Chapter 31 Faraday’s Law

Introduction This section we will focus on the last of the fundamental laws of electromagnetism, called Faraday’s Law of Induction Michael Faraday – Determined Laws of Electrolysis – Invented electric motor, generator, and transformer.

Introduction In this chapter we will look at the processes in which a magnetic field (more importantly, a change in the magnetic field) can induce an electric current.

31.1 Faraday’s Law of Induction An emf and therefore, a current can be induced in a circuit with the use of a magnet. The magnetic field by itself is not capable of inducing a current.

31.1 A change in the magnetic field is necessary. – As the magnet is moved towards the current loop a positive current is measured.

31.1 – As the magnet is moved away from loop a negative current is measured. – Note that this also applies to stationary magnets and moving coils.

31.1 Here is the basic setup of actual experiment conducted by Faraday to confirm this phenomenon.

31.1 With the use of insulated wires, the first circuit and battery is completely isolated from the second circuit with the ammeter. – With the 1 st circuit open, there is no reading in the ammeter. – With the 1 st circuit closed, there is no reading in the ammeter.

31.1 The instant the switch is open, the ammeter needle deflects to one side and returns to zero. The instant the switch is closed the ammeter needle deflects to the opposite side and returns to zero.

31.1 So its not the magnetic field that induces the current, but the change in magnetic field. Faraday’s Law of Induction – The emf induced in a circuit is directly proportional to the time rate of change of magnetic flux through the circuit.

31.1 If the circuit is a coil with N number of loops of the same area, then Assuming a uniform magnetic field the magnetic flux is equal to BAcos θ so

31.1 So there are several things that change if there is going to be an induced current. – The magnitude of B can change with time. – The area enclosed by the loop can change with time. – The angle, between B and the area vector can change with time. – Any combination of the above.

31.1 Quick quizzes p Applications of Faraday’s Law – GFI- induced current in the coil trips the circuit breaker.

31.1 – Electric Guitar Pickups- the vibrating metal string induces a current in the coil.

31.1 Example 31.1, 31.2

31.2 Motional EMFs Motional EMF- induced in a conductor moving through a constant magnetic field. Consider a conductor length ℓ, moving through a constant magnetic field B, with velocity v.

31.2 The first thing we notice is that any free electrons (charge carriers) will feel a magnetic force as per F B = qv x B This will leave one end of the conductor with extra electrons, and the other with a deficit. This creates an electric field within the conductor which enacts a force on the electrons opposite of the magnetic force.

31.2 The forces up and down will balance giving The electric field is associated with the potential difference and the length of the conductor This potential difference is maintained as long as the conductor continues to move with velocity v through the field.

31.2 A more interesting example occurs when the conducting bar is part of a closed circuit. We assume zero resistance in the bar. The rest of the circuit has resistance R.

31.2 With the magnetic field present, and the conducting bar free to slide along the conducting rails, the same potential difference or EMF is produced, which drives a current through the circuit.

31.2 This is another example of Faraday’s law where the induced current is proportional to the changing magnetic flux (increasing area). Because the area at any time is A = ℓx, the magnetic flux is given as

31.2 From Faraday’s Law, the EMF will be

31.2 From this result and Ohm’s law, the induced current will be The source of the energy is the work done by the applied force.

31.2 Quick Quizzes p 975 Ex 31.4, 31.5

31.3 Lenz’s Law Faraday’s Law indicates that the induced emf and the change in flux have opposite signs. This physical effect is known as 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.

31.3 We will look at the sliding conductor example to illustrate. In this picture the magnetic flux is increasing. Since the magnetic field is into the page, the current induced creates a magnetic field out of the page.

31.3 If we switch the direction of travel for the bar, the flux through the loop is decreasing. The current is induced to oppose that change and creates additional magnetic field into the page.

31.3 We can examine the bar magnet and loop example again.

31.3

Quick Quizzes p. 979 Conceptual Example 31.6 Induced Current – The instant the switch closes – After a few seconds – The instant the switch is opened.

31.4 Induced EMF and Electric Fields An E-field within a conductor is responsible for moving charges through circuit. Since Faraday’s law discusses induced currents, we can claim that the changing magnetic field creates an E-field within the conductor.

31.4 In fact, a changing magnetic field generates an electric field even without a conducting loop. The E-field is however non-conservative unlike electrostatic fields. The work to move a charge around the loop is given as

31.4 The electric field in the ring is given as Knowing this and the fact that We can apply Faraday’s Law to get

31.4 So if we have B as a function of time, the induced current can easily be determined. The emf for any closed path can be given as the line integral of E. ds so Faraday’s Law is often given in the general form

31.4 The most important conclusion from this is the fact that a changing magnetic field, creates and electric field. Quick quiz p 982 Example 31.8

31.5 Generators and Motors Faraday’s Law has a primary application in Generators and Motors AC Generator- – Work is done to rotate a loop of wire in a magnetic field. – The changing magnetic flux creates an emf that alternates between positive and negative.

31.5

If we look at our rotating loop, the flux through single turn is given as And assuming a constant rotational speed of ω, Where θ = 0 at t = 0.

31.5 If we have more than 1 loop, say N loops, then Faraday’s Law gives the emf produced as

31.5 The maximum emf produced is given as When ω t = 90 o and 270 o Omega is named the angular frequency and is given as ω = 2 π f, where f is the frequency in Hz. Commercial generators in the US operate at f = 60 Hz.

31.5 Quick Quiz p. 984 Example 31.9 DC Generators – Operation very similar two AC Generators – Instead of 2 rings, a DC generator uses one split ring, called a commutator.

31.5 Commutator flips the polarity of the brushes in sync with the rotating loop, ensuring all emf is of one sign. While the emf is always positive, it pulses with time.

31.5 Pulsing DC current is not suitable for most applications, so multiple coil/commutator combos oriented at different angles are used simultaneously. By superimposing the emf pulses, we get a very nearly steady value.

31.5 Motors- Make use of electrical energy to do work. Generator operating in reverse- – Current is supplied so a loop in a magnetic field. – The torque on the loop causes rotation which can be applied to work.

31.5 The problem is we also have an emf induced because the magnetic flux changes as the loop rotates. From Lenz’s law this emf opposes the current running through the loop and is typically called a “Back emf”

31.5 When the motor is initially turned on the back emf is zero. As it speeds up the back emf increases. If a load is attached to the motor (to do work) the speed will drop and therefore back emf will as well. This draws higher than normal current from the voltage source running the motor.

31.5 If the load jams the motor, and it stops the motor can quickly burn out, from the increased current draw. Example 31.10

31.6 Eddy Currents Eddy Current- A circular current induced in a bulk piece of conductor moving through magnetic field.

31.6 By Lenz’s law the induced current opposes the changing flux and therefore creates a magnetic field on the conductor, that opposes the source magnetic field. Because of this the passing conductor behaves like an opposing magnetic and the force is resistive.

31.6

The concept is applied to mass transit braking systems which combine electromagnetic induction and Eddy currents to steadly slow subways/trains etc. Quick Quiz p. 987

31.7 Maxwell’s Equations James Clerk Maxwell developed a list of equations summarizing the fundamental nature of electricity and magnetism. – Gauss’s Law (Electric Fields) The total electric flux through a closed surface is proportional to the charge contained.

31.7 – Gauss’s Law (Magnetic Fields) The total magnetic flux through a closed surface is zero. Magnetic Monopoles have never been observed. – Faraday’s Law of Induction Electric Fields are created by changing magnetic flux

31.7 – Ampere-Maxwell Law Magnetic Fields are created by current Magnetic Fields are created by changing electric flux.

31.7 These 4 equations when joined with the Lorentz Force Law (below) completely describe all classical electromagnetic interactions. They are as fundamental to the understanding of the physical world as Newton’s Laws of Motion/Universal Gravitation