Copyright © 2012 Pearson Education Inc. PowerPoint ® Lectures for University Physics, Thirteenth...

Copyright © 2012 Pearson Education Inc.

PowerPoint® Lectures forUniversity Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman

Lectures by Wayne Anderson

Chapter 30

Inductance


Energy through space for free??

• Left off with a puzzler!

Increasing current in

time

Creates increasing flux

INTO ring

Induce counter-

clockwise current and B field OUT

of ring



• But… If wire loop has resistance R, current around it generates energy! Power = i2/R!!


timeInduced current i around loop of

resistance R



• Yet…. NO “potential difference”!



resistance R



• Answer? Energy in B field!!



resistance R


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


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 current in loop 1

Increase B flux

Induce current in loop 2

Induce flux opposing change


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





Mutual inductance

• What affects mutual inductance?

– # turns of each coil, area, shape, orientation (geometry!)

– Rate of change of current!





Mutual inductance

• 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)


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?



• M = N2 B2/i1

• We need B2 from the first solenoid (B1 = oni1)

• n = N1/l

• B2 = B1A

• M = N2 oi1 AN1/li1

• M = oAN1 N2 /l

• All geometry!



• M = oAN1 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 H



• Using same system (M = 25 H)

• Suppose i2 varies with time as = (2.0 x 106 A/s)t

• At t = 3.0 s, what is average flux through each turn of coil 1?

• What is induced EMF in solenoid?



• Suppose i2 varies with time as = (2.0 x 106 A/s)t

• At t = 3.0 s, i2 = 6.0 Amps

• M = N1 B1/i2 = 25 H

• 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 = (oNi/2r)A

• L = oN2A/2r (self inductance of toroidal solenoid)

• Why N2 ??

• If N =200 Turns, A = 5.0 cm2,

r = 0.10 m

L = 40 H


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 = NdB/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


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:

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.

• The energy density in a magnetic field (Joule/m3) is

• u = B2/20 (in vacuum)

• u = B2/2 (in a magnetic material)

• Recall definition of 0 (magnetic permeability)

• B = 0 i/2r (for the field of a long wire)

0 = Tesla-m/Amp

• [u] = [B2/20] = T2/(Tm/Amp) = T-Amp/meter



• The energy stored in an inductor is U = 1/2 LI2.

• The energy density in a magnetic field (Joule/m3) is

• u = B2/20 (in vacuum)

• [u] = [B2/20] = T2/(Tm/Amp) = T-Amp/meter

• [U] = Tm2Amp = Joules

• So… energy density [u] = Joules/m3


















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 ; i = 36 mA; current limited to 4.9 mA in first 58 s.

• What is required EMF

• What is required inductor

• What is the time constant?


The R-L circuit Example

• R = 175 ; i = 36 mA; current limited to 4.9 mA in first 58 s.

• What is required EMF

• What is required inductor

• What is the time constant?

• EMF = IR = (0.36 mA)x(175 ) = 6.3 V

• i(t) = E /R (1-e -(R/L)t)

• i(58s) = 4.9 mA

• 4.9mA = 6.3V(1-e -(175/L)0.000058)

• L = 69 mH

= L/R = 390 s


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



• 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



• i(t) = i(0)e -(R/L)t



• 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



• Test yourself!


• Vab >0; Vbc >0

• Vab >0, Vbc <0

• Vab <0, Vbc >0

• Vab <0, Vbc <0

• WHY?

• Current still flows around the circuit counterclockwise



• Test yourself!


• 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!



• 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



• 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 inductorresists 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


• Initially capacitor fully charged; close switch

• Charge flows FROM capacitor, but inductorresists 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



The L-C circuit


• Now capacitor fully drained;

• Inductor keeps pushing charge in the original direction

• Capacitor charge builds up on other sideto a maximum value

• While that side charges, “back EMF” fromcapacitor tries to slow charge build-up

• Inductor keeps pushing to resist that change.

i


The L-C circuit



The L-C circuit


• Now capacitor charged on opposite side;

• Current reverses direction! System repeatsin the opposite direction

i


The L-C 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



• -Ldi/dt – qC = 0

• i(t) = dq/dt

• Ld2q/dt2 + qC = 0

• Simple Harmonic Motion!

• Pendulums

• Springs

• Standard solution!

• q(t)= Qmax cos(t+)

where 1/(LC)½



• q(t)= Qmax cos(t+)

• i(t) = - Qmax sin(t+)

(based on this ASSUMED direction!!)

• 1/(LC)½ = angular frequency


The L-C 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.

Date post:	26-Mar-2015
Category:	Documents
Upload:	stephanie-miles
View:	384 times
Download:	9 times

Copyright © 2012 Pearson Education Inc. PowerPoint ® Lectures for University Physics, Thirteenth...

Documents