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Home , Hall effect

Physics 272: and

Mark Palenik Tuesday, July 3rd Again. . . midterm

• Remember, today is the last lecture (with me, or for the week, with anyone) • Wednesday, no class • Midterm on Thursday Topics for today

• Force on a moving charge by the • Force of a current carrying wire by the magnetic field • as magnetic dipoles • Interaction of dipoles with the magnetic field • Hall effect

First. . . Magnetic field of a Wire Loop

< −푅 cos 휃 ,−푅 sin 휃 ,푧> • ∆푙 × 푟 =< −푅 sin 휃 , 푅 cos 휃 , 0 > ∆휃 × 푟 • =< 푅 cos 휃 푧, 푅 sin 휃 푧, 푅2 >/푟 → 푅2 휇 1 휇 ∆푙 휇 ∆푙 • ∆퐵 = 0 ∆푙 × 푟 = 0 = 0 4휋 푟2 4휋 푟3 4휋 (푧2+푅2)3/2 • 푑푙 × 푟 = 푅2푑휃 2휋 휇 푅2푑휃푧 • 퐵 = 0 0 4휋 (푧2+푅2)3/2

2 m0 2pR I Bz = 3/2 4p (R2 + z2 ) Magnetic force ()

• Magnetic fields act on moving charges

• Lorentz force on an object is 퐹 퐿 = 푞푣 × 퐵 • In the presence of an electric and magnetic field, 퐹 = 푞(퐸 + 푣 × 퐵) iClicker question: Magnetic force

• The fact that 푣 × 퐵 appears in the equation for force means that a) F is parallel to v b) F is parallel to B c) F can have a component parallel to v or B d) F is perpendicular to v and B e) F is only perpendicular to v iClicker question: Work done by magnetism 푏 • Recall the definition of work (푊 = 푎 퐹 ∙ 푑푥 ) • Does the magnetic field do work? a) Yes, because the direction of F is not important b) No, because F is perpendicular to B 푏 c) Yes, because even if 퐹 ∙ 푑푥 = 0, 푎 퐹 ∙ 푑푥 may not be zero d) No, because F is perpendicular to v Magnetic fields do no work (usually)

• Since the force due to the magnetic field is always perpendicular to velocity, work=0 • Acceleration is always perpendicular to v. What shape does that sound like? Charged particles in a B field often move in a circle (they will if the field is 1) uniform and 2) perpendicular to v)– Acceleration and velocity are perpendicular

Also, magnitude of acceleration is wv (constant * v) iClicker: Which field is bigger

• The same charge is place in two different uniform magnetic fields that point into the page. • The charge is initially moving to the right with a speed v in both situations. • Which circle is the path of the charge in the STRONGER field?

a) b) Magnetic force on a wire

• Remember, “I” is a sort of moving . 푞푣 • It is essentially , the amount of moving charge per 퐿 unit length. • We can find force per unit length on a wire 푞푣×퐵 퐹 • 퐼 × 퐵 = = B 퐿 퐿 F on this small • 퐹 = 퐼 × 퐵푑푙 segment of wire is IB into the page • B is the external field – NOT the field produced by the wire iClicker: Magnetic force between 2 wires • Two wires run parallel and have currents that

run in the +y direction. Do they: y z x

I I a) Attract b) Repel iClicker: Magnetic force between 2 wires • Two long wires with length L carry a current in the +y direction • Keep in mind the magnitude of the magnetic field 휇 2퐼 produced by a long wire is B= 0 4휋 푟 • What is the magnitude of the force between the wires? 휇 2퐼2 a) 0 4휋 푟 휇 2퐼2퐿 b) 0 4휋 푟 휇 2퐼퐿 I I c) 0 4휋 푟 휇 2퐼2퐿2 d) 0 4휋 푟

When magnetic fields CAN do work

• Take a dipole, like an , and place it in a magnetic field • The energy of the dipole is −휇 ∙ 퐵 • When 휇 is the same direction as 퐵, energy is minimized • When 휇 is the opposite direction of 퐵, energy is maximized • At any other point there is a torque on the dipole—the dipole wants to go to the lowest energy orientation. How will a dipole behave in a uniform field? • A magnetic dipole points at a 45 degree angle in the xy plane • It is placed in a uniform magnetic field that points in the +y direction • What happens to the dipole? y z x a) There is a torque that rotates it clockwise b) There is a torque that rotates it counter clockwise c) Nothing d) A force pushes it right e) A force pushed it left How will a dipole behave in a non- uniform field? • Remember, the energy of a dipole is −휇 ∙ 퐵 • A magnetic dipole pointing in the +y direction is placed in a non-uniform magnetic field that points in the –y direction decreases along the x axis. • Which direction is the force on the dipole? y z x a) +x B m b) -x c) +y d) -y e) There is no force, because it is a magnetic field Atoms as magnetic dipoles

• Electron and orbital angular momentum give them dipole moments. • A single produces a weak field • Several atoms can be aligned by a strong external field • Electric/fermi interactions can hold them in this alignment Bar Magnet

• A bar magnet is composed of a bunch of spin aligned atoms. It is a collection of dipoles • The field is not exactly the same as a dipole (when very close), but looks a lot like a dipole from far away • Field lines leave the north end and enter the south Other dipoles, earth, compass needle

Same as a compass needle S N

The earth is backwards! Hall effect

• Let’s think about two problems relating to the picture below that will give insight into the hall effect

+V -V • Since the positive is on the left, conventional current runs to the right. Electron current runs to the left. iClicker: Hall effect

• Assume we add a magnetic field that points out of the page. Which way would the magnetic field push the as they move through the metal block?

Metal block a) Up +V -V b) Down c) Left d) Right iClicker: Hall effect

• The magnetic field still points out of the page. If the idea that positive charges moved were correct (so positive charges move in the direction of conventional current), which way would the magnetic field push the positive charges as they move through the metal block? Metal block

+V -V a) Up b) Down c) Left d) Right Hall voltage

• If electrons are pushed down, negative charges accumulate at the bottom. points down in the block, the bottom is at a lower voltage Metal block E ------+V -V • If positive charges are pushed down, the voltage would be higher at the bottom Note: conductor is Metal block not at equilibrium, which is why we E can have an E field +++++++++ +V -V inside.