Physics 202, Lecture 13. Today s Topics. Magnetic Forces: Hall Effect (Ch. 27.8)

Transcription

1 Physics 202, Lecture 13 Today s Topics Magnetic Forces: Hall Effect (Ch. 27.8) Sources of the Magnetic Field (Ch. 28) B field of infinite wire Force between parallel wires Biot-Savart Law Examples: ring, straight wire Ampere s Law: infinite wire, solenoid, toroid The Hall Effect (1) Potential difference on current-carrying conductor in B field: positive charges moving counterclockwise: upward force, upper plate at higher potential negative charges moving clockwise: upward force Upper plate at lower potential Equilibrium between electrostatic & magnetic forces: F up = qv d B F down = qe ind = q V H w V H = v d Bw = "Hall Voltage" 1

2 The Hall Effect (2) Text: I = nqv V H = v d Bw = IB d A = nqv d wt nqt R H V H IB = 1 nqt Hall coefficient: Hall effect: determine sign,density of charge carriers (first evidence that electrons are charge carriers in most metals) Magnetic Fields of Current Distributions Recall electrostatics: Coulomb s Law) F = kq q ˆr F = q E Two ways to calculate E directly: Coulomb s Law d E = 1 4" 0 dq r 2 ˆr "Brute force" r 2 (point charges) Gauss Law "High symmetry" E i d A = q 0 = (4"k) #1 in ε " 0 = 8.85x10-12 C 2 /(N m 2 ): 0 permittivity of free space 2

3 Magnetic Fields of Current Distributions F = 2 ways to calculate B: " Id l B F = k m Id l I" d l " ˆr # # (2 circuits) r 2 Biot-Savart Law ( Brute force ) Ampere s Law ( high symmetry ) d B = µ 0I d l " ˆr 4 r 2 Bid l " = µ 0 I enclosed I AMPERIAN LOOP µ 0 = 4k m µ 0 = 4πx10-7 T m/a: permeability of free space Magnetic Field of Infinite Wire Result (we ll derive this both ways): B = µ 0 I 2r ˆ" right-hand rule closed circular loops centered on current 3

4 Quick Question 1: Magnetic Field and Current Which figure represents the B field generated by the current I? Quick Question 2: Magnetic Field and Current Which figure represents the B field generated by the current I? 4

5 Forces between Current-Carrying Wires A current-carrying wire experiences a force in an external B-field, but also produces a B-field of its own. Magnetic Force: d F F I b I a Parallel currents: attract d F F I b I a Opposite currents: repel Example: Two Parallel Currents Force on length L of wire b due to field of wire a: B a = µ 0 I a 2d F b = I b d l B µ " a = 0 I b I a L 2#d d F F I b x Force on length L of wire a due to field of wire b: B b = µ 0I b 2d F a = I a d l B b = " µ 0 I a I b L 2#d Text: 28.4, I a 5

6 Biot-Savart Law... add 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 circulates around the wire Use right-hand rule: thumb along I, fingers curl in direction of B. B Field of Straight Wire, length L (Text example 28-13) B field at point P is: B = µ 0 I (cos 1 # cos 2) 4" a When L=infinity (text example 28-11): B = µ 0I 2 a θ 1 θ 2 (return to this later with Ampere s Law) Text:

7 B Field of Circular Current Loop on Axis Use Biot-Savart Law (text example 28-12) µ0 IR 2 Bx = 2( x 2 + R 2 ) 3 / 2 Bcenter = µ0 I 2R See also: center of arc (text example 28-13) Text: B of Circular Current Loop: Field Lines B 7

8 Ampere s Law: " any closed path Ampere s Law Bid l = µ 0 I encl I ds applies to any closed path, any static B field useful for practical purposes only for situations with high symmetry Generalized form (valid for time-dependent fields): Ampere-Maxwell (later in course) Ampere s Law: B-field of Straight Wire Use symmetry and Ampere s Law: I B i d " l = µ 0 # I encl Choose loop to be circle of radius R centered on the wire in a plane to wire. Why? Magnitude of B is constant (function of R only) Direction of B is parallel to the path. B # id l " = # Brd = 2"rB = µ 0 I encl = µ 0 I B = µ I 0 2r Quicker and easier method for B field of infinite wire (due to symmetry -- compare Gauss Law) 8

9 Text example: 28-6 B Field Inside a Long Wire Total current I flows through wire of radius a into the screen as shown. What is the B field inside the wire? By symmetry -- take the path to be a circle of radius r: " Bid l = B2r Current passing through circle: I encl = r 2 Bid l = B2r = µ o I encl x x x x x x x x x x r x x a x x x x x a 2 I " B in = µ 0Ir 2a 2 B Field of a Long Wire Inside the wire: (r < a) B = µ 0 I 2 r a 2 a Outside the wire: ( r > a ) B = µ 0 I 2 r B r Text:

10 Ampere s Law: Toroid (Text example 28-10) Show: N = # of turns B in = µ 0NI 2r Ampere s Law: Toroid Toroid: N turns with current I. B θ =0 outside toroid (Consider integrating B on circle outside toroid: net current zero) B θ inside: consider circle of radius r, centered at the center of the toroid. Bid s = B2r = µ o I enclosed " B = µ 0 NI 2r x x x x x x x x x x r x x x x x x B 10

11 B Field of a Solenoid Inside a solenoid: source of uniform B field Solenoid: current I flows through a wire wrapped n turns per unit length on a cylinder of radius a and length L. L To calculate the B-field, use Biot-Savart and add up the field from the different loops. a If a << L, the B field is to first order contained within the solenoid, in the axial direction, and of constant magnitude. In this limit, can calculate the field using Ampere's Law Ampere s Law: Solenoid The B field inside an ideal solenoid is: B = µ 0nI n=n/l ideal solenoid segment 3 at 11

12 Solenoid: Field Lines Right-hand rule: direction of field lines north pole south pole Like a bar magnet (except it can be turned on and off) Text: