Last Time (Cross Products: Mathematically) Electron Current and Conventional Current Calculating the Electron Current True vs. Useful Biot-Savart Law in a Wire Relativity?? 1

Biot-Savart Law BIOT-SAVART LAW point charge = length of this chunk of wire BIOT-SAVART LAW current in a wire 2

Electric fields: produced by charges Magnetic fields: produced by moving charges charged tape Any magnetic field? Frame of Reference Biot-Savart Law is an Approximation 3 Must use the velocities of the charges as you observe them in your reference frame!

If we suddenly change the current in a wire: Magnetic field will not change instantaneously. Electron and positron collide: Produce both electric and magnetic field, these fields exist even after annihilation. Changes propagate at speed of light Why is there no time in Biot-Savart law?  We assume speed of charges is small Retardation Biot-Savart Law is an Approximation 4

Today Magnetic Field of a Straight Wire (Magnetic Field of a Current Loop) Magnetic Dipole Moment Bar Magnet Atomic Dipoles 5

Magnetic Field of a Straight Wire Calculate B in the bisecting plane BIOT-SAVART LAW current in a wire Calculate B in the bisecting plane We want to calculate B in the bisecting plane. How did we do this for E? Break the rod up into point charges. Add up E due to all point charges. 6

Electric Field in the Bisecting Plane for a point charge θ Δq i We only need the x component Blast from the Past Lecture 5 Break the rod up into point charges. Add up E due to all point charges. 7

Magnetic Field of a Straight Wire Calculate B in the bisecting plane BIOT-SAVART LAW current in a wire Calculate B in the bisecting plane Break the rod up into point charges. Add up B due to all point charges. - 8

Magnetic Field of a Straight Wire Does x change during the integration? Calculate B in the bisecting plane 9

Magnetic Field of a Straight Wire Observation point x does not change during the integration Calculate B in the bisecting plane Look up the integral B of a Long Straight Wire 10

Magnetic Field of a Straight Wire B in the bisecting plane Which direction does B point? Will the y axis look different from the x axis? B of a Long Straight Wire (Cylindrical Coordinates) No, so we can trade x  r  Always along concentric circles In cylindrical coordinates, it points in the " " direction I Cylindrical Coordinates r 11

Very Close to the Wire Very close to the wire: r << L CLOSE TO THE WIRE Another Right-Hand Rule 12

iClicker Question Current carrying wires below lie in X-Y plane. 13

B along Axis of Circular Loop of Wire BIOT-SAVART LAW current in a wire 1) Draw for one piece  Symmetry: only z component survives! NET MAGNETIC FIELD 14

B along Axis of Circular Loop of Wire BIOT-SAVART LAW current in a wire 1) Draw for one piece  Symmetry: only z component survives! 2) Write z due to one piece (lots of math!) NET MAGNETIC FIELD 15

B along Axis of Circular Loop of Wire BIOT-SAVART LAW current in a wire 1) Draw for one piece  Symmetry: only z component survives! 4) Doulbecheck ✔ UNITS Right-hand Rule ✔ 2) Write z due to one piece (lots of math!) 3) Integrate to find total B NET MAGNETIC FIELD 16

B along Axis of Circular Loop of Wire Yet Another Right Hand Rule! 17

z>>RFar from the loop: B along Axis of Circular Loop of Wire B Along Axis Far from Loop Both fall off like 1/r 3 far from the dipole B for dipole 18

Electric and Magnetic Dipole Moments B Along Axis Far from Loop B for dipole Electric Dipole Moment: Magnetic Dipole Moment: B Along Axis Far from Loop E Along Axis Far from Dipole

Magnetic field of a solenoid A constant magnetic field could be produced by an infinite sheet of current. In practice, however, it is easier and more convenient to use a solenoid. A solenoid is defined by a current I flowing through a wire that is wrapped n turns per unit length on a cylinder of radius R and length L. L R

Magnetic field of a solenoid (continued) Contribution to B at origin from length dx one turn # turns in length dx (or for L>>R) (half at ends) 21

Solenoid’s B field synopsis // to axis Long solenoid (R<<L): B inside solenoid B outside solenoid nearly zero (not very close to the ends or wires) 22

RHIC STAR Experiment STAR 23

Assume: Electrons make circular current loops What is the direction? Atomic Structure of Magnets Electrons These are electrons 24

Assume: Electrons make circular current loops S N What is the average current I per loop? current=charge/second: Atomic Structure of Magnets Electrons What is the direction? 25

Angular Momentum is Quantized: Orbital angular momentum: Quantum mechanics: L is quantized: If n=1: Atomic Structure of Magnets Assume: Electrons make circular current loops Dipole Moment of 1 Atom 26

Fuzzy electron cloud. Shapes are set by "spherical harmonics" Spherically symmetric cloud (s-orbital) has no  Only non spherically symmetric orbitals (p, d, f) contribute to  There is more than 1 electron in an atom Quantum Magnetism Magnetic Moment = Orbital Motion + "Spin" ORBITAL MOTION: 27

Electron acts like spinning charge - contributes to  Electron spin contribution to  is of the same order as one due to orbital momentum Neutrons and proton in nucleus also have spin but their  ‘s are much smaller than for electron same angular momentum: NMR, MRI – use nuclear  Quantum Magnetism Magnetic Moment = Orbital Motion + "Spin" SPIN: 28

Refrigerator Magnets Alignment of atomic dipole moments: most materialsferromagnetic materials: iron, cobalt, nickel 29

Magnetic domains Hitting or heating can also demagnetize Reality Physics - Domains Apply B-Field to Align Domains 30

Magnetic domains Why are there Multiple Domains? 31

iClicker Question 32

Today Magnetic Field of a Straight Wire (Magnetic Field of a Current Loop) Magnetic Dipole Moment Bar Magnet Atomic Dipoles 33

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Magnetic dipole moment of 1 atom: Method 2: estimate speed of electron Momentum principle: Circular motion:  – angular speed Magnetic Dipole Moment 35