Chapter 24 Inductance and Circuit Oscillations

Main Points Inductance and Inductors Energy in Inductors and in the Magnetic Field RL and RC Circuits LC Circuits RLC Circuits, Damped Oscillations

24-1 Inductance and Inductor Faraday’s Law: Changing current in a circuit will induce emf in that circuit as well as others nearby Self-Inductance: Circuit induces emf in itself Mutual Inductance: Circuit induces emf in second circuit

Self Inductance Switch closes Self-Induction: changing current through a loop inducing an opposing emf in that same loop.

The flux, therefore, is also proportional to the current. The magnetic field produced by the current in the loop shown is proportional to that current: I The flux, therefore, is also proportional to the current. We define this constant of proportionality between flux and current to be the inductance, L.

Combining with Faraday’s Law gives the emf induced by a changing current: Self-induced emf The minus sign indicates that—as the law states—the self-induced emf e  has the orientation such that it opposes the change in current I

(a) The current i is increasing and the self-induced e appears along the coil in a direction such that it opposes the increase. (b) The current i is decreasing and the self-induced emf appears in a direction such that it opposes the decrease. The inductance L is a proportionality constant that depends on the geometry of the circuit SI :H (Wb/A)

If extra material (e.g., an iron core) is added, the The inductance of an inductor (a set of coils in some geometry; e.g., solenoid) then, can be calculated from its geometry alone if the device is constructed from conductors and air (similar to the capacitance of a capacitor). If extra material (e.g., an iron core) is added, the inductance will increase (just as adding a dielectric increases capacitance) Archetypal inductor is a long solenoid, just as a pair of parallel plates is the archetypal capacitor. d A - - - - - + + + + r << l l r N turns

Example: The length and radius of a long, tightly wound solenoid with N turns are l and R respectively. Find its self-inductance? Solution: The magnitude of magnetic field in the solenoid is The magnetic flux through N turns of wires is The self-inductance of the long, tightly wound solenoid is

Example A coaxial cable is composed of two long, coaxial cylinders with radius R1 and R2 respectively. They carry equal currents I in opposite directions. What is the self-inductance per unite length? Solution: Magnitude of the magnetic field inside and outside the coaxial cable is

self-inductance per unite length In series The magnetic flux through an element area dA in a rectangular region as shown in the figure is dr self-inductance per unite length In series

The magnetic flux through circuit 2 is Mutual Induction A changing current i1 in circuit 1 causes a changing flux 21 through circuit 2. Then an induced emf appears in circuit 2 . dA 1 2 The magnetic flux through circuit 2 is M21 is a constant that depends on the geometry of the two circuits and the magnetic properties of the material.

emf induced in circuit 2 by changing currents in circuit 1, through mutual inductance:

Example Calculate the mutual inductance for two tightly wound concentric solenoids shown in figure below Solution: A current I1 in the inner solenoid sets up a magnetic field B1 the flux 21 is The mutual inductance is

We can calculate the mutual inductance by assuming that the outer solenoid carries a current I2 The mutual inductances are equal This is a general result

Example: A long, straight wire carrying a current I=I0sint and a rectangular loop whose short edges are parallel to the wire, as shown in the figure. Find the mutual inductance and the mutual induced emf in the rectangular loop. Solution: The magnetic flux  through the rectangular loop is The mutual inductance is The mutual induced emf in the rectangular is

Kirchhoff’s First Rule “Loop Rule” or “Kirchhoff’s Voltage Law (KVL)” "When any closed circuit loop is traversed, the algebraic sum of the changes in potential must equal zero." KVL:

Rules for potential differences across various circuit elements.

Our convention: We choose a direction for the current and move around the circuit in that direction. When a battery is traversed from the negative terminal to the positive terminal, the voltage increases, and hence the battery voltage enters KVL with a + sign. When moving across a resistor, the voltage drops, and hence enters KVL with a - sign. e1 R1 R2 e2 I + e1 - e2 - IR1 - IR2 = 0

Modification of Kirchhoff’s loop rule: In moving across an inductor of inductance L along the presumed direction of the current I, the potential change is e = –L dI/dt

Example A device as shown below is constructed from conductors Example A device as shown below is constructed from conductors. Find the self-inductance. Solution Suppose there is a current I in the conductor l l I a d a If there is a magnetic field which is parallel to the axis of the cylindrical shell, Find the current

l l I a d a

24-2 Energy in Inductors Loop rule An RL circuit Work done by the battery Energy stored in the inductor Energy lost in a resistor in the form of thermal energy

Where is the Energy Stored? An inductor is a device for storing energy in a magnetic field. We can integrate that term to find an expression for UL (starting from zero current) Where is the Energy Stored?

24-3 Energy in Magnetic Fields Energy is stored in the magnetic field itself (just as in the capacitor / electric field case). For an ideal solenoid We can express UL in terms of B(=nI) giving the energy density of the magnetic field: A general result

Energy is located within the electric and magnetic fields themselves

Example A long coaxial cable consists of two thin-walled concentric conducting cylinders with radii a and b. The inner cylinder carries a steady current I, the outer cylinder providing the return path for that current. The current sets up a magnetic field between the two cylinders. (a)  Calculate the energy stored in the magnetic field for a length l of the cable. (b)  Calculate the self inductance for a length l of the cable. I

Solution find H by using Ampere's law I The energy density The energy

Example An air-filled, closely wound toroidal solenoid with inner radius R1 and outer radius R2 , has N turns. Its cross-section is rectangular. Find magnetic-field energy. Solution: By using Ampere’s Law, the magnitude of the magnetic field is The magnetic energy density

The total magnetic energy The volume of a thin cylinder with radius r and thickness dr, as shown in the figure, is The total magnetic energy dr r What is the inductance of this toroid?

k I R L 24-4 Time Dependence in RL Circuits When the switch K1 is closed, the inductor keeps the current from attaining its maximum value immediately. An RL circuit e L R 1 k 2 I Current as a function of time:

An RL circuit e L R 1 k 2 I The current in an inductor never changes instantaneously, but after the current settles down to a constant value, the inductor plays no role in the circuit

L R e 1 k 2 I When the switch is thrown to K2 the inductor keeps the current from dropping to zero immediately. t I time constant Current as a function of time: The time constant determines how fast the current changes with time

General rule: inductors resist change in current Hooked to current source Initially, the inductor behaves like an open switch. After a long time, the inductor behaves like an ideal wire. Disconnected from current source Initially, the inductor behaves like a current source. After a long time, the inductor behaves like an open switch.

L i Close switch Open switch demo

ACT At t=0 the switch is thrown from position b to position a in the circuit shown: L I What is the value of the current I0 immediately after the switch is thrown? (a) I0 = 0 (b) I0 = e/2R (c) I0 = 2e/R What is the value of the current I¥ a long time after the switch is thrown? (a) I¥ = 0 (b) I¥ = e/2R (c) I¥ = 2e/R

What is the value of the current I0 just after the switch is thrown? After a long time, the switch is thrown from position a to position b as shown: What is the value of the current I0 just after the switch is thrown? (a) I0 = 0 (b) I0 = e/2R (c) I0 = 2e/R R a b e L I

If the electric field exceeds the “dielectric strength” a I I R If after a long time, the switch is opened abruptly from position a instead of being thrown to position b, What happens? e L R Just after the switch is thrown, the inductor induces an emf to keep current flowing: emf = L dI/dt (can be much larger than e) However, now there’s no place for the current to go  charges build up on switch contacts  high voltage across switch gap If the electric field exceeds the “dielectric strength” (~30 kV/cm in air)  breakdown  SPARK!

ACT 1) At time t = 0 the switch is closed. What is the current through the circuit immediately after the switch is closed? a) I = 0 b) I = V/R c) I = V/2R Initially, the inductor acts like an open switch 2) What is the current through the circuit a long time after the switch is closed? After a long time, the inductor acts like an ideal wire, a) I = 0 b) I = V/R c) I = V/2R

What is the relation between tI and tII? ACT At t=0, the switch is thrown from position b to position a as shown: Let tI be the time for circuit I to reach 1/2 of its asymptotic current. Let tII be the time for circuit II to reach 1/2 of its asymptotic current. What is the relation between tI and tII? e a b L I R II (a) tII < tI (b) tII = tI (c) tII > tI We must determine the time constants of the two circuits by writing down the loop equations. tI=L/2R tII=2L/R This confirms that inductors in series add

Time Dependence in RC Circuits RC circuit with battery and switch I + - Switch at position a: battery charges capacitor Loop rule:

Charge as a function of time: Current as a function of time: Conclusion: Capacitor reaches its final charge(Q=Ce ) exponentially with time constant t = RC. Current decays from max (e /R) with same time constant.

Suppose that the switch has been in position a for a long time, the capacity is fully charged, and there is no current. We throw the switch to position b: capacitor discharges through resistor I + -

Charge as a function of time: Current as a function of time: The negative sign indicates that the actual current is opposite in direction to the current we assumed Conclusion: Capacitor discharges exponentially with time constant t = RC Current decays from initial max value (-e/R) with same time constant

Basic principle: Capacitor resists rapid change in Q  resists rapid changes in V Charging (it takes time to put the final charge on) Initially, the capacitor behaves like a wire After a long time, the capacitor behaves like an open switch. Discharging Initially, the capacitor behaves like a battery. After a long time, the capacitor behaves like a wire.

What is the value of the current I0+ just after the switch is thrown? ACT At t=0 the switch is thrown from position b to position a in the circuit shown: The capacitor is initially uncharged. What is the value of the current I0+ just after the switch is thrown? a b e R C I (a) I0+ = 0 (b) I0+ = e /2R (c) I0+ = 2e /R What is the value of the current I¥ after a very long time? (a) I¥ = 0 (b) I¥ = e /2R (c) I¥ > 2e /R

Compare the charge on the two capacitors a short time after t = 0 ACT The two circuits shown below contain identical fully charged capacitors at t=0. Circuit 2 has twice as much resistance as circuit 1. Compare the charge on the two capacitors a short time after t = 0 Initially, the charges on the two capacitors are the same. But the two circuits have different time constants: t1 = RC and t2 = 2RC. Since t2 > t1 it takes circuit 2 longer to discharge its capacitor. Therefore, at any given time, the charge on capacitor 2 is bigger than that on capacitor 1. a) Q1 > Q2 b) Q1 = Q2 c) Q1 < Q2

24-5 Oscillations in LC Circuits Set up the circuit above with capacitor, inductor, resistor, and battery. Let the capacitor become fully charged. Throw the switch from a to b What happens?

Start with a fully charged capacitor the loop rule: Simple harmonic oscillator

solution: Q0and f are determined from initial conditions. Here, Q0 (initial condition) determines the amplitude of the oscillations The frequency of the oscillations is determined by the circuit parameters (L, C)

It undergoes simple harmonic motion, just like a mass on a spring, with trade-off between charge on capacitor (Spring) and current in inductor (Mass)

ACT At t=0 the capacitor has charge Q0; the resulting oscillations have frequency w0. The maximum current in the circuit during these oscillations has value I0. What is the relation between w0 and w2, the frequency of oscillations when the initial charge = 2Q0? L C + - Q = t=0 (a) w2 = 1/2 w0 (b) w2 = w0 (c) w2 = 2w0 What is the relation between I0 and I2, the maximum current in the circuit when the initial charge = 2Q0? (a) I2 = I0 (b) I2 = 2I0 (c) I2 = 4I0

24-6 Damped Oscillations in RLC Circuits Loop rule solution: Damped harmonic oscillator

Charge equation: Solution: Where And Determines the rate of exponential damping Q0 and f are determined from initial conditions.

Comparison of an RLC circuit with and without damping critical damping When there is no oscillator behavior We call this overdamping

The RLC circuit for various values of R. critical damping overdamping

24-7 Energy in LC and RLC Circuits No Resistance We take initial conditions: Energy in capacitor Energy in inductor Total energy is conserved

In a pure LC circuit, energy is transferred back and forth between the capacitor’s electric field and the inductor’s magnetic field.

Resistance is Introduced The power dissipated in the resistor a resistor causes energy loss, which shows up as heat. The rate of change of the energy in the capacitor and in the inductor Equal to the power loss in the resistor

Example A 1. 5 mF capacitor is charged to 57 V Example A 1.5 mF capacitor is charged to 57 V. The charging battery is then disconnected, and a 12 mH coil is connected in series with the capacitor so that LC oscillations occur at time t=0. (a) What is the maximum current in the coil? (b) What is the potential difference VL(t) across the inductor as a function of time? (c) What is the maximum rate (di/dt)max at which the current i changes in the circuit? Assume that the circuit contains no resistance. Solution (a)

Example A series RLC circuit has inductance L = 12 mH, capacitance C = 1.6 mF, and resistance R = 1.5 W. (a)  At what time t will the amplitude of the charge oscillations in the circuit be 50% of its initial value? (b)  How many oscillations are completed within this time? Solution

Summary Definition of inductance: Induced emf: emf induced in a second loop: Energy in an inductor: Energy density of a magnetic field:

RL Circuits RC Circuits LC circuit oscillations: RLC circuit oscillations: