Particle tracks What is the direction of the uniform magnetic field? electron e

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Particle tracks

Î

What is the direction of the uniform magnetic field?

electron e

– positron e

+ electron e

PHY2049: Chapter 28

1

Cosmic Ray Example

Î

Protons with energy 1 MeV move

⊥ of 0.5 gauss or 5 frequency of orbit.

× 10

-5 earth B field

T. Find radius &

K

= 1

2

mv

2

v

=

2

K m

K

m

=

=

( )(

27 kg

19

)

=1.6 10

13

J

R

=

mv eB

=

2

mK eB

R =

2900 m

f

=

1

T

=

2

π

v

R

=

2

π

(

v

)

=

eB

2

π

m f

=

760Hz

Frequency is independent of v!

PHY2049: Chapter 28

2

Helical Motion in B Field

Î

If velocity of particle has 2 components

‹

v v

||

‹

Only v

v

r

(parallel to B and perp. to B)

= v sin φ contributes to circular motion

‹ v

||

= v cos φ is unchanged

Î

So the particle moves in a helical path v

‹ v

||

‹ v

⊥ is the constant velocity along the B field is the velocity around the circle v

||

φ

v

R

=

mv

qB

B

3

PHY2049: Chapter 28

Helical Motion in Earth’s B Field

Particles moving along field lines cause Aurora Borealis and

Australis: http://science.nasa.gov/spaceweather/aurora/gallery_01oct03.html

PHY2049: Chapter 28

4

Magnetic Force on Current-Carrying Wire

Î

Magnitude of force sin

φ

‹

Easy to derive from charge, number density & drift velocity of individual charge carriers

Î

Direction of force: RHR

PHY2049: Chapter 28

5

Example

Î

A 4 m long wire carries current of 500 A in NE direction

‹

Magnitude of force (B = 0.5 gauss = 5

×

10

-5

T, pointing N) sin

φ

=

( )

(

5

)

( )( )

=

0.071N

‹

Direction of force?

Upwards, from RHR

Î

Can adjust current in wire to balance against gravity

‹

iBL

sin

φ

=

mg

Calculate mass from density, length and cross-sectional area

m

=

ρ

LA

‹

Good exam problem!

PHY2049: Chapter 28

6

Magnetic Force

Î

A vertical wire carries a current in a vertical magnetic field. What is the direction of the force on the wire?

‹

(a) left

‹

(b) right

‹

(c) no force

‹

(d) into the page

‹

(e) out of the page

B

I is parallel to B, so no magnetic force

I

PHY2049: Chapter 28

7

Torque on Current Loop

Î

Rectangular current loop in uniform magnetic field (lengths a & b)

‹

Forces in left & right branches are 0

‹

Force in top branch is into plane

‹

Force in bottom branch is out of plane

Î

Equal forces give net torque!

‹

Bottom side up, top side down (RHR)

‹

τ

Rotates around horizontal axis

=

Fd

=

( )

Î

µ = NiA

‹

⇒ “magnetic dipole moment”

Assuming N turns

‹

τ = µ B, true for any shape!!

Î

If plane tilted angle θ to B field

‹

τ

‹

θ

= µ Bsin θ is angle between normal and B

b

PHY2049: Chapter 28

a

B

a

Plane normal is ⊥

( θ = 90 ° )

B

8

b

Magnetic Force

Î

A rectangular current loop is in a uniform magnetic field.

What direction is the net force on the loop?

‹

(a) + x

‹

(b) + y

‹

(c) no force

‹

(d) – x

‹

(e) – y

B

Forces cancel on opposite sides of loop

z x

PHY2049: Chapter 28

y

9

Magnetic Dipole Moment

PHY2049: Chapter 28

10

Torque Example

Î

A 3-turn circular loop of radius 3 cm carries 5A current in a B field of 2.5 T. Loop is tilted 30 ° to B field.

30 °

Î

µ

=

3

π

2

×

(

0.03

) 2

=

2

Î

B

Î

Rotation is always in direction to align µ with B field

PHY2049: Chapter 28

11

Mass Spectrometer

Î

Originally developed by physicists. Now an important tool in chemistry, biology, environmental studies, forensics, pharmaceutics, etc.

Î

Sample is vaporized, broken into fragments of molecules, which are positively ionized. Positive ions are first accelerated by a potential difference V, and then their trajectories are bent by B. Varying B

(sometimes V) allows ions of different masses to reach the detector.

PHY2049: Chapter 28

12

Mass Spectrometer (simplified)

Î

Sample is vaporized, broken into fragments of molecules, which are positively ionized. Positive ions are first accelerated by a potential difference V, and then their trajectories are bent by B. Varying B (sometimes V) allows ions of different masses to reach the detector.

‹

Spectrometer determines mass from B

(sometimes from V)

m

=

q

(

Br

)

2

2

V

D

13

PHY2049: Chapter 28

Hall Effect: Do + or – Charges Carry Current?

Î

Î

+ charges moving counter-clockwise experience upward force

Upper plate at higher potential

Î

Î

– charges moving clockwise experience upward force

Upper plate at lower potential

¾

¾

Very quickly, equilibrium between electrostatic & magnetic forces is established and potential difference stops growing:

F

up

=

F

down

=

qE

induced

=

q

V

H w

V

H

= =

"Hall Voltage"

¾

This type of experiment led to the discovery (E. Hall, 1879) that current in conductors is carried by negative charges

Hall effect is used to measure moderate to moderately high B (10

-4

T

– 3 T)

It is also used to measure the speed of computer hard drive

PHY2049: Chapter 28

14

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