Chapter 30 – Inductance
- Mutual Inductance
- Self-Inductance and Inductors
- Magnetic-Field Energy
- The R-L Circuit
- The L-C Circuit
- The L-R-C Series Circuit 1. Mutual Inductance
- A changing current in coil 1 causes B and a changing magnetic flux through coil 2 that induces emf in coil 2. dΦ ε = −N B2 2 2 dt
Magnetic flux N2Φ B2 = M 21 i1 through coil 2:
Mutual inductance of two coils: M 21 dΦ di N B2 = M 1 2 dt 21 dt M21 is a constant that depends on geometry of di 1 N2Φ B2 ε 2 = −M 21 → M 21 = the coils = M 12 . dt i1 - If a magnetic material is present, M 21 will depend on magnetic properties. If relative permeability (K m) is not constant (M not proportional to B) ΦB2 not proportional to i 1 (exception).
di 1 ε = −M N2Φ B2 N1Φ B1 2 dt Mutual inductance: M = = i1 i2 di ε = −M 2 1 dt emf opposes the flux change
- Only a time-varying current induces an emf.
Units of inductance: 1 Henry = 1 Weber/A = 1 V s/A = 1 J/A 2
Ex. 30.1 2. 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. Lenz’s law : a self-induced emf opposes the change in current that caused it Induced emf makes difficult variations in current.
NΦ L = B Self-inductance i
dΦ di N B = L dt dt
di ε = −L Self-induced emf dt Inductors as Circuit Elements
Inductors oppose variations in the current through a circuit.
-In DC-circuit, L helps to maintain a steady current (despite fluctuations in applied emf). In AC circuit, L helps to suppress fast variations in current.
- Reminder of Kirchhoff’s loop rule: the sum of potential differences around any closed loop is zero because E produced by charges distributed around circuit is conservative E c. -The magnetically induced electric field within the coils of an inductor is non- conservative (E n). - If R = 0 in inductor’s coils very small E required to move charge through coils total E through coils Ec + E n = 0. Since Ec ≠ 0 Ec = -En accumulation of charge on inductor’s terminals and surfaces of its conductors to produce that field. di E ld = −L ∫ n dt -En ≠ 0 only within the inductor.
b di (at each point within the inductor’s ∫ En ⋅ ld = −L E c+En = 0 a dt coil) b di ∫ Ec ⋅ ld = L a dt - Self-induced emf opposes changes in current.
di Potential difference between V = V −V = L terminals of an inductor: ab a b dt
Vab is associated with conservative, electrostatic forces, despite the fact that E associated with the magnetic induction is non-conservative Kirchhoff’s loop rule can be used.
- If magnetic flux is concentrated in region with a magnetic material 0 in eqs. must be replaced by = K m 0.
3. Magnetic-Field Energy
- Establishing a current in an inductor requires an input of ener gy. An inductor carrying a current has energy stored in it. Energy Stored in an Inductor di Rate of transfer of energy into L: P = V i = L⋅i ⋅ ab dt Energy supplied to inductor during dt: dU = P dt = L i di
Total energy U supplied while the current increases from zero to I:
I 1 U = L i ⋅di = LI 2 Energy stored in ∫ an inductor 0 2
- Energy flows into an ideal (R = 0) inductor when current in inductor increases. The energy is not dissipated, but stored in L and released when current decreases. Magnetic Energy Density
-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). 2 V = (2πr) A µ N A L = 0 2π ⋅r 1 1 µ N 2 A U = LI 2 = 0 I 2 2 2 2π ⋅r
Energy per unit volume: u = U/V magnetic energy density
U U 1 N 2 I 2 u = = = µ V 2π ⋅r ⋅ A 2 0 2( π ⋅r)2 2 2 0
B µ = 2 2 r I ⋅ 2
π N 2 () Magnetic energy density in a material a in density energy Magnetic Magnetic energy density in vacuum in density energy Magnetic r ⋅ NI 0 2 2 0 π µ µ B B µ 2 2 2 = = = B u u 4. The R-L Circuit
- An inductor in a circuit makes it difficult for rapid changes in current to occur due to induced emf.
Current-Growth in an R-L Circuit
At t =0 Switch 1 closed. di v = I ⋅ R v = L ab bc dt di ε − i ⋅ R − L = 0 dt
di ε − iR ε R = = − i dt L L L di ε = ( t = 0 i = 0 Vab = 0) dt initial L di ε R ε = 0 = − I I = ( tf di/dt = 0) dt final L L R
di R = − dt i − ()ε L R
idi ' t R ∫= − ∫ dt ' i'−(ε ) L 0R 0
i − (ε R) R ln = − t −ε R L
ε −(R )⋅t i = 1− e L Current in R-L circuit: R di ε −(R )⋅t t = 0 i = 0, di/dt = ε / L = e L dt L t = ∞ i ε/R, di/dt 0
L At t = τ, the current has risen Time constant for an R-L circuit: τ = R to (1-1/e) (63 % ) of its final value.
2 di Power supplied by the source: ε ⋅i = i R + L⋅i dt Power dissipated Power stored in by R inductor Current-Decay in an R-L Circuit
−(R )⋅t L i = I0e
t = 0 I0
Τ = L/R for current to decrease to 1/e
(37 % of I 0).
Total energy in circuit: di 0 = i2 R + Li dt
No energy supplied by a source (no battery present) 5. The L-C Circuit - In L-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. L-C Circuit
t = 0 C charged Q = C Vm C discharges through inductor. Because of induced emf in L, the current
does not change instantaneously. I starts at 0 until it reaches Im. During C discharge, the potential in C = induced emf in L. When potential
in C = 0 induced emf = 0 maximum Im. During the discharge of C, the growing current in L leads to magnetic field energy stored in C (in its electric field) becomes stored in L (in magnetic field).
After C fully discharged, some i persists (cannot change instantaneously), C charges with contrary polarity to initial state.
As current decreases B decreases induced emf in same direction as current that slows decrease in current. At some point, B = 0, i =0 and C
fully charged with –Vm (-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. Electrical Oscillations in an L-C Circuit Shown is + i = dq/dt (rate of change of q in left plate). If C discharges dq/dt<0 counter clockwise “i” is negative. di q Kirchhoff’s loop rule: − L − = 0 dt C d 2q 1 + q = 0 dt 2 LC L-C circuit d 2 x k Analogy to eq. for harmonic oscillator: + x = 0 dt 2 m x = Acos( ωt +ϕ)
q Q cos ( t ) 1 = ω +ϕ ω = Angular frequency of LC oscillation dq i = = −ω ⋅Qsin( ωt +ϕ) dt ω = 2 π f
π If at t = 0 Qmax in C, i = 0 φ = 0 If at t = 0, q=0 φ = ± /2 rad Energy in an L-C Circuit
Analogy with harmonic oscillator (mass attached to spring): 2 2 2 Etotal = 0.5 k A = KE + Uelas = 0.5 m v x + 0.5 k x (A = oscillation amplitude)
k v = ± A2 − x2 Mechanical oscillations x m
2 2 Q 1 2 q - For L-C Circuit: = Li + 2C 2 2C
Total energy initially stored in C = energy stored in L + energy stored in C (at given t).
1 i = ± Q2 − q2 Electrical oscillations LC
6. The L-R-C Series Circuit Because of R, the electromagnetic energy of system is dissipated and converted to other forms of energy (e.g. internal energy of circuit materials). [Analogous to friction in mechanical system].
2 - Energy loses in R i R UB in L when C 2 completely discharged < U E = Q /2C
- Small R circuit still oscillates but with “damped harmonic motion” circuit underdamped. - Large R no oscillations (die out) critically damped.
- Very large R circuit overdamped C charge approaches 0 slowly. Analyzing an L-R-C Circuit di q − iR − L − = 0 (i = dq/dt) dt C d 2q R dq 1 + + q = 0 dt 2 L dt LC
For R < 4L/C: (underdamped)
1 R2 q = Ae −(R 2/ L)t cos − t +ϕ 2 LC 4L
A, φ are constants
1 R2 ω'= − underdamped L-R-C series circuit LC 4L2 Inductors in series: Leq = L 1 + L 2
Inductors in parallel: 1/L eq = 1/L 1 + 1/L 2