Motion of a Charged Particle in a Magnetic Field

AP Physics C

Montwood High School

R. Casao

• The magnetic force acting on a charged particle moving in a magnetic field is always perpendicular to the velocity of the particle.

• From this property, the work done by the magnetic force is 0 J since the displacement of the charge is always perpendicular to the magnetic force.

W = F·d·cos • Therefore, a static magnetic field changes

the direction of the velocity but does not change the speed or the kinetic energy of the charged particle.

• Consider a positively charged particle moving in a uniform external magnetic field with its initial velocity vector perpendicular to the magnetic field.

• The magnetic field B is into the page (as indicated by the x’s).

• The figure shows that the charged particle moves in a circle whose plane is perpendicular to the magnetic field.

• The circular path results because the magnetic force Fmag is at right angles to the velocity v and the magnetic field B and has a constant magnitude equal to q·v·B.

• The force deflects the particle and the directions of v and B change continuously.

• The force Fmag is a centripetal force, which changes only the direction of the velocity while the speed remains constant.

• The direction of the rotation is given using the right hand rule – right hand for positive charges and left hand for negative charges (Casao’s rule).

• Since Fmag= Fcentripetal:

• This reduces to:

r

vmBvq

2

r

vmBq

• Solving for the radius of curvature:

• The radius of curvature is proportional to the momentum of the particle and inversely proportional to the magnetic field B.

• The angular frequency of the rotating charged particle is:

Bq

vmr

m

Bq

r

vω

• The period T of the circular motion (time for one revolution) is equal to the circumference of the circle divided by the speed of the particle:

• The angular frequency and the period of the circular motion do not depend on the speed of the particle or the radius of the orbit.

Bq

mπ2ωπ2

vrπ2

vd

T

• The angular frequency is also called the cyclotron frequency since charged particles circulate at this frequency in a particle accelerator called a cyclotron.

• If a charged particle moves in a uniform magnetic field B with its velocity at some angle to B, its path is a helix.

• For the field B in the x-direction, there is no component of force in the x direction, therefore

ax = 0 m/s2 and the x component of v is constant.

• The magnetic force q·(v x B) causes the components of vx and vy to change in time and the resulting motion is a helix having its axis parallel to the magnetic field B.

• The projection of the path onto the yz plane (viewed along the x axis) is a circle.

• The distance between

successive rotations

in the helical path is

called the pitch, p.

• The pitch is parallel to the magnetic field B.

• The perpendicular velocity influences how much time it takes to complete the circular path.

• The parallel velocity determines the pitch.

Bq

mπ2vp

tvpt

pv

parallel

parallelparallel

• The motion of a charged particle in a nonuniform magnetic field is complex.

• If a magnetic field is strong at the ends and weak in the middle, the particles oscillate back and forth between the end points.

• Such a field can be produced by two current loops at the ends of the “bottle” to produce a strong magnetic field to pinch off the ends.

• A charged particle starting at one end will spiral along the field lines until it reaches the other end, where it reverses directions and spirals back. This configuration is known as a “magnetic bottle” because charged particles can be trapped in it.– This concept has been used to confine plasmas (hot

gases consisting of electrons and protons).– The magnetic bottle may pay a role in achieving a

controlled nuclear fusion process.– The problem is that if a large number of particles are

trapped in the magnetic bottle, collisions between the particles cause them to “leak” from the system.

• The Van Allen radiation belts consist of charged particles (e- & p+) surrounding the earth.

• The charged particles are trapped by the earth’s nonuniform magnetic field and spiral around the earth’s field lines from pole to pole.

• Most of the charged particles come from the sun.

• When the charged particles are in the atmosphere over the poles, they can collide with other atoms, causing them to emit visible light, the Aurora Borealis and Aurora Australis.