LECTURE 15 Amperes Law Gauss Law for Magnetism

LECTURE 15 • Ampere’s Law • Gauss’ Law for Magnetism • Force between Current Carrying Wires • Magnetism in Matter

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: 2

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

Magnetic Field of a Solenoid Special case: R<<L, center of the solenoid (d~0): in the middle of a long solenoid 4

Solenoid *Direction of B: Use the right hand rule. Curl fingers in direction of current flow in loop and thumb points in the direction of B. 12/20/2021 5

Magnetic Field Inside an Ideal Solenoid 12/20/2021 6

Ampere’s Law • Calculate field at distance r from wire using Ampere's Law:

Magnetic Field Inside a Long Straight Wire 12/20/2021 8

Magnetic Field Inside & Outside a Long Straight Wire Inside Outside 12/20/2021 9

Magnetic Field Inside a Toroid 12/20/2021 10

When Ampere’s Law doesn’t Help B can’t be factored out of the integral. finite length current segment 12/20/2021 insufficient symmetry current is not continuous 11

Remember Gauss’ Law for Electric Fields 12/20/2021 12

Gauss’ Law for Magnetism 12/20/2021 Since all lines of B are closed loops, any B line leaving a closed surface MUST reenter it somewhere. TRUE IN GENERAL, not just for this “dipole” example 13

Question y Two parallel horizontal wires are located in the vertical (x, y) plane as shown. Each wire carries a current of I = 1 A flowing in the directions shown. What is the direction of B at P? y I 1 = 1 A 4 cm x 4 cm I 2 = 1 A Front view 3 cm z P Side view y y y . . . z P A 12/20/2021 z P B z z P P C D z P E 14

Calculation y Two parallel horizontal wires are located in the vertical (x, y) plane as shown. Each wire carries a current of I = 1 A flowing in the directions shown. y I 1 = 1 A 4 cm x 4 cm What is the B field at point P? y y y . . A z P B 90 o z z P P C 3 cm P Side view y P z I 2 = 1 A Front view What is the direction of B at P? z . D

Force between Two Parallel Current Carrying Wires *Parallel currents attract *Anti-parallel currents repel

Force between Two Parallel Current Carrying Wires

DEMO parallel wires 13. 5 V I 1 R 1 12/20/2021 R 2 I 2 , depending on switch position 18

Force Between two Parallel Current Carrying Wires I 1 I 2 attraction I 1 I 2 repulsion Collapse or expansion? Magnetic “pressure” 12/20/2021 19

Force between a Solenoid & a Current Carrying Wire (DEMO) 12/20/2021 20

QUIZ lecture 15 The figure shows a long straight wire with current I and three wire loops, all with the same clockwise current around them. The loops are all at the same distance from the long wire and have edge lengths of either L or 2 L. Rank the loops according to the magnitude of the net force on them due to the current in the long straight wire, greatest first. A. 1 = 2 =3 I B. 1 > 2 > 3 L 2 L I (1) I I (2) (3) C. 1 > 2 = 3 D. 2 = 3 > 1 E. 3 > 2 >1 12/20/2021 21

Dipole Moments in Applied Fields 12/20/2021 22

Magnetization and “Bound Current” Net current inside the material is zero. We are left with a surface current and therefore a magnetic moment 12/20/2021 23

Magnetization and “Bound Current” 12/20/2021 24

Magnetization and Magnetic Susceptibility bismuth, 12/20/2021 25

Magnetic Materials fall into Three Categories m Km 1+ *increase in B is small *aligns with Bapp Order of (+10– 5), depends on temperature Diamagnetism Order of (– 10– 5) 1– positive and large ~ m Category Paramagnetism Decrease in B is small aligns opposite Bapp Ferromagnetism High degree of alignment even in weak Bapp 12/20/2021 26

Magnetic Susceptibility m Material m Type Bi – 1. 66 x 10– 5 diamagnetic Ag – 2. 6 x 10– 5 diamagnetic Al Fe (annealed) Permalloy mu-metal superconductor 2. 3 x 10– 5 5, 500 25, 000 100, 000 – 1 paramagnetic ferromagnetic diamagnetic (perfect) m negative for diamagnetics, small and positive for paramagnetics, large and positive for ferromagnetics. 12/20/2021 27