1. Chapter 30
Inductance

2. 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

3. 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

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

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

6. 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

7. 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

8. 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.

9. 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

10. 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

11. 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

12. 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)

13. 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

14. 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

15. 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?

16. 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!

17. 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

18. 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?

19. 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

20. 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?

21. 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

22. 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

23. 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.

24. 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

25. 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

26. 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!

27. 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

28. 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

29. 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.

30. 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.

31. 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.

32. 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.

33. 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)

34. 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)

35. 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! 

36. 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

37. 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

38. 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?

39. 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) )
L = 69 mH
 = L/R = 390 ms

40. 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

41. 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

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

43. 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

44. 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

45. 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!

46. 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

47. 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!

48. 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

49. 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

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

51. 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

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

53. 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

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

55. 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

56. 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

57. 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

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

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

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