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**Motion of a Charged Particle in a Magnetic Field**

AP Physics C Montwood High School R. Casao

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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 q 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.

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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.

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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.

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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:

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**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 w of the rotating charged particle is:

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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.

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The angular frequency w 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 q 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.

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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.

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**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.

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**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.

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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.

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**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.

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