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 1791-1867– 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 1st circuit open, there is no reading in the
ammeter.– With the 1st 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.
dtd B
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 BAcosθ so
dtdN B
cosBAdtd
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. 970-971
• 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 FB = 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.
qvBqE vBE
vBEV
31.2
• A more interesting example occurs when the conducting bar is part of a closed circuit.
• We assume zero resistancein 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
xBB
31.2
• From Faraday’s Law, the EMF will be
xBdtd
dtd B
dtdxB
vB
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.
RvB
RI
RR
vBvBIvFP A
2222
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
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
rqEFdqW 2
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
rE
2
dtdBr
dtdr
E B
221
BrBAB2
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
dtdd BsE
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
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.
cosBAB
t
tBAB cos
31.5
• If we have more than 1 loop, say N loops, then Faraday’s Law gives the emf produced as
dtdN B
tdtdNAB cos
tNAB sin
31.5
• The maximum emf produced is given as
• When ωt = 90o and 270o
• Omega is named the angular frequency and is given as ω = 2πf, where f is the frequency in Hz.
• Commercial generators in the US operate at f = 60 Hz.
NABmax
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
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.
oS
qd
AE
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
0S
dAB
dtdd B
sE
31.7
– Ampere-Maxwell Law
• Magnetic Fields are created by current• Magnetic Fields are created by changing electric flux.
dtdId E
ooo
sE
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
BvEF qq