1. Induced Voltages and Inductance
Chapter 20
Induced Voltages and Inductance

2. Electromagnetic Induction Sections 1–4

3. Michael Faraday 1791 – 1867 Great experimental scientist
Discovered electromagnetic induction
Invented electric motor, generator and transformer
Discovered laws of electrolysis

4. Faraday’s Experiment
A current can be induced by a changing magnetic field
First shown in an experiment by Michael Faraday
A primary coil is connected to a battery and a secondary coil is connected to an ammeter
When the switch is closed, the ammeter reads a current and then returns to zero
When the switch is opened, the ammeter reads a current in the opposite direction and then returns to zero
When there is a steady current in the primary circuit, the ammeter reads zero
WHY?

5. Faraday’s Startling Conclusion
An electrical current in the primary coil creates a magnetic field which travels from the primary coil through the iron core to the windings of the secondary coil
When the primary current varies (by closing/opening the switch), the magnetic field through the secondary coil also varies
An electrical current is induced in the secondary coil by this changing magnetic field
The secondary circuit acts as if a source of emf were connected to it for a short time
It is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field

6. Magnetic Flux
The emf is actually induced by a change in the quantity called the magnetic flux rather than simply by a change in the magnetic field
Magnetic flux is defined in a manner similar to that of electrical flux
Magnetic flux is proportional to both the strength of the magnetic field passing through the plane of a loop of wire and the area of the loop

7. Magnetic Flux, cont
Consider a loop of wire with area A in a uniform magnetic field
The magnetic flux through the loop is defined as
where θ is the angle between B and the normal to the plane
SI units of flux are T. m² = Wb (Weber)
Active Figure: Magnetic Flux

8. Magnetic Flux, cont
The value of the magnetic flux is proportional to the total number of lines passing through the loop
When the field is perpendicular to the plane of the loop (maximum number of lines pass through the area), θ = 0 and ΦB = ΦB, max = BA
When the field is parallel to the plane of the loop (no lines pass through the area), θ = 90° and ΦB = 0
Note: the flux can be negative, for example if θ = 180°

9. Electromagnetic Induction – An Experiment
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 in a similar manner
An induced emf is set up in the circuit as long as there is relative motion between the magnet and the loop
Active Figure: Induced Currents

10. Faraday’s Law and Electromagnetic Induction
The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit
If a circuit contains N tightly wound loops and the flux changes by ΔΦB during a time interval Δt, the average emf induced is given by Faraday’s Law:

11. Faraday’s Law and Lenz’ Law
The change in the flux, ΔΦB, can be produced by a change in B, A or θ
Since ΦB = B A cos θ
The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf, which is found by Lenz’ Law
The current caused by the induced emf travels in the direction that creates a magnetic field whose flux opposes the change in the original flux through the circuit

12. Lenz’ Law – Example 1
Consider an increasing magnetic field through the loop
The magnetic field becomes larger with time
magnetic flux increases
The induced current I will produce an induced field ind in the opposite direction which opposes the increase in the original magnetic field

13. Lenz’ Law – Example 2
Consider a decreasing magnetic field through the loop
The magnetic field becomes smaller with time
magnetic flux decreases
The induced current I will produce an induced field ind in the same direction which opposes the decrease in the original magnetic field

14. Applications of Faraday’s Law – Electric Guitar
A vibrating string induces an emf in a pickup coil
A permanent magnet inside the coil magnetizes a portion of the string nearest the coil
As the string vibrates at some frequency, its magnetized segment produces a changing flux through the pickup coil
The changing flux produces an induced emf that is fed to an amplifier

15. Applications of Faraday’s Law – Transformer
A varying voltage is applied to the primary coil
This causes a varying current in the primary coil which creates a changing magnetic field which travels from the primary coil through the iron core to the windings of the secondary coil
An electrical current is induced in the secondary coil by this changing magnetic field
The secondary circuit acts as if a voltage were connected to it

16. Application of Faraday’s Law – Motional emf
A complete electrical circuit is fashioned by a rectangular loop composed of a conductor bar, two conductor rails, and a load resistance R.
As the bar moves to the right with a given velocity, the free charges in the conductor experience a magnetic force along the length of the bar
This force sets up an induced current because the charges are free to move in the closed path of the electrical circuit

17. Motional emf, cont
As the bar moves to the right, the area of the loop increases by a factor of Δx during a time interval Δt
This causes the magnetic flux through the loop to increase with time
An emf is therefore induced in the loop given by

18. Motional emf, 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
The induced current, by Ohm’s Law, is
Active Figure: Motional emf

19. Lenz’ Law Revisited – Moving Bar Example 1
As the bar moves to the right, the magnetic flux through the circuit increases with time because the area of the loop increases
The induced current must be in a direction such that it opposes the change in the external magnetic flux
The induced current must be counterclockwise to produce its own flux out of the page which opposes the increase in the original magnetic flux

20. Lenz’ Law Revisited – Moving Bar Example 2
The bar is moving toward the left
The magnetic flux through the loop decreases with time because the area of the loop decreases
The induced current must be clockwise to produce its own flux into the page which opposes the decrease in the original magnetic flux

21. Lenz’ Law Revisited – Moving Magnet Example 1
A bar magnet is moved to the right toward a stationary loop of wire
As the magnet moves, the magnetic flux increases with time
The induced current produces a flux to the left which opposes the increase in the original flux, so the current is in the direction shown

22. Lenz’ Law Revisited – Moving Magnet Example 2
A bar magnet is moved to the left away from a stationary loop of wire
As the magnet moves, the magnetic flux decreases with time
The induced current produces an flux to the right which opposes the decrease in the original flux, so the current is in the direction shown

23. Lenz’ Law, Final Note
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