# Energy of a cyclotron

1. Sep 6, 2010

### kmoh111

1. The problem statement, all variables and given/known data
Prove that the energy of a cyclotron can be expressed as:

E (MeV) = 4.8e-3 (B * R * q)2/A

Where B is the magnetic field in Tesla
R is the maximum radius of the cyclotron in cm.
q is the charge of the particle accelerated.
A - is the mass number.

We can ignore relativistic effects.

2. Relevant equations

T = 1/2 mv2

F = mv2/R = qvB. Solve for v and plug in to kinetic energy formula T

v = qBR/m

3. The attempt at a solution

We can plug in v into T to get kinetic energy expressed in q, B, R, and m:

T = 1/2 * (qBR)2/m

Now, need to express m in terms of mass number.

We know that N (the number of particles) = 6.02e23 (particles/mole)/A(g/mole) * m

Let N = 1 since we're only interested in what the energy of a particle in the cyclotron.
Solving for m: m = A / 6.02e23.

Plugging in m in equation for T:

T = 1/2 * (qBR)2 * 6.02e23/A

Now we need to add conversion factors for getting A from g/mole to kg/mole. Also need to convert from Joules to MeV. Finally need to get R from meters to cm. This gives me:

T = 1/2 * (qBR)2 * 6.02e23/A * 1000g/kg * (100 cm/1m)2 * 1.6e-13 MeV/J

This gives me 4.816e17 MeV - which is not correct.

In order to get R in cm, should I convert to cgs units first? If so, then I need to convert the magnetic field from Guass to Tesla.

Thanks for you help.