AP C UNIT 11 ELECTROMAGNETIC

INDUCTION

Recall Electric Flux

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Welcome our newest concept… Magnetic Flux

Faraday’s Observations,1830• When a magnet moves

toward a loop of wire, the ammeter

• When the magnet is held stationary,

• When the magnet moves away from the loop, the ammeter

• If the loop is moved instead of the magnet,

Experimental Conclusions

• A current is set up in the circuit as long as there is ______________between the magnet and the loop– The same experimental results are found whether the loop

moves or the magnet moves

• The current is called ____________since there is no power source.

• An EMF is actually

Faraday’s Law & Electromagnetic Induction

• The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit.

EM induction refers to electricity deriving from magnetism whereas electromagnetism is the opposite.

Traffic light sensors

There is an inductive loop at intersection of Ft Wash & Susquehanna.

Electric Guitar• A vibrating string induces

an emf in a coil• A permanent magnet

inside the coil magnetizes a portion of the string nearest the coil

• As the string vibrates at some frequency, its magnetized segment produces a changing flux through the pickup coil

• The changing flux produces an induced emf that is fed to an amplifier

Apnea Monitor

• The coil of wire attached to the chest carries an alternating current

• An induced emf produced by the varying field passes through a pick up coil

• When breathing stops, the pattern of induced voltages stabilizes and external monitors sound an alert

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– If a leakage occurs, the field

is no longer 0 and the induced voltage triggers a circuit breaker shutting off the current

• Faraday's Law is the basic principle behind the microphone. In a microphone there is a diaphragm, around which a coil is wrapped, which can move back and forth in response to sound waves. A stationary bar magnet, placed near the coil, induces current in the coil which can then be transmitted (with amplification) to the speaker.

A long wire carries current i a distance ‘d’ from a rectangular wire loop as shown above. Determine an expression for the flux through loop.

i

w

l

d

Example

Suppose that i(t) = 3t +1. Find induced voltage in loop

Negative sign explained in Faraday’s Law

The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf, which is found by Lenz’ Law:

i

w

l

d

If i(t) = 3t +1, what was direction of induced current in loop as t increases?

Lenz’s Law examples

Determine direction of induced current in loop

Determine direction of induced current in loop as magnet approaches loop area.

Determine direction of induced current in loop as loop gets smaller.

When switch is closed, describe current flow in R

IRON CORE

Describe current through R when I goes to zero.

Which situation(s) cause(s) induced current?

Example

(a) ccw (b) cw (c) no induced current

What is the direction of the induced current in the loop?

A conducting rectangular loop moves with constant velocity v in the -y direction and a constant current I flows in the +x direction as shown x

v

Iy

Iinduced

GeneratorA coil of wire turns in a magnetic field. The flux in the coil is constantly changing, generating an emf in the coil.

Converts mechanical energy to electrical energy

If loop is made to rotate at constant rate ω in uniform B, we have from Faraday’s Law:

(e)Motional EMF

• A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field

ℓℓ

A conducting bar is placed across conducting path and pulled to right with speed v as shown.

As bar moves, a change in flux occurs which induces CCW current.

Also, a magnetic force on bar arises which acts as a resistance to the motion of the bar as it is pulled to the right

Lenz’ Law Revisited, Conservation of Energy Consequence• Assume the induced current is

clockwise instead…– The magnetic force on the bar

would be to the right– The force would cause an

acceleration and the velocity would increase

– This would cause the flux to increase and the current to increase and the velocity to increase…

exampleA metal rod of mass 0.22kg lies across two parallel conducting rails which sits on a tabletop as shown. The rod and rails have negligible resistance but significant friction where uk=0.20. A field of 0.80T points into page. A string pulls the rod to right at a constant speed of 1.8m/s.

a) Calculate the force needed to pull rod at constant speed.

b) Calculate the energy dissipated in the resistor in 2.0s.

c) Calculate the work done by string in 2.0s.

ExampleA conducting rod with mass m and length L moves on two frictionless parallel rails in the presence of a uniform magnetic field. The bar is given an initial velocity vi at time t=0. Calculate the velocity of the bar as a function of time. Bar will slow down due to resistive force.

The magnitude of the magnetic force is given by

Now, using Newton's second law, we can write the net force on the conducting rod as

Eddy currentsEddy currents are small circular or swirling currents that arise in conductors like a sheet of metal.

Eddy currents lead to heat being generated in the conductor

This is the basic principle behind induction stoves. Safe to touch unless you are metallic. Eddy currents are established in cookware causing metal to heat up.

Magnetic Braking

Analog speedometers

Rollercoaster brakes

Metal Detectors

Transformers

A transformer is a device used to change the voltage in a circuit. AC currents must be used.

75,000 V in the power lines

120 V in your house

Electrical Power Transmission• When transmitting electric power over long distances,

it is most economical to use high voltage and low current, which minimizes I2R power losses.

Ampere's Law has shown us how currents, moving electric charges, can create magnetic fields.

Faraday's Law has shown us how changing magnetic fields can induce an emf in a closed loop.

Induced E-fields

Consider a loop of wire outside of a solenoid. Current is flowing through solenoid from back to front where B-field from coils is into page.

Side View Front View

If the current is dereased gradually, the magnetic field in the solenoid's core decreases and an emf & Iinduced will be induced in the wire loop (LENZ LAW).

The force that pushes the charges around the wire is F = qE, where E is the induced electric field.

E-fields produced by either static charge or induction both exert forces on charged particles, however, there is an important difference.

Static E Induced E

b

a

ab dEV Recall we previously learned that

the potential difference between 2 points in a static field is given by:

However, in a changing B-field case…

As charge makes journey around closed loop, it must be experiencing an emf, however, to interpret that as a changing potential, it doesn’t make sense. Why?

As magnetic field increases in time through loop, an electric field is generated

timeB

Consider B-field between pole of electromagnet. Assume B to be uniform at any instant over a circular radius R. The current in the windings of the electromagnet increase with time. Beyond the circular region (r > R), assume B=0. Find E at any distance r from the center.

S

N

side view top view, looking down on N pole

r r

E

E

EXAMPLE: If B = 0.75T and B is decreasing at a rate of -0.035T/s.

A) What is the shape of the induced E-field on the conducting ring?

B) What are the magnitude and direction of this field at any point on the conducting ring with radius 0.1m?

c) Determine the current in the ring if the resistance is 4.0Ω.

d) What is the EMF between a and b?

e) If the ring is cut at some point and the ends are separated slightly, what will be the EMF between the ends?

InductanceSimilar to the idea of capacitance (holding onto charge), inductance deals with how well an inductor holds onto a magnetic field.

Energy Density of solenoid

Close switch

As soon as current appears at the first coil of the inductor, a change in magnetic flux is created, and therefore an EMF. This EMF pushes opposite to the EMF causing the flux in the first place, according to Lenz's Law

RL CIRCUIT

We expect that the current in the inductor, and hence in the entire circuit, must increase over time until it reaches its maximum value of imax = ε / R where ε is the voltage provided by the EMF source. Resistance of inductor goes to zero whereas in capacitor we say it becomes infinite R.

di/dt is positive since current is increasing, however, the EMF induced across the inductor is negative since it is pushing current in opposite direction to oppose change, therefore a minus sign is added

Going CW around circuit:

Current versus time in an R-L circuit

63%Imax

Flipping switch back causes decrease in current. Similar analysis yields:

37%Imax

Example: Close S1. After a long time:

a) Find value of current in inductor at moment S1 is closed.

b) Find value of current in inductor after long time.

Simultaneously open S1 and close S2.

At that moment, what is current in inductor?

Find current at t=2.0x10-3s in inductor.

20V

100Ω 0.10H

Example 240V

2.2H160Ω

Determine the work done by the battery from t = 0s to 0.0165s.

LC CircuitsAnalogous to an oscillating mass on a spring

The Oscillation Cycle

Prior to discharge, all energy resides in E-field of capacitor. When capacitor discharges, current flows CW and gets larger, B-field emerges in inductor resisting change.

At t=T/4, capacitor has zero charge, current has max value, B-field is max with all energy now residing in inductor’s field.

Eventually charge starts to accumulate on capacitor, current dies, B-field decreases to zero. At t=T/2, all energy is back in E-field with polarity of capacitor reversed.

Process repeats itself returning to state it was at t = T/4. At t = 3T/4 energy is back in B-field. This is called electrical oscillation.

LC Circuit Analysis:

What differentiates an LC circuit from the RC or RL circuit is the fact that current in both the RC and RL circuits changes exponentially towards a steady state.However, in the LC circuit, the current oscillates, never reaching steady state.

L C Circuit Analogy to SHM

Current and Charge variations with time for LC circuit

System oscillates according to:

Just like resonance in SHM, LC circuit has electrical resonance. Your radio is tuned in to a resonant frequency by changing the capacitance

Maxwell’s EquationsWe now gather all of the governing equations together

Maxwell retooled Ampere’s law which is only good for static case, not oscillating situations

The net magnetic flux is zero through a closed surface. B field lines cannot begin or end at any point. If they did, monopoles would exist

These four equations describe all of classical electric and magnetic phenomena

Maxwell’s Equations

Maxwell’s own contribution is just the last term of the last equation but realizing the necessity of that term had dramatic consequences. It made evident for the first time that varying electric and magnetic fields could feed off each other & these fields could propagate indefinitely through space, far from the varying charges and currents where they originated. Previously, the fields had been envisioned as tethered to the charges and currents giving rise to them. Maxwell’s new term (he called it the displacement current) freed them to move through space in a self-sustaining fashion, and even predicted their velocity it was the velocity of light!

As the E-field starts to change, that in turn induces a B-field which in turns induces E and so on.