1

Physics 202, Lecture 12Today’s Topics

Magnetic Forces (Ch. 29) Review: magnetic force, magnetic dipoles Motion of charge in uniform B field:

Applications: cyclotron, velocity selector, Hall effect

Sources of the Magnetic Field (Ch. 30, part 1) Calculating the B field due to currents (Biot-Savart)

Homework #5: due 10/15 ,10 PM. Optional reading quiz: due 10/12, 7 PM

Magnetic Fields and Forces: Recap Magnetic Force: experienced by moving charges

Magnetic Field B: sourced by moving chargesdirection: as indicated by north pole of compass

!F = q

!v !!B

!F = I d

!l !!B"

(point charges) (currents)

Field lines: closed loops!Outside magnet: N to SInside magnet: S to N

Units: 1 Tesla (T) = 1 N/(A m)

2

Torque on Current Loop in Uniform B field

Force and torque on current loop:

magnetic dipole moment:

Loop rotates to minimize I.e., until

µB

!Fnet

= 0,!!net

=!µ "!B

I

I

I

I

1

2

3

4

B X •

!µ = NI

!A

!µ "!B

U = !

!µi

!B

(N=# of turns of loop, A=area)

Charged Particle in Uniform B Field

x x x x x xx x x x x xx x x x x x

v B

q

x x x x x xx x x x x xx x x x x x

FFv

R

Magnetic force does no work on charge: kinetic energy constant

Force perpendicular to velocity: uniform circular motion

!F = q

!v !!B = m

!a

3

x x x x x xx x x x x xx x x x x x

v

F

B

qFv

R

x x x x x x

x x x x x x

x x x x x x

• Newton's 2nd Law:

F = ma ! qvB = mv

2

R

! R =mv

qB=

p

qB

• Force:

F = qvB

• centripetal acc:

a =v

2

R

Trajectory in Uniform B Field (2)

“Cyclotron” frequency: ! =v

R=qB

mT =

2!

"=2!m

qB

(an important result, with usefulexperimental consequences!)

Trajectory in Uniform B Field (3)

In the plane perpendicular to B:

Parallel to B: spacing b/w turns of helix

R =mv

!

qBT =

2!m

qB

d = v!T =

v!2!m

qB

General 3D case:

4

The drawing shows the top viewof two interconnected chambers.Each chamber has a uniquemagnetic field. A positivelycharged particle fired intochamber 1 follows the dashedpath shown in the figure.

What is the direction of the magnetic field in chamber 1?

a) Up b) Down c) Leftd) Right e) Into page f) Out of page

Question 1

What is the direction of the magnetic field in chamber 2?

a) Up b) Down c) Leftd) Right e) Into page f) Out of page

Question 2

Which field is larger, B1 or B2?

a) B1 > B2 b) B1 = B2 c) B1 < B2

5

Application: Cyclotron First Modern Particle Accelerator

First Cyclotron (1934) Lawrence & Livingston

Fixed Frequency

Application: Velocity, Mass Selectors Velocity and mass selector:

Velocity Selector

Mass Selector

speed selected:

mass selected:

v =E

B

m

q=rB

0

v=

rB0

(E B)

6

The Hall Effect (1)

positive charges movingcounterclockwise: upward force,upper plate at higher potential

negative charges movingclockwise: upward forceUpper plate at lower potential

Equilibrium between electrostatic & magnetic forces:

Potential difference on current-carrying conductor in B field:

F

up= qv

dB F

down= qE

ind= q

VH

wV

H= v

dBw = "Hall Voltage"

The Hall Effect (2)

VH = vdBw =IB

nqtI = nqvdA = nqvdwt

RH!VH

IB=1

nqtHall coefficient:

Hall effect: determine sign,density of charge carriers

(first evidence that electrons are charge carriers in most metals)

7

Magnetic Fields of charges, currents

Are there analogous equations for the Magnetic Field?

Two Ways to calculate the electric field:

"Brute force"

– Coulomb’s Law

"High symmetry"

– Gauss’ Law

d!E = k

dq

r 2r̂

!E id!A! =

qin

"0

Review: back to electrostatics:

Calculation of Magnetic Fields (Currents)

× I

– Biot-Savart Law (“Brute force”)

–AMPERIAN LOOP

– Ampere’s Law (“High symmetry”)

!B id"l = µ

0#! I

Two Ways to calculate the magnetic field:

(Tuesday’s lecture)

d!B =

µ0I

4!

d!l ! r̂

r2

8

Biot-Savart Law...

IThe magnetic field “circulates”around the wireUse right-hand rule: thumb along I,fingers curl in direction of B.

XdB

dl

rθ

…add up the pieces

d!B =

µ0I

4!

d!l ! r̂

r2

=µ

0I

4!

d!l !!r

r3

µ0= 4! "10

#7N

A2

B Field of Straight Wire, length L (Text example 30.1) Show that B at point P is:

When the length of thewire is infinity:

)cos(cos4

210

!!"

µ#=

a

IB

θ1 θ2

a

IB

!

µ

2

0=

9

Direction: another right-hand rule

closed circular loops centered on current