Lecture 21Applications of moving

charges in magnetic fields.

LHC rap

Particle accelerators

Like the newly inaugurated Large Hadron Collider (LHC) at CERN

Huge magnets keep the particles moving in circles.

In some sections, electric fields (NOT magnetic fields) accelerate the particles

E

Magnetic fields cannot do work!

E

B

mv pR

qB qB

Measuring the mass of a particle in an accelerator

Measuring the curvature of a path is the usual way of measuring momentum of particles in high energy experiments.

mv pR

qB qB

B into screen

positive particle

negative

particle

pmvR

qB qB

e+

e-

B

Thompson’s q/m experiment

mvR

qB

In 1897, Thomson measured the q/m ratio for “cathode rays” (electrons emitted by a hot filament). He found that all rays yield the same q/m ratio, for any material source.

Electrons are a fundamental constituent of all matter!

Radius of circular trajectory of charge q in uniform field B:

Charge q acquires its speed between two plates with potential difference V:

2 21

2

qVqV mv v

m

2

2 2qVm mVR

qB m qB

2 2

2

q Vm R B

With a known voltage and B-field, if we measure R we can predict q/m

Mass spectrometer

Used to identify substances

2 2

2R B

m qV

1. Electrostatically accelerated electrons knock electron(s) off the atom positive ion (q =|e|)

2. Accelerate the ion in a known potential difference V

3. Pass the ions through a known B field: Deflection depends on mass: Lighter deflects more, heavier less

4. Electrically detect the ions which “made it through”

5. Change B (or V ) and try again

DEMO: Electron beam

in Helmholtz coils

Applications: Chemical analysis (including things like identifying chemical and

biological weapons –nerve gas, anthrax, etc.).Paleoceanography: Determine relative abundances of isotopes (they

decay at different rates geological age)Space exploration: Determine what’s on the moon, Mars, etc.

Magnetic force on a current-carrying wire

qv

A current I flows in a wire with cross-section A. There are n carriers of charge q0 per unit volume.

0Force on one carrier: F qv B

0dq q nAdlCharge in a section of length dl :

0F nAdlqv B

Force on a section of length dl :

Physics 221, lecture 39: 0I nqvA

0nAvq dl B

and are paralleldl v

dF I dl B

For a straight segment of length L

F I L B

DEMO: Magnet on wire

Example: Electromagnetic rail gun

A conducting bar (orange segment) of mass m can slide without friction on the horizontal wires that are connected to a source that provides a constant current I. There is a uniform magnetic field B into the screen. The bar is initially at rest.

Find the velocity of the bar as a function of time.

I

L

F I L B

ˆI LBi

B

LI

y

x z

xx

F I LBa

m m

x x

x

v a t

I LBv t

m

v

I

LB

LI

y

x z

x

I LBv t

m

v

Wait a minute: Magnetic fields cannot do work!

Where does this additional kinetic energy come from???

Answer in chapter 29 (lectures 25-26):

We’ll see that keeping that current constant is not so obvious, even for ideal wires without resistance. When the bar moves, current tends to decrease (Lenz’s law). The extra energy comes from the additional potential energy that the battery needs to supply to keep the current the same. Ie, an electric field is doing the work.

This is still true!!

In-class example: Lifting bar

A voltage source and variable resistor are used to sweep current through a 1.0 m long rod with a mass of 100 g in a uniform, horizontal B field of 1000 G (0.1 T). The circuit is horizontal (shown from above here). If the rod is simply resting on two end supports, for what current will it lift off of the supports?

I

L

magF I L B B

L

Vmag (direction )F I LB magF

Bar lif ts off when

I LB mg

20.1 kg 9.8 m/ s9.8 A

1.0 m 0.1 T

mgI

LB

A. 3.4 A

B. 9.8 A

C. 12.4 A

D. 18.6 A

E. 32.4 A

And yes, the magnetic field is still not doing any work!

Speakers

BF

i

B

Fi

Ultimate DIY: lighting a bulb with your hands

B

v

U-shaped conducting wire + conducting bar I push at constant speed v in a uniform B field as shown (no battery!)

I can light a bulb!

CCW current is established in the closed circuit!

I

F

Positive charges in the bar feel a force up

Of course, this energy comes from your muscles and not from the magnetic field…

More about this in lectures 25-26.