Inductors and Inductance Self-Inductance RL Circuits – Current Growth RL Circuits – Current Decay

1. Physics 121 - Electricity and MagnetismLecture 12 - Inductance, RL Circuits Y&F Chapter 30, Sect 1 - 4 • Inductors and Inductance • Self-Inductance • RL Circuits – Current Growth • RL Circuits – Current Decay • Energy Stored in a Magnetic Field • Energy Density of a Magnetic Field • Mutual Inductance

2. A thin solenoid, n turns/unit length • Solenoid cross section A • Zero B field outside solenoid • Field inside solenoid: conducting loop, resistance R Flux through the wire loop: Changing current i  changing flux  EMF is induced in wire loop: Bind Current induced in the loop is: If di/dt is positive, B is growing, then Bind opposes change and i’ is counter-clockwise What makes the induced current i’ flow, outside solenoid? • B = 0 outside solenoid, so it’s not the Lorentz force • dF/dt within the loop directly induces an electric field Eind • Eind causes current to flow in the loop • Eind is there even without the wire loop (no current flowing) • Electric field lines are loops that don’t terminate on charge. • Eind is non-conservative as the line integral (potential) around a • closed path is not zero [Generalized Faraday’s Law]. Changing magnetic flux directly induces electric fields Mutual Inductance Example <–> Simple Transformer

3. N • Changing current di/dt in a single coil generates changing magnetic flux in the interior of the coil. • Changing magnetic flux induces electric field in & near the coil, which drives extra current to flow in the coils turns of wire. • That induced current in turn creates “induced magnetic flux/field” opposed to the change in flux through the coil. • The integral of the induced E field generates the “back EMF” opposing or sustaining the applied current in the coil. • For increasing current, “back” EMF limits the rate of current increase • For decreasing current, “back” EMF sustains the current (limits decrease) Inductance measures EMF opposed to the rate of change of current Eind = - L di/dt L is the inductance FARADAYS LAW EFFECTS IN COILS - MUTUAL and SELF INDUCTANCE Mutual-induction between coils: di1/dt in “primary” produces E field that induces current i2, EMF, and induced B field in “linked” secondary coil (transformer principle). Self-induction includes same effectsin a single coil:

4. Joseph Henry 1797 – 1878 Does this definition make sense? Cross-multiply……. Take time derivative • L contains all the geometry • EL is the “back EMF” Another form of Faraday’s Law! Definition: Calculating Self-inductance Recall capacitance: depends only on geometry It measures charge stored per volt Self-inductance depends only on coil geometry It measures flux created per ampere number of turns flux through one turn depends on current & all N turns self-inductance cancels current dependence in flux above SI unit of inductance:

5. where Field: + Flux in just one turn: ALL N turns contribute to self-flux through ONE turn L - Apply definition of self-inductance: RL • N turns • Area A • Length l • Volume V =Al • Depends on geometry • only, like capacitance. • Proportional to N2 ! Check: Get same formula for L if you start with Faraday’s Law for FB: Note: Inductance per unit length has same dimensions as m0 forsolenoid use above Example: Find the Self-Inductance of a solenoid

6. l Ideal inductor uses ideal solenoid abstraction: Non-ideal inductors have internal series resistance: · = - = E V ir measured terminal voltage L IND · Direction of ir depends on current r L · E Direction of depends on di/dt L · , Constant current implies induced voltage EL = 0 Inductor behaves like a wire with resistance r Vind Example: calculate self-inductance L for an ideal solenoid

7. i + If i is increasing: EL Di / Dt ELopposes increase in i - Power is being stored in B field of inductor i If i is decreasing: - ELopposes decrease in i Di / Dt EL Power is being tapped from B field of inductor + Implication for circuits: Lenz’s Law controls Back EMF across Inductors What if CURRENT i is constant (steady state)? EL = 0, inductance acts like a simple wire

8. Induced EMF in an Inductor 12 – 1: Which statement describes the current through the inductor below, if the induced EMF is as shown? • Rightward and constant. • Leftward and constant. • Rightward and increasing. • Leftward and decreasing. • Leftward and increasing.

9. EL - + i i Negative result means that induced EMF is opposed to both di/dt and i. Apply: Substitute: Inductors are Devices: • Inductors, sometimes called “coils”, are common circuit components. • Insulated wire is wrapped around a core. • They are mainly used in AC filters and tuned (resonant) circuits. Example: Current I increases uniformly from 0.0 to 1.0 A. in 0.1 seconds. Find the induced voltage EL (back EMF) across a 50 mH (milli-Henry) inductance. means that current is increasing and to the right

10. Basic series RL circuit: • Time-dependent results are reminiscent of RC circuit. • A battery with EMF E drives a current around the loop. • Changing current produces a back EMF or a sustaining EMF EL in the inductor. • Task: Derive circuit equations using Kirchhoff’s loop rule. • Convert to differential equations and solve (as for RC circuits). New term for Kirchhoff Loop Rule: When traversing an inductor in the same direction as the assumed current insert: Voltage drops by ELfor growing current Inductors in Circuits—The RL Circuit

11. Series LR circuit i a R + E b - EL L i • At t = 0, current grows rapidly but i = 0, EL= -E • L acts like a broken wire, blocks current flow • Later: Current is clockwise and growing. EL opposes E. • As t  infinity, current is large but stable, di/dt  0 • Back EMF EL 0, i  E / R, • L acts like an ordinary wire • Energy is stored in B field within L & dissipated in R Growth phase, switch to “a”. Loop equation: Decay phase, switch to “b”, exclude E, Loop equation: • Energy stored in L will be dissipated in R • EL actslike a battery maintaining previous current • At t = 0 current i = E / R, unchanged, CW • Later: current begins to collapse • As t infinity current  0. Energy is depleted • Inductance & resistance + EMF • Find time dependent behavior • Use Loop Rule NEW TERM FOR KIRCHHOFF LOOP RULE Given E, R, L: Find i, EL, UL for inductor as functions of time

12. R - b EL L + i Circuit Equation: Loop Equation is : di/dt <0 during decay, opposite to current Substitute : • First order differential equation with simple exponential decay solution • At t = 0+: large current implies large di / dt, so EL is large (now driving current) • As t  infinity: current stabilizes, di / dt and current i both  0 Current decays exponentially: i0 i t 2t 3t t EMF EL and VR also decay exponentially: Compare to RC circuit, decay LR circuit: decay phase solution • After growth phase equilibrium, switch from a to b, battery out. • Current i0 = E / R initially still flows CW through R • EMF EL tries to maintain current using stored energy • Polarity of EL isreversed versus growth. Eventually EL 0

13. Circuit Equation: Loop Equation is : Substitute : • First order differential equation again - saturating exponential solutions • At t = 0: current is small because di / dt is large. Back EMF opposes battery. • As t  infinity: current stabilizes at iinf = E / R. di / dt approaches zero, Current starts from zero, grows as a saturating exponential. iinf i • i = 0 at t = 0 in above equation  di/dt = E/L • fastest rate of change, largest back EMF t 2t 3t Back EMF EL decays exponentially t Compare to RC circuit, charging Voltage drop across resistor VR= -iR LR circuit: growth phase solution

14. S Use growth phase solution + At t = 0: current = 0 EL i - Back EMF is proportional to rate of change of current EL -E • After a very long (infinite) time: • Current stabilizes, back EMF=0 • L acts like an ordinary wire at t = infinity Example: For growth phase find back EMF EL as a function of time At t = 0, EL equals the battery potential • iR drop across R = 0 • L acts like a broken wire at t = 0

15. Current through the battery - 1 12 – 2: The three loops below have identical inductors, resistors, and batteries. Rank them in terms of current through the battery just after the switch is closed, greatest first. • I, II, III. • II, I, III. • III, I, II. • III, II, I. • II, III, I. I. II. III. Hint: what kind of wire (ordinary or broken) does L act like?

16. Current through the battery - 2 12 – 3: The three loops below have identical inductors, resistors, and batteries. Rank them in terms of current through the battery a long time after the switch is closed, greatest first. • I, II, III. • II, I, III. • III, I, II. • III, II, I. • II, III, I. I. II. III. Hint: what kind of wire (ordinary or broken) does L act like?

17. Inductors also store energy, but in their magnetic fields Magnetic Potential Energy – consider power into or from inductor derived using solenoid • UB grows as current increases, absorbing energy • When current is stable, UB and uB are constant • UB decreases as current decreases. It powers • the persistent EMF during the inductor’s decay phase • The B field stores energy with or without a conductor Energy stored in inductors Recall: Capacitors store energy in their electric fields derived using p-p capacitor

18. a) At equilibrium (infinite time) how much energy is stored in the coil? E = 12 V L = 53 mH • b) How long (tf) does it take to store 25% of this energy? take natural log of both sides Example: How much energy is stored in the magnetic field of an inductor after a growth phase?

19. Definition: number of turns in coil 2 flux through one turn of coil 2 due to all N1 turns of coil 1 mutual inductance current in coil 1 cross-multiply • M21 contains all the • geometry • E2 is EMF induced • in 2 by 1 time derivative Another form of Faraday’s Law! The linked flux is the same (smaller coil) if coils reverse roles, so….. proof not obvious Mutual Inductance • Example: a pair of co-axial, flux-linked coils • di1/dt in the first coil induces current i2 in the • second coil, in addition to self-induced effects. • M21 depends on geometry only, as did L, C, and R • Changingcurrent in primary (i1) creates E field that • drives current, induced flux and EMF through coil 2

20. Example: Calculating the mutual inductance M PRIMARY larger coil 1 N1 turns radius R1 SECONDARY smaller coil 2 N2 turns radius R2 Changing current i1 in loop 1 induces E2 in loop 2: smaller radius (R2) determines the linkage Summarizing results for mutual inductance: Larger coil (#1) is a flat loop of N1 turns (not a solenoid) Assume uniform B Use arc formula with f=2p Flux through one turn of flat smaller coil (#2) depends on B1 and smaller area A2

21. SUPPLEMENTARY MATERIAL

22. When t / tL is large: • When t/ tL is small: i = 0. Inductor acts like a wire. Inductor acts like an open circuit. • The current starts from zero and increases up to a maximum of with a time constant given by Inductive time constant Compare: • The voltage across the resistor is proportional to current Capacitive time constant • The voltage across the inductor is Extra Summarizing RL circuits growth phase

23. The switch is thrown from a to b • Kirchoff’s Loop Rule for growth was: • Now it is: • The current decays exponentially: VR (V) • Voltage across resistor also decays: • Voltage across inductor: Extra Summarizing RL circuits decay phase

24. FIND INDUCED EMF IN INNER COIL DURING THIS PERIOD • Coil diameter d = 2.1 cm = .021 m, Ainner = p d2/4, short length • Ninner = 130 turns = total number of turns in inner coil Direction: Induced B is parallel to Bouter which is decreasing Would the transformer work if we reverse the role of the coils? Example: Air core Transformer - Concentric coils • OUTER COIL – IDEAL SOLENOID • Coil diameter D = 3.2 cm • nouter = 220 turns/cm = 22,000 turns/m • Current iouter falls smoothly from • 1.5 A to 0 in 25 ms • Field uniform across outer coil is: