(1) I have completed at least 50% of the reading and study-guide assignments associated with the lecture, as indicated on the course schedule. a) True b) False

iClicker Quiz

Hint: pay attention to how to DRAW figures. Today: A. Force due to current flowing in a wire. B. torque on loops C. magnetic dipole approaches. D. Potential and Kinetic energy of dipoles. Also iLxB (cross products) to get magnitude.

Note on Monday, 17th’s reading: 30:7 is part of 29.1 in 8th edition

Lab 7 due & Prof. Campbell is teach on Friday

B

At the equator, the B field points north.

a) up (radially outward) b) down (radially inward) c) north d) south e) east f) west

Positive charge moving near the equator.

v

If charge is moving east, which way does the Lorentz force point? v

If charge is moving upward, which way does the Lorentz force point?

Differences Between Electric and Magnetic Fields

•Direction of force – The electric force acts along the direction of the electric

field. – The magnetic force acts perpendicular to the magnetic

field. •Motion

– The electric force acts on a charged particle regardless of whether the particle is moving.

– The magnetic force acts on a charged particle only when the particle is in motion.

Section 29.1

Work in Fields •The electric force does work in displacing a charged particle. •The magnetic force associated with a steady magnetic field does no work when a particle is displaced.

– This is because the force is perpendicular to the displacement of its point of application.

•The kinetic energy of a charged particle moving through a magnetic field cannot be altered by the magnetic field alone. •When a charged particle moves with a given velocity through a magnetic field, the field can alter the direction of the velocity, but not the speed or the kinetic energy.

Section 29.1

Notation Notes •When vectors are perpendicular to the page, dots and crosses are used.

– The dots represent the arrows coming out of the page.

– The crosses represent the arrows going into the page.

•The same notation applies to other vectors.

Section 29.2

Magnetic Force on a Current Carrying Conductor

•A force is exerted on a current-carrying wire placed in a magnetic field.

– The current is a collection of many charged particles in motion.

•The direction of the force is given by the right-hand rule.

Section 29.4

Force on a Wire

•In this case, there is no current, so there is no force. •Therefore, the wire remains vertical.

Section 29.4

Force on a Wire, 2

•The magnetic field is into the page •The current is up the page •The force is to the left •The wire deflects to the left

Section 29.4

Force on a Wire, 3 •The magnetic field is into the page •The current is down the page •The force is to the right •The wire deflects to the right

Force on a Wire, equation •The magnetic force is exerted on each moving charge in the wire.

– •The total force is the product of the force on one charge and the number of charges.

–

dq= ×F v B

( )dq nAL= ×F v B

Section 29.4

Force on a Wire, Equation cont.

•In terms of the current, this becomes

– I is the current. – is a vector that points in the direction of the

current. • Its magnitude is the length L of the segment.

– is the magnetic field.

B I= ×F L B

L

B

Section 29.4

Force on a Wire, Arbitrary Shape

•Consider a small segment of the wire,

•The force exerted on this segment is •The total force is

b

B d= ×∫aF s B

I

ds

Bd I d= ×F s B

Section 29.4

Total force is path independent for uniform fields.

( ) BLIBdIdI

×=×=×= ∫∫ sBsF

Which wire experiences the greatest force?

Let’s draw a diagram.

Let’s draw a diagram.

Torque on a Current Loop

•The rectangular loop carries a current I in a uniform magnetic field. •No magnetic force acts on sides 1 & 3.

– The wires are parallel to the field. So,

0× =L B

Section 29.5

Torque on a Current Loop, 2 •There is a force on sides 2 & 4 since they are perpendicular to the field. •The magnitude of the magnetic force on these sides will be:

– F2 = F4 = I a B •The direction of F2 is out of the page. •The direction of F4 is into the page.

Section 29.5

Torque on a Current Loop, 3

•The forces are equal and in opposite directions, but not along the same line of action. •The forces produce a torque around point O.

Section 29.5

What is the moment arm?

Torque on a Current Loop, Equation

•The maximum torque is found by:

•The area enclosed by the loop is ab, so τmax = IAB.

– This maximum value occurs only when the field is parallel to the plane of the loop.

2 42 2 2 2max (I ) (I )

I

b b b bτ F F aB aB

abB

= + = +

=

Section 29.5

Torque on a Current Loop, General

•Assume the magnetic field makes an angle of θ < 90o with a line perpendicular to the plane of the loop. •The net torque about point O will be τ = IAB sin θ.

Section 29.5

No net force. But there is a torque.

Summary: A current loop in a uniform field

If the field is not uniform, a net force is possible.

When is torque a maximum? pp A. moment arm parallel to B. perpedicular to B. C. half way between

Torque on a Current Loop, Summary

•The torque has a maximum value when the field is perpendicular to the normal to the plane of the loop. •The torque is zero when the field is parallel to the normal to the plane of the loop.

– is perpendicular to the plane of the loop and has a magnitude equal to the area of the loop.

Iτ = ×A B

A

Section 29.5

Direction •The right-hand rule can be used to determine the direction of . •Curl your fingers in the direction of the current in the loop. •Your thumb points in the direction of .

A

A

Section 29.5

Magnetic Dipole Moment

•The product I is defined as the magnetic dipole moment, , of the loop.

– Often called the magnetic moment •SI units: A · m2 •Torque in terms of magnetic moment:

– Analogous to for electric dipole – Valid for any orientation of the field and the

loop – Valid for a loop of any shape

τ = ×p E

τ µ= ×B

µA

Section 29.5

Aμ NI=

IabNIA ==µ

If I = 1 Amp, a = 3 m and b = 2 m, then µ = 6 A⋅m2.

Potential Energy

•The potential energy of the system of a magnetic dipole in a magnetic field depends on the orientation of the dipole in the magnetic field given by

– Umin = -µB and occurs when the dipole moment is in the same direction as the field.

– Umax = +µB and occurs when the dipole moment is in the direction opposite the field.

U µ= − B

Section 29.5

B µ µ µ

θ

BABμτ ×=×= )(NI

BABμ ⋅−=⋅−= )(NIU

Which current-carrying loop has the lowest potential energy?

Which loop experiences the greatest torque? Direction?

Electric dipoles

Points from (−) to (+).

+−= rp Q

p

30

ˆ)ˆ(34

1r

prrpE −⋅=

πε

Idealized as a “point” dipole in the limit that r+− → 0 and Q → ∞ as p stays constant.

Magnetic dipoles

Points from (S) to (N).

30 ˆ)ˆ(3

4 rB μrrμ −⋅

=π

µ

Idealized as a “point” dipole in the limit that A → 0 and I → ∞ as µ stays constant.

Aμ I=

µ

Magnetic dipole moments of subatomic particles

1836J/T109.27

224 B

Ne

B me µµµ ≈×== −Bohr magneton Nuclear magneton

NnNpBe µµµµµµ 913.1973.2 −==−=electron neutron proton

µ

L

Subatomic particles have angular momentum (L), and can also have a magnetic dipole moment µ = γL, suggesting that their moments arise due to spinning internal electric charge. Though this analogy is very limited, we often refer to magnetic dipole moments as “spins”.

Charge-particle motion in a magnetic field Magnetic force is always ⊥ to direction of motion, driving the charge in circles.

rmvqvBFB

2

==

Thus, the magnetic force is a centripetal force.

Bv×= qFB

qBmvr =Radius:

Angular frequency:

Bmq

rv

==ω

constant and || ==⊥ vm

qBrv

Only v⊥ leads to circular motion, as v|| creates no force.

Together, both components result in a spiral trajectory.

Charge-particle motion in a magnetic field

Charge-particle motion in a magnetic field

Magnetic fields do no work on electrically-charged particles, and kinetic energy is conserved!

0)( =⋅×=⋅= sBvs dqdFdW B

Because v and ds are parallel, the FB and ds are perpendicular.

mqBrmvK2

)( 22

21 ==

If B increases, then r decreases.

B

+

Charged particles trapped in the magnetosphere

http://en.wikipedia.org/wiki/Aurora_%28astronomy%29

http://en.wikipedia.org/wiki/Aurora_%28astronomy%29

Which of these trajectories belongs to a negatively charged particle?

Velocity Selector

BEv

qEqvBFF EB

=

=

=

Mass spectrometer

Brv

mq

=

Separately measure v and r.

E

B FE

FE

e- ? BEv

qEqvBFF EB

=⇒

=⇒=

Cathode ray tube

Doesn’t matter if current carriers are positive or negative. Either way, the force direction is the same.

Current flowing through conducting object in the presence of ⊥ B field. Positive and negative carriers would be forced in opposite directions.

For a metallic object, ∆V = Vtop – Vbottom should be (1) positive (2) negative (3) zero?

Case #1: Electric current flow in a wire.

B

(1) A (2) B (3) ∆V = 0

Which terminal is more positive? Case #2: Fluid mixture of positive and negative ions flowing through a tube.

Hall Effect

dEdBvV

EqBvqF

effdH

effcdc

][)(

][)(

==

==

dtqnIv

dtvqnAJI

ccd

dcc

=⇒

==

tqnIBBd

dtqnIqV

cccccH =

=

F Oct 14 20 Chap. Summaries: 26-28 Review for EXAM 3; runs to closing Monday. 28:D 16 5-6

M Oct 17 21 Serway sections 29:1, 30.7 5th-6th Ed. 29:1, 30.9

Magnetic fields and forces The magnetic field of the Earth SE:26-28

W Oct 19 22 Serway sections 29:4-5 5th-6th Ed. 29:2-3 Lab 7 description

Magnetic force on a current-carrying wire Magnetic torque on a current-carrying loop

29:A 17

F Oct 21 23 Serway sections 29:2-3, 6 5th-6th Ed. 29:4-6 Lab 8 description

Motion of charge particle in a B-field The q/m ratio of the electron The Hall effect

29:B 18 7

M Oct 24 24 Serway sections 30:1-2 Biot-Savart law Magnetic force -- current-carrying wires 29:C-D 19

W Oct 26 25 Serway sections 30:3-4 Ampere’s law Relationship to Biot-Savart law Solenoids, toroids, etc.

30:A-B 20

F Oct 28 26 Serway sections 30:5 5th-6th Ed. 30:5-6

Magnetic flux Gauss’s law of magnetism 30:C 21 8

M Oct 31 27 Serway sections 30:6 5th-6th Ed. 30:8

Magnetization (M), magnetic induction (H) Magnetic permeability (µ), susceptibility (χ) Magnetization curves and Curie’s law

30:D-F 22

W Nov 2 28 Chap. Summaries: 29-30 Review for EXAM 4 runs to closing Friday. 30:H-J 23