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But, there is a component of velocity in x-direction. The charged particle still completes a revolution, but not in the circular plane because charged particle also moves in the direction perpendicular to the circular plane. The net result is that the path of revolution is a stretched out series of circles in the form of a helix.

Helical motion

Motion of charged particle oblique to magnetic field

The expression for radius is similar as that for the circular motion under magnetic field (earlier case). The only change is that v is exchanged by v .

R = m v q B = m v sin θ q B

The expressions for time period, frequency and angular velocity etc do not change as these parameters are independent of velocity.

The distance between two consecutive points in x-direction determines the pitch of the helical path. This distance in x-direction is traveled by the particle with the parallel component of velocity in the time in which particle completes a revolution. If T be the time period of revolution, then pitch, p, of the helical path is :

p = v | | T = v T cos θ = 2 π m v cos θ q B

Problem : An electron with a kinetic energy of 10 eV moves into a region of uniform magnetic field of 5 X 10 - 4 T. The initial angle between velocity and magnetic field vectors is 60 degree. Determine the pitch of resulting helical motion.

Solution : The expression of pitch of helical path is :

p = 2 π m v cos θ q B

We notice here that speed is not directly given. However, kinetic energy is given in electron volt unit. By definition, an electron volt is equal to kinetic energy gained by an electron while passing through a potential difference of 1 V. We get kinetic energy in Joule by multiplying electron-volt value by 1.6 X 10 - 19 .

K = m v 2 2 = 10 e V = 10 X 1.6 X 10 - 19 J = 16 X - 19 J v = 2 K m = 2 X 16 X 10 - 19 9.1 X 10 - 31 = 3.52 X 10 12 = 1.88 X 10 6 m / s

Putting values in the expression of pitch :

p = 2 π X 9.1 X 10 - 31 X 1.88 X 10 6 X 0.5 1.6 X 10 - 19 X 5 X 10 - 4 p = 6.71 X 10 - 2 m = 6.71 c m

Magnetic bottle

In plasma research, one of the main tasks is to contain plasma (ions or charged elementary particles). Plasma particles can not be restrained in any material confinement because of extraordinarily high temperature associated with them. A magnetic bottle is an arrangement of two magnetic sources (solenoids or any other magnetic source) which produce magnetic fields. The arrangement is such that direction of magnetic field is from one solenoid to another. The magnetic field between two solenoids is non-uniform. It is stronger near the solenoid and weaker in the middle. See that lines of force are denser near the solenoids and rarer in the middle.

A charged particle is in the helical motion in this magnetic region. As it moves in stronger magnetic region near the solenoid, the radius of helical path is smaller. On the other hand, the radius of helical path is greater in the middle as magnetic field is weaker there.

R = m v q B

Magnetic bottle

The charged particle is trapped in magnetic field

As the particle reaches towards the solenoid i.e. end of the arrangement, it is rebounded because there is a component of magnetic force pointing towards the central part of the arrangement. See figure that how force components point toward middle. This component decelerates the particle till it stops and starts moving in opposite direction. The stronger magnetic region near the solenoid, therefore, functions as reflector of charged particles.

In this manner, plasma particles are confined within a region due to suitably designed magnetic field. The whole arrangement works like a bottle for the charged particles and hence is called magnetic bottle.

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Source:  OpenStax, Electricity and magnetism. OpenStax CNX. Oct 20, 2009 Download for free at http://cnx.org/content/col10909/1.13
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