Reference no: EM131304370

Project

A cyclolron is an electrically - powered device that is used to accelerate particles, specifically using both Electrostatic fields and Magneto static fields to accelerate charged particles outwards from a center point along a spiral path. Because a. cyclotron moves particles along a spiral, it achieves a much longer acceleration path than a straight line accelerator.

A cyclotron body consists of two hollow "D"-shaped sheet metal electrodes called "dees" inside a vacuum chamber (see Figure 11). The two dees are flat with a narrow gap between them, creating a cylindrical space within them for the particles to move. The dees are located between the poles of a large electromagnet which applies a static magnetic field perpendicular to the electrode plane. A stream of charged particles is fed into the center of the chamber and a high frequency alternating voltage is applied across the electrodes. This voltage alternately attracts and repels the charged particles causing them to accelerate.

Once the particles enter the dees the magnetic field causes the particles path to bend in a circle due to the Lorentz force perpendicular to their direction of motion.

Now let's consider the following specific cyclotron device (see Figure 2). In tbn4 device, both (lees (labeled as DI and /)2) are semi-circles with radius R., and the narrow gap between t hen) has width d (dc /?). The alternating voltage applied across the two bees is :1 periodic square-wave signal with amplitude U, which remains at U Volts for some time. then switches to -U Volts for the same amount the time. The static magnetic field created by the electromagnet has constant magnitude .B, and is pointed in the inward direction at all points. An electrically charged particle with positive charge Q awl mass in enters Di at the renter, with an initial speed Th. Please ignore the gravity awl the magnetic impact within the narrow gap.

1. In order to accelerate the particle, we need the alternating voltage U to change polarity exactly when the particle just exits one dee and is about to cross the narrow gap and enters the other dee. Please derive the expression for the required period of the alternating voltage U, by figuring out the time spent by the particle between entering and exiting the dee. (You can ignore the travel time required within the gap).

2. If the period of the alternating voltage U is set to the value that you just derived, please describe mathematically the trajectory of movement of the particle. Also, MATLAB to plot the trajectory of the particle for the following parameters:

Q = 1.6 x 10^-19C, 7n = 3.95 x 10^-25kg, B = 3.8T, U = 10517 R = lin. The. Particle starts at the center, and enters DI with an initial velocity v = 1057n/s.