1. Chapter 31
Faraday’s Law

2. Michael Faraday Great experimental physicist and chemist 1791 – 1867
Contributions to early electricity include:
Invention of motor, generator, and transformer
Electromagnetic induction
Laws of electrolysis: A method of separating bonded elements and compounds by passing an electric current through them

3. Induction An induced current is produced by a changing magnetic field
There is an induced emf associated with the induced current
A current can be produced without a battery present in the circuit
Faraday’s law of induction describes the induced emf

4. EMF Produced by a Changing Magnetic Field, 1
A loop of wire is connected to a sensitive ammeter
When a magnet is moved toward the loop, the ammeter deflects

5. EMF Produced by a Changing Magnetic Field, 2
When the magnet is held stationary, there is no deflection of the ammeter
Therefore, there is no induced current
Even though the magnet is in the loop

6. EMF Produced by a Changing Magnetic Field, 3
The magnet is moved away from the loop
The ammeter deflects in the opposite direction

7. Faraday’s Law – Statements
Faraday’s law of induction states that “the emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit”
Mathematically,

8. Faraday’s Law – Statements, cont
Remember FB is the magnetic flux through the circuit and is found by
If the circuit consists of N loops, all of the same area, and if FB is the flux through one loop, an emf is induced in every loop and Faraday’s law becomes

9. Faraday’s Law – Example
Assume a loop enclosing an area A lies in a uniform magnetic field B
The magnetic flux through the loop is FB = BA cos q
The induced emf is
e = - d/dt (BA cos q)

10. Ways of Inducing an emf The magnitude of B can change with time
The area enclosed by the loop can change with time
The angle q between B and the normal to the loop can change with time
Any combination of the above can occur

11. Lenz’s Law
Faraday’s law indicates that the induced emf and the change in flux have opposite algebraic signs
This has a physical interpretation that has come to be known as Lenz’s law
Developed by German physicist Heinrich Lenz

12. Lenz’s Law, cont.
Lenz’s law: the induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop
The induced current tends to keep the original magnetic flux through the circuit from changing

13. Problem 1
A loop of wire in the shape of a rectangle of width w and length L and a long, straight wire carrying a current I lie on a tabletop. Determine the magnetic flux through the loop due to the current I. Suppose the current is changing with time according to I = a + bt, where a and b are constants. Determine the emf that is induced. What is the direction of the induced current in the rectangle?

14. Problem 2
The square loop in the figure is made of wires with a total resistance of 20.0 W . It is placed in a uniform T magnetic field directed perpendicular into the plane of the paper. The loop, which is hinged at each corner, is pulled as shown until the separation between points A and B is 3.00 m. What is the magnitude and direction of the induced current. [0.0301] A, clockwise

15. Problem 3
A 39 turn circular coil of radius 4.80 cm and resistance 1.00 W is placed in a magnetic field directed perpendicular to the plane of the coil. The magnitude of the magnetic field varies in time according to the expression B = t t2, where t is in seconds and B is in tesla. Calculate the induced emf in the coil at t = 5.60 s. [129 mV]

16. Applications of Faraday’s Law – Electric Generator
An electric conductor, like a copper wire, is moved through a magnetic field, which causes an electric current to flow (be induced) in the conductor

17. Generators
Electric generators take in energy by work and transfer it out by electrical transmission
The AC generator consists of a loop of wire rotated by some external means in a magnetic field

18. Applications of Faraday’s Law – Pickup Coil
The coil is placed near the vibrating string and causes a portion of the string to become magnetized
When the string vibrates at the same frequency, the magnetized segment produces a changing flux through the coil
The induced emf is fed to an amplifier

19. Applications of Faraday’s Law – Transformers
An alternating current in one winding creates a time-varying magnetic flux in the core, which induces a voltage in the other windings
Is this a step-up or a step- down transformer?

20. Yamanashi maglev
The magnetized coil running along the track, called a guideway, repels the large magnets on the train's undercarriage, allowing the train to levitate between 0.39 and 3.93 inches (1 to 10 cm) above the guideway.

21. Sliding Conducting Bar
A bar moving through a uniform field and the equivalent circuit diagram
Assume the bar has zero resistance
The work done by the applied force appears as internal energy in the resistor R

22. Sliding Conducting Bar, cont.
The induced emf is
Since the resistance in the circuit is R, the current is

23. Sliding Conducting Bar, Energy Considerations
The applied force does work on the conducting bar
This moves the charges through a magnetic field
The change in energy of the system during some time interval must be equal to the transfer of energy into the system by work
The power input is equal to the rate at which energy is delivered to the resistor

24. Induced emf and Electric Fields
An electric field is created in the conductor as a result of the changing magnetic flux
Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space
This induced electric field is nonconservative
Unlike the electric field produced by stationary charges

25. Induced emf and Electric Fields, cont.
The emf for any closed path can be expressed as the line integral of E.ds over the path
Faraday’s law can be written in a general form:

26. Maxwell’s Equations, Statement
Faraday’s Law
Ampere-Maxwell Law
Gauss’s Law in Magnetism
Gauss’s Law in Electricity

27. Maxwell’s Equations, Details
Gauss’s law (electrical):
The total electric flux through any closed surface equals the net charge inside that surface divided by eo
This relates an electric field to the charge distribution that creates it

28. Maxwell’s Equations, Details 2
Gauss’s law (magnetism):
The total magnetic flux through any closed surface is zero
This says the number of field lines that enter a closed volume must equal the number that leave that volume
This implies the magnetic field lines cannot begin or end at any point
Isolated magnetic monopoles have not been observed in nature

29. Maxwell’s Equations, Details 3
Faraday’s law of Induction:
This describes the creation of an electric field by a changing magnetic flux
The law states that the emf, which is the line integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that path
One consequence is the current induced in a conducting loop placed in a time-varying B

30. Maxwell’s Equations, Details 4
The Ampere-Maxwell law is a generalization of Ampere’s law
It describes the creation of a magnetic field by an electric field and electric currents
The line integral of the magnetic field around any closed path is the given sum

31. The Lorentz Force Law
Once the electric and magnetic fields are known at some point in space, the force acting on a particle of charge q can be calculated
F = qE + qv x B
This relationship is called the Lorentz force law
Maxwell’s equations, together with this force law, completely describe all classical electromagnetic interactions

32. Maxwell’s Equations, Symmetry
The two Gauss’s laws are symmetrical, apart from the absence of the term for magnetic monopoles in Gauss’s law for magnetism
Faraday’s law and the Ampere-Maxwell law are symmetrical in that the line integrals of E and B around a closed path are related to the rate of change of the respective fluxes
Maxwell’s equations are of fundamental importance to all of science