Announcements This week:

Announcements This week: Homework due Thursday March 22: Chapter 26 sections 3-5 + Chapter 27 Recitation on Friday March 23: Chapter 27. Quiz on Friday March 23: Homework, Lectures 12, 13 and 14

Properties of magnets, and how magnets interact with each other. Visualizing magnetic field lines. No magnetic monopoles Magnetic flux thru closed surface Analyzing magnetic forces on moving charged particles Application: cyclotron motion Force on current carrying wires Current carrying loops: Magnetic moment Torque, potential energy in B-field DC Motor Summary Chapter 27 Φ B = B d A " = 0 F = q v B F = I l B µ = I A τ = µ B R = mv qb U = µ B 2

Lecture 15: March 20 Chapter 28 Sources of magnetic fields 3

Review of electrostatics Electric force on a charged particle: Φ E = τ = p E F = q E Electric flux through a closed surface: E d A Electric dipoles: " = q enclosed p = q d U = p E Electrostatics=>Magnetostatics Magnetostatics analogy Magnetic force on charged particle: ε 0 Φ B = τ = µ B F = q v B Magnetic flux through a closed surface: " B d A = 0 Magnetic dipoles: µ = I A U = µ B Electric work done during a closed loop (circuit): # E d l = 0 Work done by magnetic force around a closed loop? # B d l =?? 4

Chapter 28: Sources of magnetic fields Calculating magnetic fields Single moving charged particle Straight current-carrying wire Current-carrying wire bent into a circle Forces between current carrying wires Calculating magnetic fields from current distributions => Ampere s Law How to get a uniform magnetic field

A moving charge generates magnetic field that depends on velocity of the charge distance from the charge Direction: right hand rule Ø Cross product B(r) = µ 0 4π q v r µ 0 = 4π 10 7 Ns 2 r 2 = µ 0 4π q µ 0 is vacuum permeability Magnetic field of a moving charge v r r 3 C = 4π T m 10 7 A r Velocity of +q into the page B v B 1 r 2

A moving charge generates magnetic field that depends on velocity of the charge distance from the charge Direction: right hand rule B(r) = µ 0 4π q Magnetic field of a moving charge v r µ 0 = 4π 10 7 Ns 2 r 2 = µ 0 4π q µ 0 is vacuum permeability v r r 3 C = 4π T m 10 7 A

I clicker question A positive point charge is moving directly toward point P. The magnetic field that the point charge produces at point P A. points from the charge toward point P. B. points from point P toward the charge. C. is perpendicular to the line from the point charge to point P. D. is zero. E. The answer depends on the speed of the point charge. B(r) = µ 0 4π q v r r 2 = µ 0 4π q v r r 3 8

I clicker answer A positive point charge is moving directly toward point P. The magnetic field that the point charge produces at point P A. points from the charge toward point P. B. points from point P toward the charge. C. is perpendicular to the line from the point charge to point P. D. is zero. E. The answer depends on the speed of the point charge. B(r) = µ 0 4π q v r r 2 = µ 0 4π q v r r 3 9

Magnetic field of a current element Instead of a single moving charge, consider current carrying wire B field from superposition of charge in each segment v r B(r) = µ 0 4π q seg r 2 Charge of carriers in wire segment volume=adl" q seg = q i = neadl I = nev d A q seg vd = Id l i I B(r) = µ 0 4π I d l r r 2

Magnetic field of a current element Instead of a single moving charge, consider current carrying wire Superposition of each segment B(r) = µ 0 4π q v r seg r 2 B(r) proportional to 1/r 2 Similar to E field of a point charge Direction: current element " r Id l r B(r) = µ 0 4π I d l r r 2 Law of Biot-Savart

Example: Magnetic Field of circular current carrying loop O Rr d l B field at center of circular current carrying loop of radius R? Contribution of a small portion of the loop d l r δ B(r) = µ 0 4π I d l r R 2 all in same direction and B(0) = µ 0 NI 2R δb(r) = µ 0 4π I dl R 2 Field at center of loop If N turns in loop, B is N times stronger dl B(0) = µ I 0 2R = 2π R 12

I clicker question In the figure an irregular loop of wire carrying a current lines in the plane of the paper. Suppose that the loop is distorted into some other shape while remaining in the same plane. Point P is still within the loop. Which of the following is a TRUE statement concerning this situation? A. The magnetic field at point P will always lie in the plane of the paper. B. It is possible that the magnetic field at point P is zero C. The magnetic field at point P will not change in magnitude when the loop is distorted. D. The magnetic field at point P will not change in direction when the loop is distorted. E. None of the other statements is true. 13

I clicker answer In the figure an irregular loop of wire carrying a current lines in the plane of the paper. Suppose that the loop is distorted into some other shape while remaining in the same plane. Point P is still within the loop. Which of the following is a TRUE statement concerning this situation? A. The magnetic field at point P will always lie in the plane of the paper. B. It is possible that the magnetic field at point P is zero C. The magnetic field at point P will not change in magnitude when the loop is distorted. D. The magnetic field at point P will not change in direction when the loop is distorted. E. None of the other statements is true. 14

Magnetic field of a straight current-carrying conductor B field a distance x from long wire Consider a small element of wire dl" Choose origin at point P d B(r) = µ 0 4π I d l ( r) r 3 db points into the page To solve for B need to express all variables in terms of angle φ To calculate B: -90 φ +90 At point P, the field db from dl points into the page Total field B from the entire conductor also points into the page

Magnetic field of a straight current-carrying conductor B field a distance x from wire Consider a small element of wire dl" Choose origin at point P d B(r) = µ 0 4π I d l ( r) r 3 db points into the page To solve for B need to express all variables in terms of angle φ What is dl in terms of φ? What is r in terms of φ?

Magnetic field of a straight current-carrying conductor db field a distance x from a small element of wire dl" Choose origin at point P d B(r) = µ 0 4π I d l ( r) r 3 db points into the page To solve for B need to express all variables in terms of angle φ tanφ = y x y = x tanφ cosφ = x r r = x cosφ dy = dl = xsec 2 φdφ = xdφ cos 2 φ

Magnetic field of a straight current-carrying conductor db field a distance x from a small element of wire dl" Choose origin at point P d B(r) = µ 0 4π I d l ( r) r 3 db points into the page To solve for B need to express all variables in terms of angle φ tanφ = y x y = x tanφ dy = dl = xsec 2 φdφ = xdφ cos 2 φ cosφ = x r r = x cosφ At point P, the field db from dl points into the page Total field B from the entire conductor also points into the page d B(r) = µ 0 4π I d l ( r) = µ 0 xdφ φ I( r 3 4π cos 2 φ )(cos2 ) = µ 0 x 2 4π x I cosφdφ

Magnetic field of a straight current-carrying conductor To solve for B need variables in terms of φ To calculate B: -90 φ +90 db(r) = µ 0 φ= +π 2 cosφ dφ 4π x I φ= π 2 B(r) = µ 0I 4π x [1 ( 1)] = µ 0I 2π x B(r) = µ 0 I 2πr Total field B from the entire conductor given by right hand rule: thumb in direction of current, hand in direction of B

Demo: B from current carrying wire 20

Example Consider a wire bent in the hairpin shape. The wire carries a current I. What is the approximate magnitude of the magnetic field at point a? B(0) loop = µ 0 I 2R R a B(r) line = µ 0I 2πr Apply superposition: the net field at point a is superposition of three B fields produced by the current the semi-circle plus two straight wires. Semi-circle: Top wire: Bottom wire: B(r) = ( 1 2 ) µ 0I 2R = µ 0I 4R B(r) = ( 1 2 ) µ 0I 2π R = µ 0I 4π R B(r) = ( 1 2 ) µ I 0 2π R = µ I 0 4π R B(r) = µ 0 I 4R + 2( µ 0 I 4π R ) = µ 0 I (1/2 of the field of a circular loop) (1/2 of the field of an infinite wire) (1/2 of the field of an infinite wire) 4R (1+ 2 π )

Interaction between 2 current carrying wires I 2 I I 2 I 1 I 1 Magnetic field due to I 1 at wire I 2 B(r) = µ 0I 1 2πr Magnetic force on wire I 2 F = I 2 l B F L = I 2B = I 2 µ 0 I 1 2πr F L = µ 0 I 1 I 2 2πr Force/unit length between current carrying wires Attract: when parallel currents Repel: when anti-parallel currents 22

Interaction between 2 current carrying wires I 2 I 1 Magnetic field due to I 1 at wire I 2 B(r) = µ 0I 2πr Magnetic force on wire I 2 F = I l B F L = I 2 B = I 2 µ 0 I 1 2πr F L = µ 0 I 1 I 2 2πr Force/unit length between current carrying wires Attract: when parallel currents Repel: when anti-parallel currents 23

A wire consists of two straight sections with a semicircular section between them. If current flows in the wire as shown, what is the direction of the magnetic field at P due to the current? I clicker question A. to the right B. to the left C. out of the plane of the figure D. into the plane of the figure E. misleading question the magnetic field at P is zero 24

A wire consists of two straight sections with a semicircular section between them. If current flows in the wire as shown, what is the direction of the magnetic field at P due to the current? I clicker answer A. to the right B. to the left C. out of the plane of the figure D. into the plane of the figure E. misleading question the magnetic field at P is zero 25

Calculating magnetic fields Single moving charged particle Straight current-carrying wire Current-carrying wire bent into a circle Forces between current carrying wires Attract: parallel Repe: anti-parallel Chapter 28: Summary today B(r) = µ 0 4π q v r r 2 = µ 0 4π q B(r) line = µ 0I 2πr B(0) loop = µ I 0 2R F L = µ 0I 1 I 2 2πr Next: Calculating magnetic fields from current distributions => Ampere s Law v r r 3