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Name: Mike
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RF Cavities and Particle Accelerators Particle accelerators use RF cavities to ramp up the velocity of particles. I believe the particles ride on the negative or positive parts the RF waves depending on whether the charge is + or - The questions I have are:
1. Is there a relationship between the force the particle feels and the intensity of the rf wave the particle is "attached" to while riding the wave?
2. Is the velocity of the particle determined by the RF frequency or the speed at which the rf wave is travelling (i.e that of light)?
3. I am hoping there is some sort of simple mathematical relationship that describes the force the particle feels in relation to its velocity, Rf frequency, rf field intensity and where the particle is at any given moment of time?

Actually, your questions ARE pretty simple, but there might be a slight misconception lurking under there. So I will proceed with some caution.

1. A charged particle "perceives" an electromagnetic wave as traveling and oscillating electric and magnetic fields. The force on a charged particle from an electric field is simply the particle's charge multiplied by the field (which, being a vector, has a direction as well as intensity). The force from a magnetic field is a bit more complicated, as it depends on the relative velocities of the particle and field as well as on the charge and field intensity, but basically again it is proportional to the field strength and the charge.

2. Neither. The velocity of the particle depends on what the velocity was a short time ago and on the force. This is because force determines acceleration, not velocity. Acceleration is the RATE of CHANGE of velocity, which is quite different from velocity (though of course acceleration and velocity are obviously related). This is Newton's second law, which states that the acceleration of any body is directly proportional to the sum of all forces acting on the body and inversely proportional to its mass, a = F/m. (At the very high speeds attained by particles in accelerators, there is a substantial relativistic correction to this formula, but the idea that force determines acceleration, not velocity directly, still stands.)

3. Force formulas: For a particle of charge q in an electric field E, the force on the particle is qE. For a particle of charge q moving at velocity v relative to a magnetic field B, the force on the particle is qv x B, where "x" denotes the vector cross product. (In English, the force is perpendicular to both v and B, and has a magnitude qvB sin(theta), where theta is the angle between the "electric momentum" qv and the magnetic field B.

Richard E. Barrans Jr., Ph.D., M.Ed.
Department of Physics and Astronomy
University of Wyoming

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