Chapter 30 Inductance

Energy through space for free?? Left off with a puzzler! Creates increasing flux INTO ring Increasing current in time Induce counter-clockwise current and B field OUT of ring

Energy through space for free?? But… If wire loop has resistance R, current around it generates energy! Power = i2/R!! Increasing current in time Induced current i around loop of resistance R

Energy through space for free?? Yet…. NO “potential difference”! Increasing current in time Induced current i around loop of resistance R

Energy through space for free?? Answer? Energy in B field!! Increasing current in time Induced current i around loop of resistance R

To relate the induced emf to the rate of change of the current Goals for Chapter 30 To learn how current in one coil can induce an emf in another unconnected coil To relate the induced emf to the rate of change of the current To calculate the energy in a magnetic field

Introduce circuit components called INDUCTORS Goals for Chapter 30 Introduce circuit components called INDUCTORS To analyze circuits containing resistors and inductors To describe electrical oscillations in circuits and why the oscillations decay

Introduction How does a coil induce a current in a neighboring coil. A sensor triggers the traffic light to change when a car arrives at an intersection. How does it do this? Why does a coil of metal behave very differently from a straight wire of the same metal? We’ll learn how circuits can be coupled without being connected together.

Mutual inductance Mutual inductance: A changing current in one coil induces a current in a neighboring coil. Increase B flux Induce flux opposing change Increase current in loop 1 Induce current in loop 2

Mutual inductance EMF induced in single loop 2 = - d (B2 )/dt Caused by change in flux through second loop of B field Created by the current in the single loop 1 B2 is proportional to i1 Induce flux opposing change Increase current in loop 1 Induce current in loop 2

Mutual inductance What affects mutual inductance? # turns of each coil, area, shape, orientation (geometry!) Rate of change of current! Induce flux opposing change Increase current in loop 1 Induce current in loop 2

Mutual inductance N2 B = M21i1 IF you have N turns, each with flux B , total flux is Nx larger Total Flux = N2 B2 and B2 proportional to i1 So N2 B = M21i1 Where M21 is the “Mutual Inductance” constant (Henrys = Wb/Amp)

Because geometry is “shared” Mutual inductance EMF2 = - N2 d [B2 ]/dt and N2 B = M21i1 so EMF2 = - M21d [i1]/dt Because geometry is “shared” M21 = M12 = M

Mutual inductance Define Mutual inductance: A changing current in one coil induces a current in a neighboring coil. M = N2 B2/i1 M = N1 B1/i2

Mutual inductance examples Long solenoid with length l, area A, wound with N1 turns of wire N2 turns surround at its center. What is M?

Mutual inductance examples M = N2 B2/i1 We need B2 from the first solenoid (B1 = moni1) n = N1/l B2 = B1A M = N2 moi1 AN1/li1 M = moAN1 N2 /l All geometry!

Mutual inductance examples M = moAN1 N2 /l If N1 = 1000 turns, N2 = 10 turns, A = 10 cm2, l = 0.50 m M = 25 x 10-6 Wb/A M = 25 mH

Mutual inductance examples Using same system (M = 25 mH) Suppose i2 varies with time as = (2.0 x 106 A/s)t At t = 3.0 ms, what is average flux through each turn of coil 1? What is induced EMF in solenoid?

Mutual inductance examples Suppose i2 varies with time as = (2.0 x 106 A/s)t At t = 3.0 ms, i2 = 6.0 Amps M = N1 B1/i2 = 25 mH B1= Mi2/N1 = 1.5x10-7 Wb Induced EMF in solenoid? EMF1 = -M(di2/dt) -50Volts

Calculating self-inductance and self-induced emf Toroidal solenoid with area A, average radius r, N turns. Assume B is uniform across cross section. What is L?

Calculating self-inductance and self-induced emf Toroidal solenoid with area A, average radius r, N turns. L = N B/i B = BA = (moNi/2pr)A L = moN2A/2pr (self inductance of toroidal solenoid) Why N2 ?? If N =200 Turns, A = 5.0 cm2, r = 0.10 m L = 40 mH

Self-inductance Self-inductance: A varying current in a circuit induces an emf in that same circuit. Always opposes the change! Define L = N B/i Li = N B If i changes in time: d(Li)/dt = NdB/dt = -EMF or EMF = -Ldi/dt

Inductors as circuit elements! Inductors ALWAYS oppose change: In DC circuits: Inductors maintain steady current flow even if supply varies In AC circuits: Inductors suppress (filter) frequencies that are too fast.

Potential across an inductor The potential across an inductor depends on the rate of change of the current through it. The self-induced emf does not oppose current, but opposes a change in the current. Vab = -Ldi/dt

Inductors store energy in the magnetic field: Magnetic field energy Inductors store energy in the magnetic field: U = 1/2 LI2 Units: L = Henrys (from L = N B/i ) N B/i = B-field Flux/current through inductor that creates that flux Wb/Amp = Tesla-m2/Amp [U] = [Henrys] x [Amps]2 [U] = [Tesla-m2/Amp] x [Amps]2 = Tm2Amp But F = qv x B gives us definition of Tesla [B] = Teslas= Force/Coulomb-m/s = Force/Amp-m

Inductors store energy in the magnetic field: Magnetic field energy Inductors store energy in the magnetic field: U = 1/2 LI2 [U] = [Tesla-m2/Amp] x [Amps]2 = Tm2Amp [U] = [Newtons/Amp-m] m2Amp = Newton-meters = Joules = Energy!

The energy stored in an inductor is U = 1/2 LI2. Magnetic field energy The energy stored in an inductor is U = 1/2 LI2. The energy density in a magnetic field (Joule/m3) is u = B2/20 (in vacuum) u = B2/2 (in a magnetic material) Recall definition of 0 (magnetic permeability) B = 0 i/2pr (for the field of a long wire) 0 = Tesla-m/Amp [u] = [B2/20] = T2/(Tm/Amp) = T-Amp/meter

The energy stored in an inductor is U = 1/2 LI2. Magnetic field energy The energy stored in an inductor is U = 1/2 LI2. The energy density in a magnetic field (Joule/m3) is u = B2/20 (in vacuum) [u] = [B2/20] = T2/(Tm/Amp) = T-Amp/meter [U] = Tm2Amp = Joules So… energy density [u] = Joules/m3

Potential across an inductor The potential across an inductor depends on the rate of change of the current through it. The self-induced emf does not oppose current, but opposes a change in the current.

Potential across an inductor The potential across an inductor depends on the rate of change of the current through it. The self-induced emf does not oppose current, but opposes a change in the current.

Potential across an inductor The potential across an inductor depends on the rate of change of the current through it. The self-induced emf does not oppose current, but opposes a change in the current.

Potential across an inductor The potential across an inductor depends on the rate of change of the current through it. The self-induced emf does not oppose current, but opposes a change in the current.

The R-L circuit An R-L circuit contains a resistor and inductor and possibly an emf source. Start with both switches open Close Switch S1: Current flows Inductor resists flow Actual current less than maximum E/R E – i(t)R- L(di/dt) = 0 di/dt = E /L – (R/L)i(t)

The R-L circuit Close Switch S1: E – i(t)R- L(di/dt) = 0 di/dt = E /L – (R/L)i(t) Boundary Conditions At t=0, di/dt = E /L i() = E /R Solve this 1st order diff eq: i(t) = E /R (1-e -(R/L)t)

Current growth in an R-L circuit i(t) = E /R (1-e -(R/L)t) The time constant for an R-L circuit is  = L/R. [ ]= L/R = Henrys/Ohm = (Tesla-m2/Amp)/Ohm = (Newtons/Amp-m) (m2/Amp)/Ohm = (Newton-meter) / (Amp2-Ohm) = Joule/Watt = Joule/(Joule/sec) = seconds! 

Current growth in an R-L circuit i(t) = E /R (1-e -(R/L)t) The time constant for an R-L circuit is  = L/R. [ ]= L/R = Henrys/Ohm EMF = -Ldi/dt [L] = Henrys = Volts /Amps/sec Volts/Amps = Ohms (From V = IR) Henrys = Ohm-seconds [ ]= L/R = Henrys/Ohm = seconds!  Vab = -Ldi/dt

The R-L circuit E = i(t)R+ L(di/dt) Power in circuit = E I E i = i2R+ Li(di/dt) Some power radiated in resistor Some power stored in inductor

The R-L circuit example R = 175 W; i = 36 mA; current limited to 4.9 mA in first 58 ms. What is required EMF What is required inductor What is the time constant?

The R-L circuit Example R = 175 W; i = 36 mA; current limited to 4.9 mA in first 58 ms. What is required EMF What is required inductor What is the time constant? EMF = IR = (0.36 mA)x(175W ) = 6.3 V i(t) = E /R (1-e -(R/L)t) i(58ms) = 4.9 mA 4.9mA = 6.3V(1-e -(175/L)0.000058) L = 69 mH  = L/R = 390 ms

Current decay in an R-L circuit Now close the second switch! Current decrease is opposed by inductor EMF is generated to keep current flowing in the same direction Current doesn’t drop to zero immediately

Current decay in an R-L circuit Now close the second switch! –i(t)R - L(di/dt) = 0 Note di/dt is NEGATIVE! i(t) = -L/R(di/dt) i(t) = i(0)e -(R/L)t i(0) = max current before second switch is closed

Current decay in an R-L circuit i(t) = i(0)e -(R/L)t

Current decay in an R-L circuit Test yourself! Signs of Vab and Vbc when S1 is closed? Vab >0; Vbc >0 Vab >0, Vbc <0 Vab <0, Vbc >0 Vab <0, Vbc <0

Current decay in an R-L circuit Test yourself! Signs of Vab and Vbc when S1 is closed? Vab >0; Vbc >0 Vab >0, Vbc <0 Vab <0, Vbc >0 Vab <0, Vbc <0 WHY? Current still flows around the circuit counterclockwise

Current decay in an R-L circuit Test yourself! Signs of Vab and Vbc when S1 is closed? Vab >0; Vbc >0 WHY? Current still flows around the circuit counterclockwise through resistor EMF generated in L is from c to b So Vb> Vc!

Current decay in an R-L circuit Test yourself! Signs of Vab and Vbc when S2 is closed, S1 open? Vab >0; Vbc >0 Vab >0, Vbc <0 Vab <0, Vbc >0 Vab <0, Vbc <0

Current decay in an R-L circuit Test yourself! Signs of Vab and Vbc when S2 is closed, S1 open? Vab >0; Vbc >0 Vab >0, Vbc <0 WHY? Current still flows counterclockwise di/dt <0; EMF generated in L is from b to c! So Vb> Vc!

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit. Initially capacitor fully charged; close switch Charge flows FROM capacitor, but inductor resists that increased flow. Current builds in time. At maximum current, charge flow now decreases through inductor Inductor now resists decreased flow, and keeps pushing charge in the original direction i

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit. Initially capacitor fully charged; close switch Charge flows FROM capacitor, but inductor resists that increased flow. Current builds in time. Capacitor slowly discharges At maximum current, no charge is left on capacitor; current now decreases through inductor Inductor now resists decreased flow, and keeps pushing charge in the original direction i

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit.

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit. Now capacitor fully drained; Inductor keeps pushing charge in the original direction Capacitor charge builds up on other side to a maximum value While that side charges, “back EMF” from capacitor tries to slow charge build-up Inductor keeps pushing to resist that change. i

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit.

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit. Now capacitor charged on opposite side; Current reverses direction! System repeats in the opposite direction i

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit.

Electrical oscillations in an L-C circuit Analyze the current and charge as a function of time. Do a Kirchoff Loop around the circuit in the direction shown. Remember i can be +/- Recall C = q/V For this loop: -Ldi/dt – qC = 0

Electrical oscillations in an L-C circuit -Ldi/dt – qC = 0 i(t) = dq/dt Ld2q/dt2 + qC = 0 Simple Harmonic Motion! Pendulums Springs Standard solution! q(t)= Qmax cos(wt+f) where w = 1/(LC)½

Electrical oscillations in an L-C circuit q(t)= Qmax cos(wt+f) i(t) = - w Qmax sin(wt+f) (based on this ASSUMED direction!!) w = 1/(LC)½ = angular frequency

The L-C circuit An L-C circuit contains an inductor and a capacitor and is an oscillating circuit.

Electrical and mechanical oscillations Table 30.1 summarizes the analogies between SHM and L-C circuit oscillations.

The L-R-C series circuit An L-R-C circuit exhibits damped harmonic motion if the resistance is not too large.