Chapter 30 Inductance


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1 Chapter 30 Inductance  Mutual Inductance  SelfInductance and Inductors  MagneticField Energy  The R Circuit  The C Circuit  The RC Series Circuit
2 . Mutual Inductance  A changing current in coil causes B and a changing magnetic flux through coil that induces emf in coil. ε N dφ B Magnetic flux through coil : N B M Φ i Mutual inductance of two coils: M ε dφ B N M M M N Φ i B M is a constant that depends on geometry of the coils M.
3  If a magnetic material is present, M will depend on magnetic properties. If relative permeability (K m ) is not constant (M not proportional to B) Φ B not proportional to i (exception). ε ε M M Mutual inductance: emf opposes the flux change M N Φ i B NΦ i B  Only a timevarying current induces an emf. Units of inductance: Henry Weber/A V s/a J/A Ex. 30.
4 . Self Inductance and Inductors  When a current is present in a circuit, it sets up B that causes a magnetic flux that changes when the current changes emf is induced. enz s law: a selfinduced emf opposes the change in current that caused it Induced emf makes fficult variations in current. NΦ i B Selfinductance dφ N B ε Selfinduced emf
5 Inductors as Circuit Elements Inductors oppose variations in the current through a circuit. In DCcircuit, helps to maintain a steady current (despite fluctuations in applied emf). In AC circuit, helps to suppress fast variations in current.  Reminder of Kirchhoff s loop rule: the sum of potential fferences around any closed loop is zero because E produced by charges stributed around circuit is conservative E c. The magnetically induced electric field within the coils of an inductor is nonconservative (E n ).  If R 0 in inductor s coils very small E required to move charge through coils total E through coils E c + E n 0. Since E c 0 E c E n accumulation of charge on inductor s terminals and surfaces of its conductors to produce that field. E n dl
6  E n 0 only within the inductor. b a b a E E n c dl dl E c E + n 0 (at each point within the inductor s coil)  Selfinduced emf opposes changes in current. Potential fference between terminals of an inductor: Vab Va Vb V ab is associated with conservative, electrostatic forces, despite the fact that E associated with the magnetic induction is nonconservative Kirchhoff s loop rule can be used.  If magnetic flux is concentrated in region with a magnetic material µ 0 in eqs. must be replaced by µ K m µ 0.
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8 3. MagneticField Energy  Establishing a current in an inductor requires an input of energy. An inductor carrying a current has energy stored in it. Energy Stored in an Inductor Rate of transfer of energy into : P Vabi i Energy supplied to inductor during : du P i Total energy U supplied while the current increases from zero to I: U I i 0 I Energy stored in an inductor  Energy flows into an ideal (R 0) inductor when current in inductor increases. The energy is not ssipated, but stored in and released when current decreases.
9 Magnetic Energy Density r A N π µ 0 0 ) ( r I N A r U V U u π µ π 0 I r A N I U π µ The energy in an inductor is stored in the magnetic field within the coil, just as the energy of a capacitor is stored in the electric field between its plates. Ex: toroidal solenoid (B confined to a finite region of space within its core). V (πr) A Energy per unit volume: u U/V magnetic energy density
10 µ µ 0 B u B u ( ) 0 µ π B r I N r NI B π µ 0 Magnetic energy density in vacuum Magnetic energy density in a material
11 4. The R Circuit  An inductor in a circuit makes it fficult for rapid changes in current to occur due to induced emf. CurrentGrowth in an R Circuit At t 0 Switch closed. v ab I R v bc ε i R 0 ε ir ε R i
12 initial final ε 0 ε R I ( t 0 i 0 V ab 0) ε I R ( t f / 0) i ε R ( ) R i ' i' ( ε ) R t 0 0 i ln ( ε R) ε R R ' R Current in R circuit: t ε i e R ( R ) t
13 ε e ( R )t t 0 i 0, / ε / t i ε/r, / 0 Time constant for an R circuit: τ R At t τ, the current has risen to (/e) (63 % ) of its final value. Power supplied by the source: ε i i R Power ssipated by R + i Power stored in inductor
14 CurrentDecay in an R Circuit i I 0 e ( R )t t 0 I 0 Τ /R for current to decrease to /e (37 % of I 0 ). Total energy in circuit: 0 i R + i No energy supplied by a source (no battery present)
15 5. The C Circuit  In C circuit, the charge on the capacitor and current through inductor vary sinusoidally with time. Energy is transferred between magnetic energy in inductor (U B ) and electric energy in capacitor (U E ). As in simple harmonic motion, total energy remains constant.
16 C Circuit t 0 C charged Q C V m C scharges through inductor. Because of induced emf in, the current does not change instantaneously. I starts at 0 until it reaches I m. During C scharge, the potential in C induced emf in. When potential in C 0 induced emf 0 maximum I m. During the scharge of C, the growing current in leads to magnetic field energy stored in C (in its electric field) becomes stored in (in magnetic field). After C fully scharged, some i persists (cannot change instantaneously), C charges with contrary polarity to initial state. As current decreases B decreases induced emf in same rection as current that slows decrease in current. At some point, B 0, i 0 and C fully charged with V m (Q on left plate, contrary to initial state). If no energy loses, the charges in C oscillate back and forth infinitely electrical oscillation. Energy is transferred from capacitor E to inductor B.
17 Electrical Oscillations in an C Circuit Shown is + i dq/ (rate of change of q in left plate). If C scharges dq/<0 counter clockwise i is negative. Kirchhoff s loop rule: d q x + q C 0 Acos( ω t + ϕ) C circuit q C Analogy to eq. for harmonic oscillator: 0 d x k + x m 0 q Q cos t ( ω +ϕ) dq i ω Qsin( ωt + ϕ) ω C Angular frequency of oscillation ω π f If at t 0 Q max in C, i 0 φ 0 If at t 0, q0 φ ±π/ rad
18 Energy in an C Circuit Analogy with harmonic oscillator (mass attached to spring): E total 0.5 k A KE + U elas 0.5 m v x k x (A oscillation amplitude) v x ± k m A x Mechanical oscillations  For C Circuit: Q C i + q C Total energy initially stored in C energy stored in + energy stored in C (at given t). i ± C Q q Electrical oscillations
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20 6. The RC Series Circuit Because of R, the electromagnetic energy of system is ssipated and converted to other forms of energy (e.g. internal energy of circuit materials). [Analogous to friction in mechanical system].  Energy loses in R i R U B in when C completely scharged < U E Q /C  Small R circuit still oscillates but with damped harmonic motion circuit underdamped.  arge R no oscillations (e out) critically damped.  Very large R circuit overdamped C charge approaches 0 slowly.
21 Analyzing an RC Circuit ir q C 0 (i dq/) d q R dq + + q C 0 For R < 4/C: (underdamped) q Ae ( R / ) t cos C R 4 t + ϕ A, φ are constants ω ' C R 4 underdamped RC series circuit
22 Inductors in series: eq + Inductors in parallel: / eq / + /
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