Last time Result of Cross Product is Perpendicular to both and MAGNETIC FORCE point charge Result of Cross Product is Perpendicular to both and Right-Hand Rule: 2) 1)

Today Magnet force on currents Hall effect Relativity effect

iClicker Question Small metal ball has charge q = +0.05C and mass, m = 0.025kg. Ball enters a region of magnetic field B = 0.5 T that is perpendicular to its velocity v = 200m/s. Centripetal acceleration = v2/r. What is radius of the curve ball will move thru in magnetic field? 200m 100m 250m 120m 50m . A

. a) F = qv1B = (0.05)(200)(0.5) = 5N b) (see diagram) Force in –z direction c) F = ma = qv1B = 5N a = 5N/0.025kg = 200 m/s2 d) v2/r = 200m/s2 , r = v2/(200m/s2) = (200m/s)2/200m/s2 = 200m

Biot-Savart Law BIOT-SAVART LAW point charge We need to understand how these are related BIOT-SAVART LAW point charge BIOT-SAVART LAW current in a wire Blast from the Past: Lecture 12 = length of this chunk of wire

GOAL: Show (point charge) (wire) (POSITIVE) POINT CHARGE MANY POINT CHARGES N particles particles per volume ΔV ΔV Move the vector symbol CHUNK OF WIRE = length of this chunk of wire Blast from the Past: Lecture 12

Magnetic Force on a charge or wire point charge We just showed how these are related MAGNETIC FORCE current in a wire = length of this chunk of wire

Magnetic Force on a Wire Current-carrying wire in an applied magnetic field B Bapplied Bapplied I Which way will the wire move?

Very Close to the Wire B I Very close to the wire: r << L http://physick.wikispaces.com/Electric+Current B I Blast from the Past: Lecture 13

iClicker question: Two long parallel wires carry currents of 5 A and 10 A in the same direction as shown. What is the direction of the magnetic force acting on the 10-A wire? Perpendicular to the plane of the page and into the page Perpendicular to the plane of the page and out of the page Downward Inward toward the other wire Outward away from the other wire d

Force Between Parallel Wires Bapplied = Magnetic field applied by the red wire. Blue wire feels a force down. Fon blue wire Bapplied I Bapplied I What about reciprocity? (Equal and opposite forces)

iClicker question: Two long parallel wires carry currents of 5 A and 10 A in opposite directions as shown. What is the direction of the magnetic force on the 10-A wire? Perpendicular to the plane of the page and into the page Perpendicular to the plane of the page and out of the page Upward Downward Outward away from the other wire e

Force Between (Anti) Parallel Wires Bapplied = Magnetic field applied by the red wire. Blue wire feels a force up. x Bapplied Fon blue wire I x Bapplied x Bapplied I Fon red wire x Bapplied Bapplied = Magnetic field applied by the blue wire. Red wire feels a force down.

Hall Effect By measuring the Hall effect for a particular material, we can determine the sign of the moving particles that make up the current Why would it be anything other than electrons? (Negative charges) Semiconductors: sometimes current is carried by electrons, but sometimes it is carried by the "holes". In semiconductors, "holes" (missing electrons) in the electron sea behave like positive charges.

Hall Effect Inside a Material: Bapplied Bapplied - - - - - - - - - - - MAGNETIC FORCE point charge A hole in the electron sea behaves like a proton. Inside a Material: + + + + + + + + + + + Moving holes get pushed to the top Proton or "Hole" x Bapplied x Bapplied Electron EC got here Moving electrons get pushed to the bottom - - - - - - - - - - -

Hall Effect Inside a Material: ΔV = Hall Voltage - - - - - - - - - - - MAGNETIC FORCE point charge A hole in the electron sea behaves like a proton. Inside a Material: + + + + + + + + + + + Moving holes get pushed to the top  Until the Hall Voltage is strong enough to balance the magnetic force ΔV = Hall Voltage - - - - - - - - - - - How long does this go on? x Bapplied EC got here + + + + + + + + + + + Moving electrons get pushed to the bottom - - - - - - - - - - -

Measuring the Hall Effect http://www.machinerylubrication.com 1. Apply B-Field 2. Apply Current I 3. Measure "Hall Voltage" EC got here

Currents Due to Magnetic Forces Metal bar polarization Fm How much force do we need to apply to keep the bar moving at constant speed? What is direction of E?

Moving Bar and Energy Conservation P=IV=I(emf) Are we getting something for nothing? Bar – current I: FI F Fm Work: Power: x Main principle of electric generators: Mechanical power is converted to electric power

Reference Frame Any magnetic field? charged tape What is direction of E? No work done – static equilibrium, net force is zero.

iClicker Question charged tape Jessie Jack In case Jack have two positively changed tapes side by side. For the interaction between the two tapes : A). Jack and Jessie: purely through E force. B). Jack: only through E force. Jessie: only through B force. C). Jack and Jessie: through a mixture of B and E forces. D). Jack: though only E forces. Jessie: through a mixture of B and E forces. What is direction of E? No work done – static equilibrium, net force is zero.

Magnetic Forces in Moving Reference Frames Electric force: (assume independent of frame?) Two protons +e r 1 2 v Magnetic field: F21,m B1 Magnetic force: F21,e E1 Coulomb force on moving charges: approximation, E is not exactly the same (address later). Recall that we have already said that two current carrying wires experience an attractive magnetic force if current direction is the same.

Magnetic Forces in Moving Reference Frames Electric force: +e r 1 2 v E1 F21,e B1 F21,m Magnetic force: Ratio: Coulomb force on moving charges: approximation, E is not exactly the same (address later)

Magnetic Forces in Moving Reference Frames 1 2 v E1 F21,e B1 F21,m For v<<c the magnetic force is much smaller than electric force How can we detect the magnetic force on a current carrying wire? Ask students to reconcile this with the fact that two wires carrying current in the same direction are attracted to each other. Full Lorentz force: downward

Magnetic Forces in Moving Reference Frames 20 ns +e r 1 2 v E1 F21,e B1 F21,m 15 ns Who will see protons hit floor and ceiling first? Time must run slower in moving frame. Einstein 1905: “On the electrodynamics of moving bodies” This is a kind of phenomenon which led Einstein to develop theory of relativity Twin paradox http://en.wikipedia.org/wiki/Twin_paradox

Relativistic Field Transformations Our detailed derivations are not correct for relativistic speeds, According to the theory of relativity:

Magnetic Field of a Moving Particle Still: Moving: Slow case: v<<c  Field transformation is consistent with Biot-Savart law What about B below particle? Electric and magnetic fields are interrelated Magnetic fields are relativistic consequence of electric fields

Electric Field of a Rapidly Moving Particle For a single charged particle Ex, Ey, Ez are different when moving rapidly.

Interpretation depends on reference frame v Uniform B field in lab frame. For observers in the metal bar rest frame, the charge build up because of non-zero E fieid. For lab frame observers, the charge is build up by magnetic force.