Faraday's Law of Electromagnetic Induction: Induced Voltages and Inductance

1. 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 inducedemf 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