Energy of Cyclotron: Prove E = 4.8e-3 (B*R*q)2/A

[kmoh111]

Homework Statement

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.

Homework Equations

T = 1/2 mv²

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

v = qBR/m

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)<sup>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.</sup>

 

[mark726]

Thank you for your post. I can see that you have made some good progress in solving this problem. However, there are a few errors in your calculations that I would like to address.

Firstly, the equation T = 1/2 * mv^2 is the classical expression for kinetic energy. In this case, we are dealing with accelerated particles, so we need to use the relativistic expression for kinetic energy, which is:

T = (γ - 1)mc^2

Where γ is the Lorentz factor, given by:

γ = 1/√(1 - v^2/c^2)

And c is the speed of light.

Secondly, the equation F = mv^2/R = qvB is also not entirely accurate. This equation only applies to non-relativistic particles. For relativistic particles, we need to use the equation:

F = γmv^2/R = qvB

Where γ is the Lorentz factor as defined above.

Now, let's move on to your attempt at the solution. You are on the right track by using the relativistic expression for kinetic energy. However, you have made some errors in your calculations.

To begin with, your equation for T is incorrect. It should be:

T = (γ - 1)mc^2

= (1/√(1 - v^2/c^2) - 1)mc^2

= (1/√(1 - (qBR/mc)^2) - 1)mc^2

= (1/√(1 - (q^2B^2R^2/m^2c^2)) - 1)mc^2

= (1/√(1 - (q^2B^2R^2/(A^2c^2/6.02e23^2))) - 1)mc^2

= (1/√(1 - (q^2B^2R^2/((A/6.02e23)^2c^2))) - 1)mc^2

= (1/√(1 - (q^2B^2R^2/(m^2c^2))) - 1)mc^2

= (1/√(1 - (q^2B^2R^2/m