circuit, with the unit henry (H)and the inductance can be also considered as the magnetic flux linkageper unit current.In linear media, the inductance of a circuit depends only on theshape and the sizes,but not on the current.The magnetic flux linkage is different from the magnetic flux, anditisassociatedwithacurrentU
2. Inductances In a linear medium, the magnetic flux through the closed circuit is also proportional to the current I. I L = It is called the inductance of the circuit, with the unit henry (H), and the inductance can be also considered as the magnetic flux linkage per unit current. In linear media, the inductance of a circuit depends only on the shape and the sizes, but not on the current. The magnetic flux linked with the current I is called the magnetic flux linkage with the current I, and it is denoted as . The ratio of to I is denoted by L, hence The magnetic flux linkage is different from the magnetic flux, and it is associated with a current
If the magnetic flux is linked with a current N times, then themagnetic flux linkage will be increased by N times. If only a part of themagnetic flux is linked to a current, the magnetic flux linkage must bereducedproportionatelyA loop coil with N turns the magnetic flux linkage with the currentis y =N@,and theinductanceof theloopcoil withNturnsisNd下111Ii1212Supposewe havetwo loopcurrents.athe magnetic flux linkage P, linked withr2-r1dl2currentI,consistsoftwoparts:oneisr2ygenerated by the magnetic flux caused byOcurrent I, itself, andit is denoted as Y,Another Y 2 is produced by the magnetic flux at loopl, by current I,uK
If the magnetic flux is linked with a current N times, then the magnetic flux linkage will be increased by N times. If only a part of the magnetic flux is linked to a current, the magnetic flux linkage must be reduced proportionately. I N I L = = Suppose we have two loop currents, the magnetic flux linkage 1 linked with current I 1 consists of two parts: one is generated by the magnetic flux caused by current I 1 itself, and it is denoted as 11 . dl1 O z y x dl2 l2 l1 I2 I1 r2 - r1 r2 r1 A loop coil with N turns the magnetic flux linkage with the current is = N , and the inductance of the loop coil with N turns is Another 12 is produced by the magnetic flux at loopl 1 by current I 2
Hence, the magnetic flux linkage Y linked with current I isYl =4 +4l2The magnetic flux linkage P, linked with current I, is4, =421 + 422n4nIf the surrounding medium is linear, then all the ratios.422andare independent of the currents since all the magnetic7Pflux linkages are proportional to the current generating them.Yi2PLetM12 =12where Ln is called the self-inductance of loop li, and M, is called themutualinductancefromloopl,to loop l42Similarly,wedefineM.011,1where L22 is called the self-inductance of loop l2, and M2, is called themutualinductance fromloop l, to loopl2u7
The magnetic flux linkage 2 linked with current I 2 is 2 =21 +22 If the surrounding medium is linear, then all the ratios, , , and are independent of the currents since all the magnetic flux linkages are proportional to the current generating them. 1 11 I 2 12 I 2 22 I 1 21 I where L11 is called the self-inductance of loop l 1 , and M12 is called the mutual inductance from loop l 2 to loop l 1 . Similarly, we define 2 22 22 I L = 1 21 21 I M = where L22 is called the self-inductance of loop l 2 , and M21 is called the mutual inductance from loop l 1 to loop l 2 . Hence, the magnetic flux linkage 1 linked with current I 1 is 1 =11 +12 1 11 11 I L = 2 12 12 I M Let =
Substitute the above parameters Lu, L22 , M2 , and M2, intothe above equation, we haveY, = M21l, + L2212Y = L.I, + M1212Forlinearhomogeneous media, we can prove thatM12 = M21Since we can find the mutualinductances between any two loopcircuits as follows:dl, :dldl,.dlMa-M12 =r-r1J124元Consideringdl, dl, = dl, dl, 2 -rl=|r -rl , we haveM12 = M21u√
Substitute the above parameters L11，L22，M12 ，and M21 into the above equation, we have 1 11 1 12 2 = L I + M I 2 21 1 22 2 = M I + L I For linear homogeneous media, we can prove that M12 = M21 Since we can find the mutual inductances between any two loop circuits as follows: − = 2 1 2 1 1 2 21 d d 4π l l M r r l l − = 1 2 1 2 2 1 12 d d 4π l l M r r l l Considering ，we have 1 2 2 1 2 1 1 2 dl dl = dl dl , r −r = r −r M12 = M21
dl, dldl, -dlM-IJMo-司If dl, I dl, everywhere, the mutual inductances be zeroIf dl, /l dl, everywhere, the mutual inductances will be maximumIn electronic devices,if we need to increase the magnetic couplingbetween two coils,the two coils should be placed parallelto each other.If the magnetic coupling needs to be eliminated, they should beperpendiculartoeach other.The mutualinductance could be positive or negative, while theself-inductanceis always positiveU7
− = 2 1 2 1 1 2 21 d d 4π l l M r r l l − = 1 2 1 2 2 1 12 d d 4π l l M r r l l In electronic devices, if we need to increase the magnetic coupling between two coils, the two coils should be placed parallel to each other. If the magnetic coupling needs to be eliminated, they should be perpendicular to each other. The mutual inductance could be positive or negative, while the self-inductance is always positive. If everywhere, the mutual inductances be zero. d 1 d 2 l ⊥ l If everywhere, the mutual inductances will be maximum. 1 d 2 dl // l