Fig.1 Conceptual diagrm showing operation of a cyclotron.

The operation of a cyclotron is based on the fact that the period of the motion of a charged particle in a uniform magnetic field is independent of the velocity of the particle, as can be seen in the following derivation:

**F = ma**

**q vB**

Solve for r:

**r = m v/qB**

Now find the period, T:

**T = 1/f = 2π/ω = 2πr/ v = 2πmv/qBv **

The ** v**’s cancel:

**T = 2πm/qB**

The “Cyclotron Frequency” ** f**
immediately follows:

*f*** = qB/2πm**

Fig. 1 is a schematic drawing of a cyclotron. The particles move in two semicircular metal
containers called **ΔV**
that alternates in time with a period T, which is chosen to be equal to the
cyclotron period that was found in the above derivation. This potential difference creates an electric
field across the gap between the

The charged particles are initially injected into DEE 1 with
a small velocity from an ion source near the center of the ** ΔV**. Each time the particle arrives at the gap, it
is accelerated and gains kinetic energy q**
ΔV**. Thus, it moves in larger and larger semicircular orbits until it
eventually leaves the magnetic field. In
the typical cyclotron, each particle may make up to 50 to 100 revolutions
before reaching its final energy.

The kinetic energy of a particle leaving a cyclotron can be calculated by the following derivation:

1^{st} set the following equal to the **r _{max}** of the

**r _{max}**

*v*** = qBr/m**

Next solve for the Kinetic Energy:

**KE = ½ m v^{2} = ½ m (q^{2}B^{2}r^{2}/m^{2})**

Cancel the m’s and finally get:

**KE = q ^{2}B^{2}r^{2}/2m**

Now for an example with the

__12-INCH CYCLOTRON PARAMETERS:__

**r _{max}**

** B _{max} = 1.2 Tesla**

** q
= -1.6E-19 Coulombs**

** m
= 1.67E-27 kg**

1^{st} Determine the
Cyclotron Frequency:

*f*** = qB/2πm
= (1.6E-19 C)(1.2 T)/( 2π)(1.67E-27 kg) = 18.3
MHz**

2^{nd} Find the maximum Kinetic Energy of the protons:

**KE = (1.6E-19 C) ^{2}(1.2
T)^{2}(0.127 m)^{2}/2(1.67E-27 kg) **

**= 1.78E-13 Joules or**

**= 1.1 MeV **

_{ }

* - adapted from Tipler's Physics for Scientists and Engineers, Vol. 2. *

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