Chapter 21

Magnetic Flux and Faraday’s Law of Induction

What is E/M Induction?Electromagnetic Induction is the process of

using magnetic fields to produce voltage, and in a complete circuit, a current.

Michael Faraday first discovered it, using some of the works of Hans Christian Oersted. His work started at first using different combinations of wires and magnetic strengths and currents, but it wasn't until he tried moving the wires that he got any success.

It turns out that electromagnetic induction is created by just that - the moving of a conductive substance through a magnetic field.

The Space Tether Experiment

•  The Space Tether Experiment

• The space tether experiment, a joint venture of the US and Italy, called for a scientific payload--a large, spherical satellite--to be deployed from the US space shuttle at the end of a conducting cable (tether) 20 km (12.5 miles) long. The idea was to let the shuttle drag the tether across the Earth's magnetic field, producing one part of a dynamo circuit. The return current, from the shuttle to the payload, would flow in the Earth's ionosphere, which also conducted electricity, even though not as well as the wire.

•     One purpose of such a set-up might be to produce electric power, generating current to run equipment aboard the space shuttle. That electric comes at a price: it is taken away from the motion energy ("kinetic energy") of the shuttle, since the magnetic force on the tether opposes the motion and slows it down. In principle, it should also be possible to reverse this process: a future space station could use solar cells to produce an electric current, which would be pumped into the tether in the opposite direction, so that the magnetic force would boost the orbital motion and would raise the orbit to a higher altitude.

Magnetic InductionAs the magnet moves back and forth a current is said to be

INDUCED in the wire.

Faraday’s experiment: closing the switch in the primary circuit induces a current in the secondary circuit, but only while the current in the primary circuit is changing.

• The current in the secondary circuit is zero as long as the current in the primary circuit, and therefore the magnetic field in the iron bar, is not changing.

• Current flows in the secondary circuit while the current in the primary is changing. It flows in opposite directions depending on whether the magnetic field is increasing or decreasing.

• The magnitude of the induced current is proportional to the rate at which the magnetic field is changing.

Note the motion of the magnet in each image:

Magnetic FluxThe first step to understanding the complex nature of

electromagnetic induction is to understand the idea of magnetic flux.

Flux is a general term associated with a FIELD that is bound by a certain AREA. So MAGNETIC FLUX is any AREA that has a MAGNETIC FIELD passing through it.

AB

We generally define an AREA vector as one that is perpendicular to the surface of the material. Therefore, you can see in the figure that the AREA vector and the Magnetic Field vector are PARALLEL. This then produces a DOT PRODUCT between the 2 variables that then define flux.

Wb)or Weber( Tm :Unit

cos2

BA

AB

B

B

Magnetic Flux

How could we CHANGE the flux over a period of time? We could move the magnet away or towards (or the wire). We could increase or decrease the area. We could ROTATE the wire along an axis that is PERPENDICULAR to the field thus changing the angle between the area and magnetic field vectors.

Magnetic flux is used in the calculation of the induced emf.

Faraday’s LawFaraday learned that if you change any part of the flux over time

you could induce a current in a conductor and thus create a source of EMF (electromotive force, voltage, potential difference). Since we are dealing with time here, we are a talking about the RATE of CHANGE of FLUX, which is called Faraday’s Law.

wireof turns#

)cos(

Nt

BAN

tN B

Faraday’s law: An emf is induced only when the magnetic flux through a loop changes with time.

There are many devices that operate on the basis of Faraday’s law.

An electric guitar pickup:

The Forever Flashlight uses the Faraday Principle of Electromagnetic Energy to eliminate the need for batteries. The Faraday Principle states that if an electric conductor, like copper wire, is moved through a magnetic field, electric current will be generated and flow into the conductor.

Tape recorder:

AC motors use Faraday’s law to produce rotation and thus convert electrical and magnetic energy into rotational kinetic energy. This idea can be used to run all kinds of motors. Since the current in the coil is AC, it is turning on and off thus creating a CHANGING magnetic field of its own. Its own magnetic field interferes with the shown magnetic field to produce rotation.

DC motors use a commutator to change the direction of the current flow. This keeps torque applied in one rotational direction.

Commutator

TransformersProbably one of the greatest inventions of all time is the

transformer. AC Current from the primary coil moves quickly BACK and FORTH (thus the idea of changing!) across the secondary coil. The moving magnetic field caused by the changing field (flux) induces a current in the secondary coil.

If the secondary coil has MORE turns than the primary you can step up the voltage and runs devices that would normally need MORE voltage than what you have coming in. We call this a STEP UP transformer.

We can use this idea in reverse as well to create a STEP DOWN transformer.

MicrophonesA microphone works when

sound waves enter the filter of a microphone. Inside the filter, a diaphragm is vibrated by the sound waves which in turn moves a coil of wire wrapped around a magnet. The movement of the wire in the magnetic field induces a current in the wire. Thus sound waves can be turned into electronic signals and then amplified through a speaker.

Sample Problem: A coil of radius 0.5 m consisting of 1000 loops is placed in a 500 mT magnetic field such that the flux is maximum. The field then drops to zero in 10 ms. What is the induced potential in the coil?

Sample Problem: A coil of radius 0.5 m consisting of 1000 loops is placed in a 500 mT magnetic field such that the flux is maximum. The field then drops to zero in 10 ms. What is the induced potential in the coil?

Sample Problem: A coil with 200 turns of wire is wrapped on an 18.0 cm square frame. Each turn has the same area, equal to that of the frame, and the total resistance of the coil is 2.0 . A uniform magnetic field is applied perpendicularly to the plane of the coil. If the field changes uniformly from 0 to 0.500 T in 0.80 s, find the magnitude of the induced emf in the coil while the field has changed as well as the magnitude of the induced current.

Sample Problem: A coil with 200 turns of wire is wrapped on an 18.0 cm square frame. Each turn has the same area, equal to that of the frame, and the total resistance of the coil is 2.0 . A uniform magnetic field is applied perpendicularly to the plane of the coil. If the field changes uniformly from 0 to 0.500 T in 0.80 s, find the magnitude of the induced emf in the coil while the field has changed as well as the magnitude of the induced current.

Lenz’s LawAn induced current always flows in a direction that opposes the change that caused it.

Therefore, if the magnetic field is increasing, the magnetic field created by the induced current will be in the opposite direction; if decreasing, it will be in the same direction.

23-4 Lenz’s Law

This conducting rod completes the circuit. As it falls, the magnetic flux decreases, and a current is induced.

23-4 Lenz’s Law

The force due to the induced current is upward, slowing the fall.

23-4 Lenz’s Law

Currents can also flow in bulk conductors. These induced currents, called eddy currents, can be powerful brakes.

Lenz’s LawLenz's law gives the direction of the induced emf and current

resulting from electromagnetic induction. The law provides a physical interpretation of the choice of sign in Faraday's law of induction, indicating that the induced emf and the change in flux have opposite signs.

tN B

Lenz’s Law

In the figure above, we see that the direction of the current changes. Lenz’s Law helps us determine the DIRECTION of that current.

Lenz’s Law & Faraday’s Lawt

N B

Let’s consider a magnet with it’s north pole moving TOWARDS a conducting loop.

DOES THE FLUX CHANGE?

DOES THE FLUX INCREASE OR DECREASE?

WHAT SIGN DOES THE “” GIVE YOU IN FARADAY’S LAW?

DOES LENZ’S LAW CANCEL OUT?What does this mean?

Yes!

Increase

Positive

NO

This means that the INDUCED MAGNETIC FIELD around the WIRE caused by the moving magnet OPPOSES the original magnetic field. Since the original B field is downward, the induced field is upward! We then use the right hand rule to determine the direction of the current.

Binduced

Lenz’s Law

A magnet is dropped down a conducting tube.

The magnet INDUCES a current above and below the magnet as it moves.

The INDUCED current creates an INDUCED magnetic field of its own inside the conductor that opposes the original magnetic field.

Since the induced field opposes the direction of the original it attracts the magnet upward slowing the motion caused by gravity downward.

Lenz’s LawLet’s consider a magnet with it’s north pole moving AWAY from a conducting loop.

DOES THE FLUX CHANGE?

DOES THE FLUX INCREASE OR DECREASE?

WHAT SIGN DOES THE “” GIVE YOU IN FARADAY’S LAW?

DOES LENZ’S LAW CANCEL OUT?

What does this mean?

tN B

Yes!

Decreases

negativeyes

In this case, the induced field DOES NOT oppose the original and points in the same direction. Once again use your curled right hand rule to determine the DIRECTION of the current.

Binduced

23-5 Mechanical Work and Electrical Energy

This diagram shows the variables we need to calculate the induced emf.

23-5 Mechanical Work and Electrical Energy

Change in flux:

Induced emf:

Electric field caused by the motion of the rod:

23-5 Mechanical Work and Electrical Energy

If the rod is to move at a constant speed, an external force must be exerted on it. This force should have equal magnitude and opposite direction to the magnetic force:

23-5 Mechanical Work and Electrical Energy

The mechanical power delivered by the external force is:

Compare this to the electrical power in the light bulb:

Therefore, mechanical power has been converted directly into electrical power.

Summary of Chapter 23

• A changing magnetic field can induce a current in a circuit. The magnitude of the induced current depends on the rate of change of the magnetic field.

• Magnetic flux:

• Faraday’s law gives the induced emf:

Summary of Chapter 23

• Lenz’s law: an induced current flows in the direction that opposes the change that created the current.

• Motional emf:

• emf produced by a generator:

• An electric motor is basically a generator operated in reverse.

• Inductance occurs when a coil with a changing current induces an emf in itself.

Summary of Chapter 23

• Definition of inductance:

• Inductance of a solenoid:

• An RL circuit has a characteristic time constant:

Summary of Chapter 23

• Current in an RL circuit after closing the switch:

• Magnetic energy density:

• Transformer equation: