Class 31 Today we will: learn about EMF learn how Faraday’s law works learn Lenz’s Law and how to apply it

Last Time -- Induced Current Accelerating charges produce electric fields in the opposite direction to the acceleration. i i

Faraday’s Law If the number of magnetic field lines through a loop is changing, we produce a looping electric field.

Induced Current Current increases in a wire… i

Induced Current … so the magnetic field increases… i

Induced Current … so the number of magnetic field lines passing through the loop (flux) increases… i

Induced Current … so there is an induced EMF around the loop … i EMF

Induced Current … so current flows around the loop. i i EMF

Faraday’s Law of Induction

=EMF

What is EMF? 1) Any voltage, as from a battery. 2) An effective voltage produced by induced electric fields.

What is EMF? 1) Any voltage, as from a battery. 2) An effective voltage produced by induced electric fields. I usually reserve the term EMF for induced voltages.

Two Ways to Produce Induced EMF 1) Acceleration of charges – changing current in a circuit. 2)Motional EMF – charges in a conductor moving in a B field.

Motional EMF A wire of length L moves through a magnetic field. The wire is perpendicular to B. What happens?

Motional EMF Charges along the wire feel a Lorentz Force.

Motional EMF Charges doesn’t increase indefinitely. Eventually a voltage develops across the wire.

Motional EMF Add three other fixed wires to make a loop. Now current will continue to flow around the loop.

Clicker Question 1 What happens if all sides of the loop move together? A. Current flows. B. Current doesn’t flow.

What happens both ways? i i

The magnetic flux – the number of magnetic field lines – passing through the loop changes. i i

Faraday’s Law of Induction …works for both motional EMF and the EMF of accelerating charges!

What is EMF? We can think of an induced EMF as a voltage produced all along a wire segment.

What is EMF? Let’s take a square loop with an increasing magnetic field passing through it. Assume the wire has a small resistance. V

What is EMF? Assume that the EMF around the loop is 40 V. What would a voltmeter read? V

What is EMF? The voltmeter would read zero! V = 0V

What is EMF? The voltmeter would read zero because the voltage drop due to resistance in the wire segment is exactly the same as the voltage increase due to induction in the wire segment. V = 0V

What is EMF? Another way of putting it is that the wire segment is like a lot of little batteries and resistors in series. The voltage goes up through each battery, but down by the same amount through each resistor. V = 0V

What is EMF? Each electron that goes around the full loop once gains 40eV of energy from the EMF and loses 40eV of energy to heat! V =0V

What if there’s a resistor in the loop? The total EMF is 40V.

What if there’s a resistor in the loop? The total EMF is 40V. Ohm’s Law gives i = 8A. The voltage across the resistor is 40V.

Where is the EMF being produced?

Everywhere, including through the resistor.

Where is the voltage dropping?

Primarily in the resistor – just a little in the wire. The resistor only lets a little current trickle through the wire – as compared to having no resistor.

Lenz’s Law To determine the direction induced current will flow in a circuit or to determine the direction of the induced electric field, we use Lenz’s Law.

Lenz’s Law First, we ask two questions: 1) What is the direction of the external B field? 2) Is the external B field increasing or decreasing?

Lenz’s Law 1) What is the direction of the external B field? 2) Is the external B field increasing or decreasing? 3) Find the direction of the induced magnetic field. The induced magnetic field opposes change in the external magnetic field.

Lenz’s Law 1) What is the direction of the external B field? 2) Is the external B field increasing or decreasing? 3) Find the direction of the induced magnetic field. The induced magnetic field opposes change in the external magnetic field. 4) Find the direction the induced current using the right-hand rule.

Lenz’s Law The external B is into the screen and increasing. x x x x x x x x

Lenz’s Law The external B is into the screen and increasing. To oppose change, the induced B, must be out of the screen. x x x x x x x x

Lenz’s Law The external B is into the screen and increasing. To oppose change, the induced B, must be out of the screen. To produce this B, the current is ccw. x x x x x x x x i

Lenz’s Law The external B is into the screen and decreasing. x x x x x x x x

Lenz’s Law The external B is into the screen and decreasing. To oppose change, the induced B, must be into the screen. x x x x x x x x i

Lenz’s Law The external B is into the screen and decreasing. To oppose change, the induced B, must be into the screen. To produce this B, the current is cw. x x x x x x x x i

Class 32 Today we will: work several Faraday’s law problems learn about Eddy currents

Maxwell’s Equations in Integral Form Gauss’s Law of Electricity Gauss’s Law of Magnetism Ampere’s Law Faraday’s Law

Maxwell’s Equations in Integral Form Gauss’s Law of Electricity Gauss’s Law of Magnetism Ampere’s Law Faraday’s Law Flux through a Gaussian surface

Maxwell’s Equations in Integral Form Gauss’s Law of Electricity Gauss’s Law of Magnetism Ampere’s Law Faraday’s Law Flux through a Gaussian surface Line integral around an Amperian loop

Maxwell’s Equations in Integral Form Gauss’s Law of Electricity Gauss’s Law of Magnetism Ampere’s Law Faraday’s Law Flux through a Gaussian surface Line integral around an Amperian loop Flux through the Amperian loop

Calculating Flux through a Loop We will assume that the magnetic field is constant over a loop. Then, the flux is: Remember that points in the direction of the normal to the loop.

Three Ways to Generate an EMF 1) The magnetic field changes in time. 2) The area changes in time. 3) the angle changes in time.

Eddy Currents A wire loop is moved into a region where there is a magnetic field. In what direction does current flow?

Eddy Currents A wire loop is moved into a region where there is a magnetic field. In what direction does current flow? i

Eddy Currents Viewing the same thing from the side, the loop becomes a magnet that is repelled by the external field. (Remember field lines come out of the N pole.) N S S S N

Eddy Currents It takes force to push the loop into the field. Where does this energy go? N S S S N

Eddy Currents If the wire loop is moved out of the magnetic field, in what direction does current flow?

Eddy Currents If the wire loop is moved out of the magnetic field, in what direction does current flow? i

Eddy Currents Viewing the same thing from the side, the loop becomes a magnet that is attracted by the external field. N S N S N S

Eddy Currents A disk is even more effective at producing induced currents. Such currents are called “eddy currents.”

Changing Area Put a moveable length of wire on a fixed u- shaped wire in a uniform magnetic field. a x R

Lenz’s law Which way does the induced current flow? a x R

Lenz’s law Which way does the induced current flow? a x R i

Find the Flux a x R

a x R

Find the EMF a x R

a x R Not worrying about the minus sign:

Find the Current a x R

a x R

Find the Power Dissipated in the Resistor a x R

a x R

Find the Force on the Moveable Wire a x R

a x R

Find the Work Done in Moving Δx a x R

a x R

Find the Mechanical Power a x R

a x R

Changing the Magnetic Field a b

a b Lenz’s law Which way does the induced current flow?

a Lenz’s law Which way does the induced current flow? i b

Find the Flux a i b

a i b

Find the EMF a i b

a i b

Where does the energy come from this time? a i b

a i b If the external field comes from a permanent magnet, the magnetic field of the loop attracts the permanent magnet, making it more difficult to move away.

Where does the energy come from this time? a i b If the external field comes from an electromagnet, the interaction of the loop with the electromagnet takes some energy from the electromagnet’s circuit.

We can use Faraday’s Law to calculate the electric field as well as the EMF in one problem only!

A Circular Loop in the Field of an Electromagnet with Circular Pole Faces

Flux through the Amperian loop of radius r.

A Circular Loop in the Field of an Electromagnet with Circular Pole Faces Flux through the Amperian loop of radius r. Line integral around the Amperian loop of radius r.

A Circular Loop in the Field of an Electromagnet with Circular Pole Faces

Flux through the Amperian loop of radius r.

A Circular Loop in the Field of an Electromagnet with Circular Pole Faces Flux through the Amperian loop of radius r. But the field stops at R!

A Circular Loop in the Field of an Electromagnet with Circular Pole Faces Flux through the Amperian loop of radius r. Line integral around the Amperian loop of radius r. But the field stops at R!

Changing the Angle Attach a handle to a circular loop of wire.

Changing the Angle Place the loop in a magnetic field with the shaft perpendicular to.

Changing the Angle We rotate the handle with angular speed. The flux is: The EMF is:

Class 33 Today we will: learn how motors and generators work learn about split commutators and their use in DC motors and generators

Humphrey Davy ( ) First to isolate potassium, sodium, barium, calcium, strontium, magnesium, boron, and silicon. 1813: Discovers Michael Faraday

Michael Faraday ( ) 1831: Discovers electromagnetic induction independently of Henry. Develops the transformer, motor, and generator. Discovers the Faraday Effect of light - the rotation of the plane of polarization in magnetic fields. Develops the First and Second Laws of Electrochemistry. Discovers paramagnetism and diamagnetism.

James Clerk Maxwell b. 1831, Edinburgh, Scotland d. 1879, Cambridgeshire, England 1861: Proposes "displacement current" and creates Maxwell's Equations. Recognizes light as electromagnetic radiation.

James Clerk Maxwell “What is done by what is called ‘myself’ is, I feel, done by something that is greater than myself within me.”

Changing the Angle Attach a handle to a circular loop of wire.

Changing the Angle Place the loop in a magnetic field with the shaft perpendicular to.

Changing the Angle We rotate the handle with angular speed. The flux is: The EMF is:

Engineering Considerations We usually want to get current out of the coils as they turn, so rather than use a complete loop, we use an open loop with each end connecting to a wire.

Engineering Considerations But fixed wires would twist and break. -- So we use commutators and brushes.

Commutators and Brushes These allow electrical connections to be made by pressing conductors together. wire shaft graphite brush spring steel clip wire shaft commutator

Commutators A typical AC generator or motor uses double commutators. Each end of the loop is connected to a separate commutator.

Motors A simple motor consists of a current-carrying coil in a uniform magnetic field. N S

Motors A torque on the coil tends to align the magnetic dipole moment of the loop with the external field. N S

Motors A torque on the coil tends to align the magnetic dipole moment of the loop with the external field. N S +

Motors As the dipole moment rotates past the magnetic field, however, the torque reverses. N S +

Motors We ’ ve created a vibrator instead of a motor. N S +

Motors But, we could keep the loop (armature) turning in the same direction, if we could reverse the magnetic dipole moment. This can be done by changing the direction of the battery. N S +

Motors It ’ s a little hard to keep moving the leads on the battery back and forth by hand. So we need a better way of doing it. N S +

AC Current The easiest way to do this is to use AC current instead of a battery. The direction of the current through the loop automatically changes sign periodically.

AC Current Note that the speed of such a motor is closely tied to the frequency of the AC power supply.

The Split Commutator Another clever solution is to use a “ split commutator. ” A split commutator automatically changes the end of the loop connected to the positive terminal of the battery every half cycle.

DC Motors Thus, split commutators allow motors to be operated by DC power sources.

Generators Generators are just motors operated in reverse. N S load

AC Generators With double commutators, we get a sinusoidal current out of a generator. N S load

Quasi-DC Generators With split commutators, we get a sinusoidal current that changes sign each half cycle. N S load

DC Generators Adding a second loop can give something that is closer to a DC output. N S load