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

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 ⊥ earth B field of 0.5 gauss or 5 × 10 -5 T. Find radius & frequency of orbit.

K = 1

2 mv

2 ⇒ v =

2 K m

K m

=

( )(

×

= 1.67 10

− 27 kg

− 19

)

× − 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 = µ Bsin θ



θ is angle between normal and B b

PHY2049: Chapter 28 a

B a

Plane normal is ⊥ B

( θ = 90 ° )

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

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

F up

= qv B F down

= qE induced

= q

V

H w

V

H

= v Bw = "Hall Voltage"

¾ This type of experiment led to the discovery (E. Hall, 1879) that current in conductors is carried by negative charges

¾ Hall effect is used to measure moderate to moderately high B (10 -4 T – 3 T)

¾ It is also used to measure the speed of computer hard drive

PHY2049: Chapter 28 14