Physics 121 - Electricity and Magnetism Lecture 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

Recap: Faraday’s Law of Induction Magnetic Flux: Faraday’s Law: A changing magnetic flux through a coil of wire induces an EMF in the wire, proportional to the number of turns, N. Bind & iind oppose changes in FB Lenz’s Law: The current driven by an induced EMF creates an induced magnetic field that opposes the flux change. Induction and energy transfer: The forces on the loop oppose the motion of the loop, and the power required to sustain motion provides electrical power to the loop. i1 i2 Transformer principle: changing current i1 in primary induces EMF and current i2 in secondary coil. Generalized Faraday Law: A changing magnetic flux creates non-conservative electric field.

conducting loop, resistance R Changing magnetic flux directly induces electric fields conducting loop, resistance R Bind A thin solenoid, n turns/unit length Solenoid cross section A Zero B field outside solenoid Field inside solenoid: Flux through the wire loop: Changing current i  changing flux  EMF is induced in wire loop: Current induced in the loop is: If di/dt is positive, B is growing, then Bind opposes change and i’ is counter-clockwise B = 0 outside solenoid, so it’s not the Lorentz force An electric field Eind is induced directly by dF/dt within the loop 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 a non-conservative (non-electrostatic) field as the line integral (potential) around a closed path is not zero What makes the induced current I’ flow, outside solenoid? Generalized Faradays’ Law (hold loop path constant)

Example: Find the electric field induced by a region of changing magnetic flux Find the magnitude E of the induced electric field at points within and outside the magnetic field. Assume: dB/dt = constant across the circular shaded area. E must be tangential: Gauss’ law says any normal component of E would require enclosed charge. |E| is constant on a circular integration path due to symmetry. For r > R: For r < R: The magnitude of induced electric field grows linearly with r, then falls off as 1/r for r>R

Eind = - L di/dt L is the inductance RELATED FARADAYS LAW EFFECTS IN COILS: Mutual-induction between coils: di1/dt in “transformer primary” induces EMF and current i2 in “linked” secondary coil (transformer principle). N Self-induction in a single Coil: di/dt produces “back EMF” due to Lenz & Faraday Laws. Find (or Bind) opposes dF/dt due to current change. Eind opposes di/dt. ANY magnetic flux change is resisted (analogous to inertia) Changing current in a single coil generates changing magnetic field/flux in itself. Changing flux induces flux opposing the change, along with opposing EMF, current, and induced field. The “back EMF” limits the rate of current (and therefore flux) change in the circuit For increasing current, back EMF limits the rate of increase For decreasing current, back EMF sustains the current Inductance measures opposition to the rate of change of current Eind = - L di/dt L is the inductance

Definition of Self-inductance Joseph Henry 1797 – 1878 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: Why choose this definition? Cross-multiply……. Take time derivative Another form of Faraday’s Law! L contains all the geometry EL is the “back EMF”

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

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

Lenz’s Law applied to Back EMF If i is increasing: EL opposes increase in i Power is being stored in B field of inductor + - EL i Di / Dt If i is decreasing: EL opposes decrease in i Power is being tapped from B field of inductor EL - + i Di / Dt What if CURRENT i is constant? EL = 0, inductance acts like a simple wire

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.

means that current is increasing and to the right Example: Current I increases uniformly from 0 to 1 A. in 0.1 seconds. Find the induced voltage (back EMF) across a 50 mH (milli-Henry) inductance. i + - EL means that current is increasing and to the right Negative result means that induced EMF is opposed to both di/dt and i. Apply: Substitute: 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. Devices:

Inductors in Circuits—The RL Circuit Basic series RL circuit: A battery with EMF E drives a current around the loop Changing current produces a back EMF or sustaining EMF EL in the inductor. Derive circuit equations using Kirchhoff’s loop rule. Convert to differential equations and solve (as for RC circuits). New for Kirchhoff rule: When traversing an inductor in the same direction as the assumed current insert:

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

LR circuit: decay phase solution b EL + - After growth phase equilibrium, switch from a to b, battery out Current i0 = E / R initially still flows CW through R Inductance tries to maintain current using stored energy Polarity of EL reversed versus growth. Eventually EL 0 Loop Equation is : Substitute : Circuit Equation: di/dt <0 during decay, opposite to current Current decays exponentially: t 2t 3t i0 i 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 EMF EL and VR also decay exponentially: Compare to RC circuit, decay

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

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

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?

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?

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

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

Energy stored in inductors Recall: Capacitors store energy in their electric fields derived using p-p capacitor Inductors also store energy, but in their magnetic fields Magnetic Potential Energy – consider power into or from inductor 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 derived using solenoid

Sample problem: Energy storage in magnetic field of an inductor during growth phase a) At equilibrium (infinite time) how much energy is stored in the coil? L = 53 mH E = 12 V b) How long (tf) does it take to store half of this energy? take natural log of both sides

flux through one turn of coil 2 due to all N1 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 Changing current in primary (i1) creates varying flux through coil 2  induced E field in coil 2 mutual inductance number of turns in coil 2 flux through one turn of coil 2 due to all N1 turns of coil 1 current in coil 1 Definition: cross-multiply time derivative Another form of Faraday’s Law! M21 contains all the geometry E2 is EMF induced in 2 by 1 The linked flux is the same (smaller coil) if coils reverse roles, so….. proof not obvious

determines the linkage results for mutual inductance: Calculating the mutual inductance M Outer coil 1 is a flat loop of N1 turns (not a solenoid) PRIMARY large coil 1 N1 turns radius R1 SECONDARY small coil 2 N2 turns radius R2 Assume uniform B Use arc formula with f=2p Flux through one turn of flat inner coil 2 depends on B1 and smaller area A2 Current i1 in Loop 1 is changing: smaller radius (R2) determines the linkage Summarizing results for mutual inductance:

Air core Transformer Example: 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: 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?