Physics 202, Lecture 13. Today s Topics

Transcription

1 Physics 202, Lecture 13 Tody s Topics Sources of the Mgnetic Field (Ch. 30) Clculting the B field due to currents Biot-Svrt Lw Emples: ring, stright wire Force between prllel wires Ampere s Lw: infinite wire, solenoid, toroid Displcement Current: Ampere-Mwell Mgnetism in Mtter On WebAssign Tonight: Homework #6: due 10/22,10 PM. Optionl reding quiz: due 10/19, 7 PM Mgnetic Fields of Current Distributions First, review: bck to electrosttics: Two wys to clculte the electric field directly: Coulomb s Lw d E = 1 4" 0 dq r 2 Guss Lw E i d A = q in " 0 ˆr "Brute force" "High symmetry" ε 0 = C 2 /(N m 2 ): permittivity of free spce 1

2 Mgnetic Fields of Current Distributions Two Wys to clculte the mgnetic field: Biot-Svrt Lw ( Brute force ) Ampere s Lw ( high symmetry ) d B = µ I 0 d l " ˆr 4 r 2 Bid s " = µ 0 I enclosed AMPERIAN LOOP I µ 0 = 4π10-7 T m/a: permebility of free spce Biot-Svrt Lw... dd up the pieces dl d B = µ 0 I 4 d l ˆr r 2 = µ 0 I 4 d l r r 3 θ r X db I B field circultes round the wire Use right-hnd rule: thumb long I, fingers curl in direction of B. 2

3 B Field of Circulr Current Loop on Ais (Tet emple 30.3) B field on is of current loop is: µ0 IR 2 B = 2( 2 + R 2 ) 3 / 2 Bcenter = µ0 I 2R See lso: center of rc (tet emple 30.2) B of Circulr Current Loop: Field Lines B 3

4 B Field of Stright Wire, length L (Tet emple 30.1) B field t point P is: B = µ 0 I (cos 1 # cos 2) 4" When the length of the wire is infinity: B = µ 0I 2 θ 1 θ 2 (return to this lter with Ampere s Lw) Superposition: emple Ch. 30, #3 Quick Question 1: Mgnetic Field nd Current Which figure represents the B field generted by the current I

5 Quick Question 2: Mgnetic Field nd Current Which figure represents the B field generted by the current I. Forces between Current-Crrying Wires A current-crrying wire eperiences force in n eternl B-field, but lso produces B-field of its own. Mgnetic Force: I b d F F I Prllel currents: ttrct d F F I b I Opposite currents: repel 5

6 Emple: Two Prllel Currents Force on length L of wire b due to field of wire : B = µ I 0 2d F b = I b d l B µ " = 0 I b I L 2#d d F F I b I Force on length L of wire due to field of wire b: B b = µ 0 I b 2d F = I d l B µ " b = 0 I I b L 2#d Ampere s Lw: " ny closed pth Ampere s Lw Bid s = µ 0 I enclosed I ds pplies to ny closed pth, ny sttic B field useful for prcticl purposes only for situtions with high symmetry Ampere s Lw cn be derived from Biot-Svrt Lw Generlized form: Ampere-Mwell (lter in lecture) 6

7 Ampere s Lw: B-field of Stright Wire Use symmetry (nd Ampere s Lw) I Bi d " s = µ 0 # I Choose loop to be circle of rdius R centered on the wire in plne to wire. Why? Mgnitude of B is constnt (function of R only) Direction of B is prllel to the pth. # Bi d s " = # Brd = 2"rB = µ 0 I B = µ I 0 2r Quicker nd esier method for B field of infinite wire (due to symmetry -- compre Guss Lw) B Field Inside Long Wire Totl current I flows through wire of rdius into the screen s shown. Wht is the B field inside the wire? By symmetry -- tke the pth to be circle of rdius r: r Bid s = B2r " Current pssing through circle: I enclosed = r 2 2 I Therefore: Ampere s Lw gives " Bid s = B2r = µ o I enclosed B = µ 0Ir 2 2 7

8 B Field of Long Wire Inside the wire: (r < ) B = µ 0 I 2 r 2 Outside the wire: ( r > ) B = µ 0 I 2 r B r Ampere s Lw: Toroid (Tet emple 30.5) Using Ampere s Lw, B field inside toroid is found to be: B = µ 0NI 2 r 8

9 Ampere s Lw: Toroid Toroid: N turns with current I. B φ =0 outside toroid (Consider integrting B on circle outside toroid: net current zero) B φ inside: consider circle of rdius r, centered t the center of the toroid. Bid s = B2r = µ o I enclosed " B = µ 0 NI 2r r B B Field of Solenoid Solenoid: source of uniform B field (inside of it) Solenoid: current I flows through wire wrpped n turns per unit length on cylinder of rdius nd length L. L To clculte the B-field, use Biot-Svrt nd dd up the field from the different loops. If << L, the B field is to first order contined within the solenoid, in the il direction, nd of constnt mgnitude. In this limit, cn clculte the field using Ampere's Lw 9

10 Ampere s Lw: Solenoid The B field inside n idel solenoid is: (see bord) B = µ 0nI n=n/l idel solenoid segment 3 t Solenoid: Field Lines Right-hnd rule: direction of field lines. In this emple, since the field lines leve the left end, the left end is the north pole. Like br mgnet (ecept it cn be turned on nd off) 10