Physics 202, Lecture 12. Today s Topics

Transcription

1 Physics 202, Lecture 12 Today s Topics Magnetic orces (Ch. 27) Review: magnetic force, magnetic dipoles Motion of charge in uniform field: Applications: cyclotron, velocity selector, Hall effect Sources of the Magnetic ield (Ch. 28) Calculating the field due to currents (iot-savart) Magnetic ields and orces: Recap Magnetic orce: experienced by moving charges = q v = d l (point charges) (currents) Magnetic ield : sourced by moving charges direction: as indicated by north pole of compass " Units: 1 Tesla (T) = 1 N/(A m) ield lines: closed loops Outside magnet: N to S nside magnet: S to N 1

2 orces on a Current Loop or current loops in a uniform magnetic field as shown, what is the direction of the force on each side? = X = Case 1 recall:σ =0 Case 2 Torque on Current Loop in Uniform ield Though the net magnetic force on a closed current loop in a uniform field is zero, there can be a net torque. Case 1: A θ=π/2 X Loop has length dimension a (normal to ), width b. = 2 b sin" = absin" 2 = A " Define the magnetic moment µ # A = µ " More general: A θ X µ = N A (N turns) 2

3 Magnetic Dipole Moments Magnetic dipole moment µ. S N µ Macroscopic µ=a Microscopic µ L angular momentum of orbiting or spin definition of magnetic moment = 0 " = " µ # U = - µi µ in ield Text: Torque on Current Loop in Uniform field orce and torque on current loop: net = 0, net = µ " magnetic dipole moment: µ = N A Loop rotates to minimize.e., until µ " U = µi 4 1 (N=# of turns of loop, A=area) X 3 2 Text: µ 3

4 Charged Particle in Uniform ield = q v = m a v x x x x x vx q R orce perpendicular to velocity: uniform circular motion Magnetic force does no work on charge: kinetic energy constant Trajectory in Uniform ield orce: = qv centripetal acc: a = v2 R Newton's 2nd Law: v x x x x x vx q R = ma qv = m v2 R = mv q = p q Cyclotron frequency: R (an important result, with useful experimental consequences) = v R = q m T = 2 " = 2m q 4

5 Trajectory in Uniform ield: 3D case General 3D case: n the plane perpendicular to : R = mv q T = 2m q Parallel to : spacing b/w turns of helix d = v T = v 2m q Question 1 The drawing shows the top view of two interconnected chambers. Each chamber has a unique magnetic field. A positively charged particle fired into chamber 1 follows the dashed path shown in the figure. What is the direction of the magnetic field in chamber 1? a) Up b) Down c) Left d) Right e) nto page f) Out of page 5

6 Question 2 What is the direction of the magnetic field in chamber 2? a) Up b) Down c) Left d) Right e) nto page f) Out of page Which field is larger, 1 or 2? a) 1 > 2 b) 1 = 2 c) 1 < 2 Text examples: 27.19, 26, 30 Application: Cyclotron irst Modern Particle Accelerator ixed requency Text example: irst Cyclotron (1934) Lawrence & Livingston 6

7 Application: Velocity, Mass Selectors Velocity and mass selector: speed selected: Velocity Selector v = E mass selected: m q = r 0 v = r 0 (E ) Mass Selector Text: The Hall Effect (1) Potential difference on current-carrying conductor in 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: up = qv d down = qe ind = q V H w V H = v d w = "Hall Voltage" Text:

8 The Hall Effect (2) = nqv V H = v d w = d A = nqv d wt nqt R H V H = 1 nqt Hall coefficient: Hall effect: determine sign,density of charge carriers (first evidence that electrons are charge carriers in most metals) Magnetic ields of charges, currents Review: back to electrostatics: Two Ways to calculate the electric field: Coulomb s Law d E = k dq r 2 ˆr "rute force" Gauss Law E i d A = q in " 0 "High symmetry" Are there analogous equations for the Magnetic ield? 8

9 Calculation of Magnetic ields (Currents) Two Ways to calculate the magnetic field: iot-savart Law ( rute force ) d = µ 0 4 d l ˆr r 2 Ampere s Law ( High symmetry ) # i d l " = µ 0 AMPERAN LOOP (Tuesday s lecture) iot-savart Law... θ dl r add up the pieces d = µ 0 4 X d d l ˆr r 2 = µ 0 4 µ 0 = 4 " 10 #7 N A 2 d l r r 3 The magnetic field circulates around the wire Use right-hand rule: thumb along, fingers curl in direction of. 9

10 ield of Straight Wire, length L Show that at point P is: = µ 0 (cos 1 # cos 2) 4" a When the length of the wire is infinity: = µ 0 2 a θ 1 θ 2 Direction: another right-hand rule closed circular loops centered on current 10

11 Quick Question: Magnetic ield And Current Which figure represents the field generated by the current Quick Question: Magnetic ield And Current Which figure represents the field generated by the current. 11

12 orces between Current-Carrying Wires A current-carrying wire can experience force from -field. A current-carrying wire also produces a -field. Thus: one current-carrying wire exerts a force on another current-carrying wire: d Current goes together wires come together Current goes opposite wires go opposite = µ L 2d b a Exercise: Circular Current Loop Show the field is: µ R = x ( x R ) 3/ 2 center = µ 0 2R 12

13 of Circular Current Loop: ield Lines 13