中国科学技术大学：《电磁学》课程教学资源（PPT课件讲稿）Chapter 12 Magnetic Fields：V

Chapter 12 Magnetic Fields: V Mutual induction m a Induced electromotance in terms of Mutual inductance Self-Inductance l Coefficient of Coupling a Transients in rc circuits

Chapter 12 Magnetic Fields:V ◼ Mutual Induction M ◼ Induced Electriomotance in Terms of Mutual Inductance ◼ Self-Inductance L ◼ Coefficient of Coupling ◼ Transients in RC Circuits

12.1 Mutual Inductance m See fig 12-1

The current Ia in circuit a produces a magnetic field and a flux linKage a6 In the circuit b This Aab is proportional to la ab Similarly, a current Ib in b also produces a flux link- age /ab in a ba Mbale It can be shown that Mab= Mb For any shape of a and b

The factor of proportionality M called the mutual inductance btwn the two circuits Remark M has the unit weber/ampere= henrys M depends on the geometrical configurations of the two circuits, including their sizes, shape, orienta tions, positions M is positive if the flux pr roduced one circuit is related to the current in the other circuit by the ht-hand rule. For instance, Fig. 12-1 gives a pos itive m the directions for the currents are chosen arbitrarily

Example: M Between T wo Coaxial Solenoids See Fig 12-2 The number of turns per meter is N the same for both solenoids Figure 12-2 Coaxial solenoids. The two radii are taken to be approximately equal

The coil a produces a magnetic field uoNla Its flux linking for each turn in the coil b ab=p0Na·丌B and its total flux linking in the coil b is ab=NbΦab=07B2N"Nbla The mutual inductance is TRNM wor- NaNb/la

12.2 Induced electromotance in Terms of mutual Inductance a change in Ia induces the electromotance in b ab This equation is convenient to calculate vh, since M and dIa/dt are easy to measure Note that we can write aA Db=fbE·d1=fb(-+)·dl, where a is the vector potential at b produced by la Similarly, a change in Ib induces the electromotance n a

Example: The Vector Potential of The Inner solenoid The inner solenoid produces b=0 ouside itself, but a change in la induces an electromotance in the outer solenoid b

Ex xplanation: The inner solenoid a produces the vector potential A outside, (even thoughB-VXA-0 and thus a change in Ia gives aA/ at outside This gives rise to e outside da aA E=-VV at at and an electromotance b=力E·dl in the outer solenoid b Calculatⅰon

Calculation: Since the mutual inductance m is known the elec tromotance induced in the outer solenoid b is dl MOT R N NE On the other hand OA OA at at). 2T RNb Thus 0A1 RN at (0 le vector potential is A RNl whose direction is azimuthal as the inner current i