1. Induction and Inductance
Chapter 30
Induction and Inductance

2. Magnetic Flux, 1 Flux: the flow of a quantity through a surface.
Magnetic Flux is proportional to the number of field lines that pass through a surface.
Also:
Magnetic flux is proportional to both the strength of the magnetic field passing through the plane of a wire loop wire and the area of the loop

3. Magnetic Flux, 2
(a) When the field is perpendicular to the plane of the loop, θ = 0 and ΦB = ΦB, max = BA
(b) When the field is parallel to the plane of the loop, θ = 90° and ΦB = 0
The flux can be negative, for example if θ = 180°
SI units of flux are T m² = Wb (Weber)

4. Magnetic Flux 3
The flux can be visualized with respect to magnetic field lines
The value of the magnetic flux is proportional to the total number of lines passing through the loop
When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum
When the area is parallel to the lines, no lines pass through the area and the flux is 0

5. Magnetic Flux Formula Φ measured in Weber (Wb) 1 Wb = 1T x 1 m2
B- magnetic field, n- normal, θ = angle between n and B, A surface area θ θ
Φ measured in Weber (Wb)
1 Wb = 1T x 1 m2

6. First Experiment:
1. A current appears only if there is relative motion between the loop and the magnet (one must move relative to the other); the current disappears when
the relative motion between them ceases.
2. Faster motion produces a greater current.
3. If moving the magnet’s north pole toward the loop causes, say, clockwise current, then moving the north pole away causes counterclockwise current.
Moving the south pole toward or away from the loop also causes currents, but in the reversed directions.
The current thus produced in the loop is called induced current.

7. Electromagnetic Induction
When a magnet moves toward a loop of wire, the ammeter shows the presence of a current (a)
When the magnet is held stationary, there is no current (b)
When the magnet moves away from the loop, the ammeter shows a current in the opposite direction (c)
If the loop is moved instead of the magnet, a current is also detected

8. An ammeter is connected in a circuit of a conducting loop
When a bar magnet is moved closer to, or farther from, the loop, an electromotive force (emf) is induced the loop
The ammeter indicates currents in different directions depending on the relative motion of magnet and loop
When the magnet stops moving, the current returns to zero as indicated by the ammeter

9. Second Experiment:
For this experiment we use the apparatus of Fig. 30-2, with the two conducting loops close to each other but not touching. If we close switch S, to turn on a current in the right-hand loop, the meter suddenly and
briefly registers a current—an induced current—in the left-hand loop. If we then
open the switch, another sudden and brief induced current appears in the left hand
loop, but in the opposite direction. We get an induced current (and thus an induced emf) only when the current in the right-hand loop is changing (either turning on or turning off) and not when it is constant (even if it is large).

10. Faraday's Law
Define the flux of the magnetic field through an open surface as: dA n B
Faraday's Law:
The emf e induced in a circuit is determined by the time rate of change of the magnetic flux through that circuit.
So what is
this emf??
The minus sign indicates direction of induced current (given by Lenz's Law).

11. Faraday’s Law of Induction:

12. Michael Faraday
Faraday is often regarded as the greatest experimental scientist of the 1800s. His contributions to the study of electricity include the invention of the electric motor, generator, and transformer.

13. Example Faraday 1. Write magnetic flux and solve for B: 2. Derive B:
An 80n coil that has the radius of 5 cm and a resistance of 30 Ω sits in a region that has a uniform magnetic field perpendicular to the coil. At what rate must the magnitude of the magnetic field change to produce a current of 4 A in the coil?
1. Write magnetic flux and solve for B:
2. Derive B:
3. Use Farday’s law to express the rate of of change of flux:
4. Use Ohm’s Law to calculate the induced emf required to produce a current of 4 A through a 30 ohm resistor:
5. Substitute the numerical values to calculate dB/dt:

14. The plane of a circular, 200-turn coil of radius 5
The plane of a circular, 200-turn coil of radius 5.25 cm is perpendicular to a uniform magnetic field produced by a large electromagnet. This field is changed at a steady rate from T to T in s. What is the magnitude of the emf induced in the coil?
110 V
170 V
1.7 V
26 V
87 V

15. The plane of a circular, 200-turn coil of radius 5
The plane of a circular, 200-turn coil of radius 5.25 cm is perpendicular to a uniform magnetic field produced by a large electromagnet. This field is changed at a steady rate from T to T in s. What is the magnitude of the emf induced in the coil?
110 V
170 V
1.7 V
26 V
87 V

16. a time-varying magnetic flux. a time-varying magnetic field.
According to Faraday's law, a necessary and sufficient condition for an electromotive force to be induced in a closed circuit loop is the presence in the loop of
a magnetic field.
magnetic materials.
an electric current.
a time-varying magnetic flux.
a time-varying magnetic field.

17. a time-varying magnetic flux. a time-varying magnetic field.
According to Faraday's law, a necessary and sufficient condition for an electromotive force to be induced in a closed circuit loop is the presence in the loop of
a magnetic field.
magnetic materials.
an electric current.
a time-varying magnetic flux.
a time-varying magnetic field.

18. The instantaneous induced emf in a coil of wire located in a magnetic field
depends on the time rate of change of flux through the coil.
depends on the instantaneous value of flux through the coil.
is independent of the area of the coil.
is independent of the number of turns of the coil.
is determined by the resistance in series with the coil.

19. The instantaneous induced emf in a coil of wire located in a magnetic field
depends on the time rate of change of flux through the coil.
depends on the instantaneous value of flux through the coil.
is independent of the area of the coil.
is independent of the number of turns of the coil.
is determined by the resistance in series with the coil.

20. Example, Induced emf in a coil due to a solenoid:

21. Faraday’s Law and Lenz’ Law
The minus sign is included because of the polarity of the emf. The induced emf in the coil gives rise to a current whose magnetic field OPPOSES ( Lenz’s law) the change in magnetic flux that produced it

22. Lenz’ Law
When applying Lenz’ Law, there are two magnetic fields to consider
The external changing magnetic field that induces the current in the loop
The magnetic field produced by the current in the loop

23. Lenz’ Law, Moving Magnet Example
B induced
(a) A bar magnet is moved to the right toward a stationary loop of wire. As the magnet moves, the magnetic flux increases with time
(b) The induced current produces a flux to the left to counteract the increasing external flux to the right

24. Lenz's Law B B v v B increasing Conservation of energy considerations:
The induced current will appear in such a direction that it opposes the change in flux that produced it.
B increasing v B N S v B S N
B induced
B induced
Conservation of energy considerations:
Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated.
Why???
If current reinforced the change, then the change would get bigger and that would in turn induce a larger current which would increase the change, etc..

25. Binduced
Binduced

26. Lenz’s Law:
Opposition to Pole Movement. The approach of the magnet’s north pole in Fig increases the magnetic flux through the loop, inducing a current in the loop. To oppose the magnetic flux increase being caused by the approaching magnet, the loop’s north pole (and the magnetic moment m) must face toward the approaching north pole so as to repel it. The current induced in the loop must be counterclockwise in Fig If we next pull the magnet away from the loop, a current will again be induced in the loop. Now, the loop will have a south pole facing the retreating north pole of the magnet, so as to oppose the retreat. Thus, the induced current will be clockwise.

27. Lenz’s Law:
Fig The direction of the current i induced in a loop is such that the current’s magnetic field Bind opposes the change in the
magnetic field inducing i. The field is always directed opposite an increasing field (a) and in the same direction (b) as a decreasing
field B. The curled–straight right-hand rule gives the direction of the induced current based on the direction of the induced field.
If the north pole of a magnet nears a closed conducting loop with its magnetic field directed downward, the flux through the loop increases. To oppose this increase in flux, the induced current i must set up its own field Bind directed upward inside the loop, as shown in Fig. 30-5a; then the upward flux of the field Bind opposes the increasing downward flux of field . The curled– straight right-hand rule then tells us that i must be counterclockwise in Fig. 30-5a.

28. Example, Induced emf and current due to a changing uniform B field:

29. Example, Induced emf and current due to a changing nonuniform B field:

30. Motional EMF

31. Application of Faraday’s Law – Motional emf
A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field
The electrons in the conductor experience a magnetic force
F = q v B
The electrons tend to move to the lower end of the conductor ℓ

32. Motional emf
As the negative charges accumulate at the base, a net positive charge exists at the upper end of the conductor
As a result of this charge separation, an electric field is produced in the conductor
Charges build up at the ends of the conductor until the downward magnetic force is balanced by the upward electric force
There is a potential difference between the upper and lower ends of the conductor

33. ε =Bℓv, voltage across the conductor
Motional emf, cont.
ε =Eℓ
F=qvB  
ε =Bℓv, voltage across the conductor  
If the motion is reversed, the polarity of the potential difference is also reversed
F=qE=q (ε/ℓ ) =qvB

34. Magnitude of the Motional emf using Faraday’s Law

35. Motional emf in a Circuit, cont.
The changing magnetic flux through the loop and the corresponding induced emf in the bar result from the change in area of the loop
The induced, motional emf, acts like a battery in the circuit

36. Motional emf in a Circuit
A conducting bar sliding with v along two conducting rails under the action of an applied force Fapp. The magnetic force Fm opposes the motion.

37. Motional emf general formula
If θ is the angle between the velocity of the conductor and the magnetic field than:
ε = Blvsinθ v θ B

38. Example: Operating a light bulb
Rod and rail have negligible resistance but the bulb has a resistance of 96 W, B=0.80 T, v=5.0 m/s and ℓ =1.6 m.
Calculate (a) emf in the rod, (b) induced current (c) power delivered to the bulb and (d) the energy used by the bulb in 60 s.
(a) ξ=vBℓ
e =(5.0 m/s)(0.80 T)(1.6 m)=6.4 V
(b) I=ξ/R
I=(6.4V)/(96 W)=0.067 A
(c) P=ξI
P=ξI=(6.4 V)(0.067 A)=0.43 W
(d) E=Pt
E=(0.43 W)(60 s)=26 J (=26 Ws)

39. Induction and Energy Transfers:
If the loop is pulled at a constant velocity v, one must apply a constant force F to the loop since an equal and opposite magnetic force acts on the loop to oppose it. The power is P=Fv.
As the loop is pulled, the portion of its area within the magnetic field, and therefore the magnetic flux, decrease. According to Faraday’s law, a current is produced in the loop. The magnitude of the flux through the loop is FB =BA =BLx.
Therefore,
The induced current is therefore
The net deflecting force is:
The power is therefore

40. Induction and Energy Transfers: Eddy Currents

41. Induced Electric Field:

42. Induced Electric Fields, Reformulation of Faraday’s Law:
Consider a particle of charge q0 moving around the circular path. The work W done on it in one revolution by the induced electric field is W =Eq0, where E is the induced emf.
From another point of view, the work is
Here where q0E is the magnitude of the force acting on the test charge and 2pr is the distance over which that force acts.
In general,

43. When a changing magnetic flux is present, the integral
Induced Electric Fields, A New Look at Electric Potential:
When a changing magnetic flux is present, the integral
is not zero but is dFB/dt.
Thus, assigning electric potential to an induced electric field leads us to conclude that electric potential has no meaning for electric fields associated with induction.

44. Example, Induced electric field from changing magnetic field:

45. Example, Induced electric field from changing magnetic field, cont.:

46. Self Inductance I ΦB ΦB is proportional to I
ΦB =L I, where L is a constant. I
Is by definition the SELF INDUCTANCE OF a circuit. Depends only on the “geometry” of the circuit..
Units: [L]SI = H (Henry)

47. Inductors and Inductance:
An inductor (symbol ) can be used to produce a desired magnetic field.
If we establish a current i in the “N” windings (turns) of the solenoid which can be treated as our inductor, the current produces a magnetic flux FB through the central region of the inductor.
The inductance of the inductor is then
The SI unit of inductance is the tesla–square meter per ampere (T m2/A). We call this the henry (H), after American physicist Joseph Henry,

48. Example: L of a Solenoid
Total magnetic flux through a solenoid with N turns and length l:
ΦB=N x (flux through one turn) = NBA B A
But the magnetic field inside a solenoid traveled by the current I is uniform, and given by the formula:
Using the definition of L:
we get:
Example: for a solenoid of length 10 cm, area 5 cm2, and 100 turns:

49. Self-Induced EMF:

50. RL Circuits L Loop rule: RC®RL: RC: Þ Therefore, Þ I R ea b L
At t=0, the switch is closed and the current I starts to flow.
Loop rule:
Note that this equation is identical in form to that for the RC circuit with the following substitutions: Þ
RC:
RC®RL: Þ
Therefore,

51. RL Circuits:
If we suddenly remove the emf from this same circuit, the charge does not immediately fall to zero but approaches zero in an exponential fashion:

53. RL Circuits:

54. RL Circuit - mechanical analogy
If we replace I with v, and dI/dt with a, and R with b (air resistance), and L with m, and identify Vbattery = constant with the constant force of gravity, mg, we have an equivalent equation:
mg = bv + ma, or mg – bv = ma (with down being the positive direction for a falling object).
From the mechanical analogy, this should be like having a mass with air resistance. If we have a constant force (like gravity), the object will accelerate up to a terminal speed (due to force of air resistance increasing up to the point where it balances the gravity).
SF=ma  -bv + mg = ma, or
m dv/dt + bv = mg

55. Plumber’s RL Analogy Valve Constriction P1 P2 Pump Flywheel
The “plumber’s analogy” of an RL circuit is a pump (=battery) pumping water in a closed loop of pipe that includes a valve (=switch), a constriction (=resistor), and rotating flywheel turned by the flow. When the valve starts the flow, the flywheel begins to spin until a steady flow is achieved and the pressure difference across the flywheel (P2-P3) goes to zero. P3
Pump = Battery
Valve = Switch
Constriction = Resistor
Flywheel = Inductor
Pressure = Potential Water Flow = Current

56. Capacitors: Early and Late
Initially, when a switch closes there is a potential difference of 0 across an uncharged capacitor. After a long time, the capacitor reaches its maximum charge and there is no current flow through the capacitor. Therefore, at t=0 the capacitor behaves like a short circuit (R=0), and at t=∞ the capacitor behaves like at open circuit (R=∞).
Example:
100 V
100 V
12.5 A
100 V
2.5 A
10 A
0 V
Circuit
at t=0
at t=∞
Calculate initial currents. IB = 100 V/8 W = 12.5 A
Calculate final potentials.

57. Inductors: Early and Late
Initially, when a switch closes an inductor appears to have an infinite resistance and has a maximum potential drop across it. Ultimately, the inductor reaches a steady current flow with no potential drop across it. Therefore, at t=0 the inductor behaves like at open circuit (R=∞), and at t=∞ the inductor behaves like a short circuit (R=0). This behavior is opposite that of a conductor.
Example:
V=10V
I = 3 A
I = 1 A
V=20V
I = 0 A
I = 1 A
Circuit
at t=0
at t=∞
Calculate initial potentials.
Calculate final currents.

58. Example, RL circuit, immediately after switching and after a long time:

59. Example, RL circuit, during a transition:

60. Energy Stored in a Magnetic Field:
Power dissipated by resistor
Power delivered by source
This is the rate at which magnetic potential energy UB is stored in the magnetic field.
This represents the total energy stored by an inductor L carrying a current i.

61. Example, Energy stored in a magnetic field:

62. Energy Density of a Magnetic Field:
Consider a length l near the middle of a long solenoid of cross-sectional area A carrying current i; the volume associated with this length is Al.
The energy UB stored by the length l of the solenoid must lie entirely within this volume because the magnetic field outside such a solenoid is approximately zero. Also, the stored energy must be uniformly distributed within the solenoid because the magnetic field is (approximately) uniform everywhere inside.
Thus, the energy stored per unit volume of the field is

63. Mutual Induction (optional):
The mutual inductance M21 of coil 2 with respect to coil 1 is defined as
The right side of this equation is, according to Faraday’s law, just the magnitude of the emf E2 appearing in coil 2 due to the changing current in coil 1.
Similarly,

64. Example, Mutual Inductance Between Two Parallel Coils:

65. Example, Mutual Inductance Between Two Parallel Coils, cont.: