Charged Particles in Electric Charged Particles in Electric and and

Magnetic FieldsMagnetic Fields

• Motion of charged particles• Lorentz Force• Examples: cyclotron, mass spectrometer

Recall:Recall:

In general:

“Lorentz Force”

EqF

EqBvqF

in a magnetic fieldBvqF

in an electric field

Magnetic Fields: Magnetic Fields: FF = = qqvvBBq

x x x

x x

x x x

vv

FF

+The force is perpendicular to the velocity (and path), so:

• no work is done by B• kinetic energy is constant• speed is constant

Only the direction of the motion changes due to the magnetic force.

BB

1) Uniform1) Uniform BB, , vv perpendicular to perpendicular to BB

+q

v

x x x

x x

x x x

Motion is a circle.

F Magnetic force (v ) is

Fm = qvB (constant)

But Fc = mv2/r , so the radius of

circle is :mv

rqB

r

Example:Example:

A proton is moving in a circular orbit of radius 14cm in a uniform magnetic field of 0.35T directed perpendicular tothe velocity of the proton. Find the orbital speed of the proton.

2) Uniform B, v not 2) Uniform B, v not perpendicular to perpendicular to BB

-path is a circular helix along a field line-the component v (parallel to B) is constant

On a large scale, the particles follow the field lines.

+

x

z

y+v

FrequenFrequencycy

Angular frequency [rad/s]

= "cyclotron frequency f "2

is independent of speed and radius. = 1/T (period)

2 2 2 2

21 12 2 2

sin

qBr q B rKE mv m

m mqBr

ce vm

The kinetic energy of a particle at radius r is

v qBr qBr mr m

x x x

x x

x x x

x x

+

-

The CyclotronThe Cyclotron : accelerates particles to very : accelerates particles to very high high velocities velocities

The electric field across the gap is reversed each time the proton arrives, so that its speed in the gap continually increases. Because the time for each half-circle is the same for any proton speed, the voltage supply can just be set to a constant frequency.

+

“dees”

AC Voltage Supply

B

Quiz

A cyclotron can accelerate protons to a maximum kinetic energy of 1 MeV. You want to design a new, improved model with a higher maximum proton energy. Which of the following changes would help?

A) Double the magnetic fieldB) Double the diameter of the deesC) Double the proton currentD) Double the voltage of the voltage supply

Example:Example:

A cyclotron needs to accelerate electrons to a speed of 0.95c for experiments. Given that it can provide a magnetic field of 0.5T:

a) what is the cyclotron frequency?b) what is the radius of the cyclotron?

The Velocity SelectorThe Velocity Selector: allows to select particles : allows to select particles that that move at the same move at the same velocityvelocity

x x x

x x

x x x

x x

Source Detector+ v

B

E

all , , vBE

Find: Conditions so that the path is straight.

Free-body diagram:

qvBFM

+

qEFE

vB

E

xNeed for straight path0F

qEqvB

BE

v for straight path

Question:Question:

qvBFM

+

qEFE

vB

E

x

How are the charges deflected that are either travelingtoo slow or too fast to travel in a straight line?

Mass SpectrometerMass Spectrometer : : separates ions according separates ions according to to their mass:charge ratio their mass:charge ratio

IonSource

x x x Uniform B

x x x

x x x

x x x

radius depends onmass

Assume 12C and 14C have same charge. Find the ratio of the diameters of the circular paths if the ions have the same speeds

Exercise: