Our Study of Magnetism Lorentz Force Equation Motion

Our Study of Magnetism • Lorentz Force Equation • Motion in a uniform B-field • Forces on charges moving in wires • Magnetic dipole N S • Today: fundamentals of how currents generate magnetic fields 10/29/2020 2

LECTURE 14 • Fundamental Laws for Calculating B-field • Biot-Savart Law (“brute force”) • Ampere’s Law (“high symmetry”) • Example: B-field of an Infinite Straight Wire • from Biot-Savart Law • from Ampere’s Law • Other examples

Biot-Savart Law: B-field due to a moving charge Biot-Savart Law: B-field due to a current in a wire Right Hand Rules The magnetic field “curls” or “loops” around the wire. No d. B field at P 2 since d is parallel to r.

Observations about Biot-Savart Law 10/29/2020 5

DEMO 6 B-01 Oersted’s Experiment Switch open: I = 0 Compass points north. 10/29/2020 Switch closed: I 0 6

Magnetic Field of an Infinite Straight Wire • Calculate field at point P using Biot-Savart Law: x dl φ I R P y dl r φ

Calculation of Electric Field • Two ways to calculate Coulomb’s Law "Brute force" Gauss’ Law "High symmetry" What are the analogous equations for the Magnetic Field?

Calculation of Magnetic Field • Two Ways to calculate Biot-Savart Law I "Brute force" Ampere’s Law AMPERIAN LOOP INTEGRAL "High symmetry“ also, only the ENCLOSED Current These are the analogous equations

Magnetic Field of an Infinite Straight Line B • Calculate field at distance R from wire using Ampere's Law: dl I Ampere's Law simplifies the calculation thanks to symmetry around the current! (axial/cylindrical) R

Magnetic Field Lines of a Straight Wire DEMO – 6 B-03 10/29/2020 11

Example A current I flows in an infinite straight wire in the +z direction (towards you). A concentric infinite cylinder of radius R carries current 2 I in the -z direction. What is the magnetic field Bx(a) at point a, just outside the cylinder as shown? (a) Bx(a) < 0 (b) Bx(a) = 0 y a x b x x x 2 I I x x x (c) Bx(a) > 0 x x

Example A current I flows in an infinite straight wire in the +z direction (towards you). A concentric infinite cylinder of radius R carries current 2 I in the -z direction. What is the magnetic field Bx(b) at point b, just inside the cylinder as shown? (a) Bx(b) < 0 (b) Bx(b) = 0 y a x b x x x 2 I I x x x (c) Bx(b) > 0 x x

QUIZ lecture 14 An infinitely long hollow conducting tube carries current I in the direction shown. What is the direction of the magnetic field inside the tube? (A) The magnetic field is zero (B) Radially outward to the center (C) Radially inward from the center (D) Counterclockwise (E) Clockwise 10/29/2020 14

QUIZ lecture 14 Two loops are placed near current carrying wires as shown in Case 1 and Case 2. In both cases, the direction of the current in the two wires are opposite to each other. For which loop is greater? CASE 2 CASE 1 R R I I (A) Case 1 (B) Case 2 (C) The integral is the same for both cases. 10/29/2020 15

Magnetic Field at the Center of a Current Loop Practice both right hand rules here: 10/29/2020 16

Line Segment Combinations I 2 1 R I 10/29/2020 Find B at point P. P 3 I 18

Double Arc Find B at point P. Only the arcs contribute. 10/29/2020 19

Magnetic Field on Axis of Current Loop 10/29/2020 20

Magnetic Field Lines of a Current Loop DEMO – 6 B-04 & 05 10/29/2020 21

Solenoid (DEMO) 10/29/2020 DEMO – 6 B-04 & 05 22

Magnetic Field of a Solenoid Step 1: Cut up the distribution into pieces Step 2: Contribution of one piece origin: center of the solenoid one loop: B Number of loops per meter: N/L Number of loops in z: (N/L) z Field due to z: 23

Magnetic Field of a Solenoid Step 3: Add up the contribution of all the pieces B Magnetic field of a solenoid: 24

Magnetic Field of a Solenoid Special case: R<<L, center of the solenoid (d~0): in the middle of a long solenoid 25

Demos 6 B-13 26

Earth’s Magnetic Field 10/29/2020 32

Earth’s Magnetic Field (continued) How much current must the loop carry to produce a magnetic field of 0. 7 gauss at one of the Earth’s poles? 10 k. G = 1 T 10/29/2020 33

Earth’s Magnetic Field (continued) 1. It has decreased by ~ 6% in the last century. 2. Last polarity reversal was ~ 750, 000 years ago. 3. Time between reversals has varied between 20 K and 37 M years. Average is 500 K years. 4. Inner temperature is between 3, 900 and 7, 2000 C degrees. It has cooled about 1100 C in the last 4 billion years. 5. There is some agreement that convective motion of the molten part of the earth is generating the magnetic field. 10/29/2020 34