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 la in circuit a produces a magnetic field and a flux linkage Mab in the circuit b is Mab is proportional to la ab Similarly, a current Ib in b also produces a flux link age Aab in a Aba= Mbale lt can be shown that ab=M for any shape of a and b
The factor of proportionality M=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 d pr roduced one circuit is related to the current in the other circuit by the right-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 uon'la Its flux linking for each turn in the coil b is ab=10Nln·丌R2 and its total flux linking in the coil b is ∧ab=NbΦab=10丌B2 N'Nbla The mutual inductance is M=△ab/la=10xR2NNb uoR NaNb/la
12.2 Induced Electromotance in Terms of mutual Inductance a change in Ia induces the electromotance in b d b dt This equation is convenient to calculate vb, since M and dIa/dt are easy to measure Note that we can write aA D=fbE·d=fb(一)·dl, at where a is the vector potential at b produced by 1 Similarly, a change in Ib induces the electromotance In 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
Explanation The inner solenoid a produces the vector potential A outside,(even though B=VXA=0 and thus a change in la gives aA/at outside This gives rise to e outside E-I OA OA ot ot and an electromotance b=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 Vb=-N C borR NNb dt On the other hand aA OA 1= (-x+)·dl=(-x).2RN at at Thu 0A1 coRral at le vector potential is A= HORN C whose direction is azimuthal as the inner current i
