Electromagnetic Induction

emf – electromotive force When you studied electric circuits, you

learned that a source of electrical energy, such as a battery is needed to produce a continuous current.

The potential difference, or voltage given to the charges by the battery is called the electromotive force, or emf.

Emf is not actually a force, it is a potential difference measured in volts.

Faraday’s Experiment - Induction

A current can be produced by a changing magnetic field (not a constant B field!) First shown in an experiment by Michael

Faraday A primary coil is connected to a battery A secondary coil is connected to an

ammeter

• From Ch. 24, a B field is produced by a moving charge. Well it works the other way too.

Faraday’s Experiment• The secondary circuit has a

current that is induced by the magnetic field that is created by current through the primary coil.

When the switch is closed, the ammeter deflects in one direction and then returns to zero. It responds to the changing magnetic field as the primary current ramps up.

When there is a steady current in the primary circuit, the ammeter reads zero.

When the switch is opened, the ammeter deflects in the opposite direction and then returns to zero (as the current ramps down.)

Faraday’s Conclusions

An electrical current is produced by a 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

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.

Magnetic Flux, 2

The box is a loop of wire. The wire is in a uniform

magnetic field B. The loop has an area A. The flux is defined as ΦB = BA = B A cos θ

θ is the angle between B and the normal to the plane

Flux ONLY counts the component of the B field that is perpendicular to the loop

Magnetic Flux, 3

When the field is perpendicular to the plane of the loop, as in (a), θ = 0 and ΦB = ΦB, max = BA

When the field is parallel to the plane of the loop, as in b, θ = 90° and ΦB = 0 The flux can be negative, for example if θ =

180° SI units of flux are T m² = Wb (Weber) (so if you see the unit of Weber/m2, that is a

Tesla)

ΦB = BA = B A cos θθ is the angle between B and the normal to the plane of the loop (one loop).

Magnetic Flux

The flux can be visualized with respect to magnetic field lines The value of the magnetic flux is

proportional to the total number of lines passing through the loop

When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum

When the area is parallel to the lines, no lines pass through the area and the flux is 0

emf and magnetic flux

Faraday’s law of induction states that the instantaneous emf induced in a circuit equals the rate of change of magnetic flux through the circuit:

or N = number of loops of wire emf = ε = induced voltage You will need to judge the – sign

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

Induction

Note the direction of the induced current.

Top picture, N end magnet moves towards loop.

Use RHR B on the current in the loop. Your thumb will point in opposite direction of motion of magnet.

Induced current always opposes change in B field (that’s why there’s a negative sign in the eq.)

Electromagnetic Induction – Results of the Experiment

A current is set up in the circuit as long as there is relative motion between the magnet and the loop The same experimental results are

found whether the loop moves or the magnet moves

The current is called an induced current because is it produced by an induced emf

PhET simulations on induction

faraday_en.jar

Go to “pickup coil” tab Note that these are electrons

moving in the coil, so use your left hand.

Applications of Faraday’s Law – Ground Fault Interrupters The ground fault

interrupter (GFI) is a safety device that protects against electrical shock Wire 1 leads from the wall

outlet to the appliance Wire 2 leads from the

appliance back to the wall outlet

The iron ring confines the magnetic field, which is generally 0 (B fields from currents in opposite directions cancel out)

If a leakage occurs, the field is no longer 0 and the induced voltage triggers a circuit breaker shutting off the current

Application of Faraday’s Law – Motional emf A straight conductor of

length ℓ moves perpendicularly with constant velocity through a uniform field

RHR says + charges go up LHR says - charges go down The electrons in the

conductor experience a magnetic force F = q v B

The electrons move to the lower end of the conductor (LHR)

B field goes into page

l

Motional emf

As the - charges accumulate at the base, that is like + charge moving upwards.

As a result of this charge separation, an electric field is produced in the conductor (see above)

Charges build up at the ends of the conductor until the downward magnetic force is balanced by the upward electric force F=qE = qvB so E = vB

There is a potential difference between the upper and lower ends of the conductor V=E*l (recall that E = Volts/meter, so V = E*length)

E

Motional emf Therefore the potential difference

between the ends of the conductor can be found by emf = ΔV = E l = B l v (ΔV = voltage or emf, L= length of wire, v=velocity) The upper end is at a higher potential than

the lower end A potential difference is maintained

across the conductor as long as there is motion through the field If the motion is reversed, the polarity of the

potential difference is also reversed

• A wire of length L, moving through a magnetic field, B, induces an electromotive force (EMF=Voltage).

• If the wire is part of a circuit, then there will be a current in the direction shown (+ charge moves downwards by RHR)

• This current will interact with the magnetic field and produce a force, F = BIL.

• Notice that the resulting force F= BIL, opposes the original motion, v, of the wire.

Motional emf in a closed loop circuit Assume all of the circuit except Rhas zero resistance. As the bar is pulled to the right with velocity v under the influence of an appliedforce, Fapp, the resulting emf is:

׀ along the length of the bar This force sets up an induced current (see direction

of I) because the charges are free to move in the closed path

EMF

Motional emf in a closed loop circuit 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

SO ׀ε׀ = EMF = B l v

Lenz’ Law Revisited-opposing the motion

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

This is why there is a negative sign in Faraday’s law

Original: Use RHR A: fingers into, thumb right, palm up, therefore current flows up. Induced: Use RHR A: fingers in, thumb up, palm left, induced force goes left.

Lenz’ Law, Opposing the Motion If the magnetic flux

due to the external field is increasing into the page, then the flux due to the induced current must be out of the page

Therefore the current must be counterclockwise when the bar moves to the right

(RHR B: Put thumb in direction of current, curl fingers, they point out of page!)

Lenz’ Law, Opposing the Motion

Now imagine the bar is moving toward the left (less area)

The magnetic flux through the loop is decreasing with time

The induced current must be clockwise to to produce its own flux into the page

RHR B: Thumb in direction of I, curl fingers, and they go into page (within the loop) adding to flux.

Lenz’ Law, Moving Magnet Example

A bar magnet is moved to the right toward a stationary loop of wire (a) (the diagram on the left) As the magnet gets closer, magnetic flux

increases The induced current (note direction of I)

produces a flux to the left, so the current is in the direction shown for loop (b)

Note B field for (b) opposes change in B field for (a)

Eddy Current Videos

2’ Neodymium magnet in a copper pipe http://www.youtube.com/watch?v=E97CYWlALEs

4’ Eddy current demo w/ Alum pendulum http://www.youtube.com/watch?v=OJvEOXsSuaQ

2’ Russian guy neodymium magnet/Eddy current http://www.youtube.com/watch?v=tDBigHmn-FQ

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 first loop

The magnetic field produced by the current in the second loop

Generators

Alternating Current (AC) generator Converts mechanical energy to

electrical energy Consists of a wire loop rotated by

some external mechanical means There are a variety of sources that

can supply the energy to rotate the loop

These may include falling water, heat by burning coal to produce steam, nuclear power, wind power, etc.

AC Generators As the loop rotates, the

magnetic flux through it changes with time (max when it’s perpendicular to the B field)

This induces an emf, which creates a current in the external circuit that runs the toaster

The ends of the loop are connected to slip rings that rotate with the loop

Connections to the external circuit are made by stationary brushes in contact with the slip rings

Generators The emf generated by the

rotating loop can be found by

The reason for the 2 above is that there is an emf coming from segment BC and one from AD (both with the same sense/direction of rotation).

Since the axis of rotation r= a/2 (see graphic >) and v=rω and θ=ωt, [ω = 2f, f= freq. in Hz)

Generators (cont.)

If the coil has N turns, the emf is N times as large because each loop has the same emf in it.

Also using the fact that A=la (see diagram), the total emf is

And since the sin function is a maximum at 90 or 270 degrees, at those angles,

When loop is parallel to B field

AC Generators – Summary If the loop rotates with a constant angular speed, ω, and has N turns

ε = N B A ω sin ω t A=area of loop, N = no. of loops ε = εmax when loop is parallel to B because the

component of velocity perpendicular to the B field is the greatest at that point.

ε = 0 when when the loop is perpendicular to B

where f is the frequency in Hertz (60 Hz in USA) so is in radians/sec and =angular velocity.

DC Generators

Components are essentially the same as that of an AC generator

The major difference is the contacts to the rotating loop are made by a split ring or commutator.

DC Generators

The output voltage always has the same polarity. The current is a pulsingcurrent To produce a steady current, many loops

and commutators around the axis of rotation are used

The multiple outputs are superimposed and the output is almost free of fluctuations (not like the graph above)

Motor example / commutator

http://www.youtube.com/watch?v=qqkUeQ0nsF8

About 46 minutes into video MIT 8.02 Lecture 11

RMS voltage and current

For the sine wave signal from an AC generator, the effective voltage (also known as root-mean-square voltage) is calculated by

Vrms = 0.707 Vmax (.707 = )

and likewise for current Irms = 0.707 I max

QUICK QUIZ 1

The figure below is a graph of magnitude B versus time t for a magnetic field that passes through a fixed loop and is oriented perpendicular to the plane of the loop. Rank the magnitudes of the emf generated in the loop at the three instants indicated (a, b, c), from largest to smallest.

QUICK QUIZ 1 ANSWER

(b), (c), (a). At each instant, the magnitude of the induced emf is proportional to the rate of change of the magnetic field (hence, proportional to the slope of the curve shown on the graph).