Applications of the Motion of Charged Particles in a Magnetic

Field

AP Physics C

Montwood High School

R. Casao

Lorentz Force• In many devices that involve the motion of

charged particles in uniform magnetic fields, the charge under consideration will be moving with velocity v in the presence of both an electric field E and a magnetic field B.

• The charge will experience both an electric force q·E and a magnetic force q·(v × B).

• The total force, called the Lorentz force, on the charge is:

)Bx(vqEqF

Velocity Selector• Experiments involving the motion of charged

particles often require particles that move in a straight line with the same velocity.

• A combination of an electric field and a magnetic field can produce this stream of particles.

• A uniform electric field E is provided by a pair of charged parallel plates.

• A uniform magnetic field B is applied perpendicular to E such that Fmag is equal to and opposite to Felectric.

B

EvBvqEq

The two fields, called crossed fields can be manipulated to

produce velocities in other directions.

Mass Spectrometer• Separates atomic and molecular ions based on their

mass-to-charge ratio.• A beam of ions first passes through a velocity selector

and then enters a uniform magnetic field Bo, where the ions move in a semicircle of radius r before striking a photographic plate at P.

• Mass-to-charge ratio:

vBr

qm o

• If the magnetic field in the velocity selector is B, then:

• Charge-to-mass ratio for electrons:– Electrons are accelerated from the cathode to the

anode, passing through slits in the anodes, and allowed to drift into a region of perpendicular electric and magnetic fields.

– The crossed fields are first adjusted to produce an undeflected beam.

E

BBrqm o

• The magnetic field B is turned off and the electric field E produces a measureable beam deflection on the screen.

• From the size of the deflection and the values for E and B, the charge-to-mass ratio can be determined.

Cyclotron

• Cyclotron accelerates charged particles to very high velocities using both electric and magnetic fields.

• The high energy particles that emerge from the cyclotron are used to bombard other nuclei to produce nuclear reactions for researchers to study.

• Hospitals use cyclotrons to produce radioactive substances used in diagnosis and treatment.

• Motion of the charges occurs in two semicircular containers D1 and D2 (called dees).

• The dees are evacuated to prevent energy losses in collisions with the ions and air molecules.

• A high frequency alternating voltage is applied to the dees and an electromagnet provides a uniform magnetic field directed perpendicular to the dees.

• Positive ions released at P near the center of the electromagnet move in a semicircle and arrive back at the gap in a time T/2.

• The frequency of the applied voltage is adjusted so that the polarity of the dees is reversed in the same time it takes for the ions to complete one half of a revolution.

• If the phase of the applied voltage is adjusted so that D2 is a a lower potential than D1 by an amount V, the ion will accelerate across the gap to D2 and its kinetic energy will increase by an amount q·V.

• Ion continues to move in D2 in a semicircular path of larger radius due to the velocity increase.

• After a time T/2 it arrives at the gap.

• The voltage across the gap is reversed so that D1 is now negative and the ion is again accelerated across the gap.

• The ion is accelerated at each half revolution, gaining a kinetic energy equal to q·V each time.

• When the radius of the orbit is nearly that of the dees, the ion leave the system through an exit slit.

• The maximum kinetic energy of the ion can be obtained upon the exit of the ion from the cyclotron:

• Frequncy of oscillation of the cyclotron:

m2rBq

vm0.5Km

rBqv

2222

π2ω

f

• Period:

• Angular frequency (rad/s):

• Cyclotron frequency (Hz):

Bqmπ2

vBqvmπ2

vrπ2

T

mBq

ωcyclotron

mπ2Bq

f

• When the cyclotron principle is used to accelerated electrons, it has been historically called a betatron. The cyclotron principle as applied to electrons is illustrated below.