March 9 Physics 54 Lecture Professor Henry Greenside

Reminder: Answers to PRS Questions in Notes Section of PowerPoint Slides A BC D x L E FG H

Key Points From Previous Lecture Make sure you understand how to use the right hand rule to deduce the direction of the magnetic field inside a solenoid from the direction of the current in a single loop of the solenoid. Also make sure you know how to deduce the value of the line integral around the path abcda by breaking the path integral into different short paths that correspond to where the magnetic field changes value or direction compared to the direction along the path! A straight wire of length L moving with speed v in a magnetic field of constant strength B (with v perpendicular to the length of the wire and perpendicular to B) acts like a battery with emf E =vLB across the ends of the wire. See Eq. (29-3) on page 739 of Giancoli and especially Example 29-3 of Giancoli.

Getting Ready for Spring Break I  What does this picture correspond to?

Getting Ready for Spring Break II What does this animation correspond to? (Click mouse to animate.) 

Getting Ready for Spring Break III: Why did the chicken cross the Moebius strip?

At the Whiteboard… 1.Finale of Ampere’s law: the toroid. 2.Application of solenoids, toroids, and Faraday’s law to magnetic confinement of a thermonuclear fusion plasma: the ultimate environmentally and politically acceptable power source for the human race? 3.Faraday’s law of induction: an electromotive force emf is produced by a time varying magnetic flux. 4.Mnemonic: First letter of Faraday sounds like “flux”, which helps you remember when to use Faraday’s law versus Ampere’s law. 5.Several worked examples of Faraday’s law such as a wire falling down vertical conducting rails under gravity.

Magnetic Field in A Toroid: Another Application of Ampere’s Law The magnetic field in a toroid is stronger near the inner surface than near the outer surface. This makes sense: the windings are closer together near the center. Magnetic field is everywhere zero outside the interior of an ideal toroid.

Easiest Fusion Reaction on Earth: Hydrogen Isotopes Deuterium and Tritium About 450 times more energy comes out than goes in

A Day in the Life of a Gigawatt Reactor In: 9,000 tons of coal Out: Electricity and heat 30,000 tons of CO tons of SO 2 80 tons of NO 2 In: 1 lb deuterium, 3 lb lithium, 1.5 lb tritium Out: Electricity and heat 4 lb helium gas Some moderately radioactive materials (short lifetimes, none useful for atomic weapons!) Note: meltdown impossible for fusion reactor since only 1g or so of fuel present at any moment.

Where to Get Deuterium? 1 barrel water = 42 gallons water = ¾ oz D 2 O ~ 32,000 gallons oil via D-T

Big insight: magnetic confinement Which way is B pointing in diagram on right?

Tokamak: A Fusion Machine That Creates a Donut-Shaped Star "тороидальная камера в магнитных катушках"

ITER: International Thermonuclear Experimental Reactor Being built in Cadarache, France

Transition to Chapter 29 Lenz

PRS Review Question: Direction of Current? If conducting ring is falling downward toward north pole of bar magnet, then 1. A current will flow left to right, out of the page toward you. 2. A current will flow right to left, out of the page toward you. 3. No current will flow since there is no battery.

Relativity of Motion Implies Creation of E Without Charge Motion! Instead of ring falling, we can imagine ring fixed and bar magnet moving upward as shown. This has to be physically the same situation as the ring descending so a current will start to flow, even though there is no initial motion of the charges in the ring! This is something quite new, discovered by Faraday. In fact, this implies that an electric field that is oriented tangentially along the circumference of the ring must appear, there is no other way to explain how the charges in the ring can begin moving. Conclusion: somehow a time varying magnetic field causes an electric field to appear. This in turn accelerates the charges in the loop, causing an emf and a current.

Demo: Magnet, Coil, and Galvanometer

Demo of currents being generated by changing magnetic fields: magnet falling down a conducting tube

Demo of currents generated by changing magnetic field: a ring launcher

Transcranial Magnetic Stimulation: Changing MagneticField Causes Currents to Flow in Brain

The Jewel in the Crown: Faraday’s Law E induced electromotive force (emf) with units of volts V E magnitude of constant electric field along loop L length along some closed loop for which the electric field has a constant magnitude and orientation w.r.t. the loop N number of turns of wire in loop (N=1 for simple loop)  flux of magnetic field through a single coil of the loop Minus sign “–” on right side says induced EMF opposes change in flux, so- called “Lenz’s law”. We need to understand when and how flux  changes.

Magnetic Flux   = B A cos(  ) Flux through area can change because: 1.magnetic field B changes 2.area A changes 3.orientation  of normal of area w.r.t B changes 4.any combination of these.

PRS Question: Flux Through Solenoid a b c d Order of loops from largest to smallest magnetic fluxes is (assume B=0 outside of solenoid to good accuracy) : 1. a > b > c > d. 2. a = b > c > d. 3. d > a > b > c. 4. c > b > a > d. 5. Some other order.

Faraday Law Example: Loop Moving from High to Low Magnetic Field 1.What is the total flux  through this yellow loop when there is length L-x in the left region of strength B 1 and length x in right region of strength B 2 < B 1 ? 2.How does this flux change with time when the loop moves with constant speed v to right as shown by the pink arrow, i.e., what is the value of the quantity d  /dt? 3.What is the current through this loop if loop has resistance R? 4.What is the electrical power consumed by the resistance R? How does this compare with the mechanical power associated with pulling the loop to the right with force F and speed v? 5.What is the force F needed to pull this loop at constant speed to the right? x h L-x v

Worked Example of Faraday’s Law: Electrical Power from Falling Bar

Noninvasive Brain Studies: SQUID MEG Superconducting Quantum Interference Device (SQUID) Magnetoencephalography (MEG)