Animated Solution for Physics - Magnetic Effects of Current: If the maximum value of accelerating potential provided by a radio frequency oscillator is $12\text{ kV}$. The number of revolution made by a proton in a cyclotron to achieve one sixth of the speed of light is ........... . [Given, $m_p=1.67\times 10^{-27}\text{ kg}$, $e=1.6\times 10^{-19}\text{ C}$, $c=3\times 10^8\text{ m/s}$]

Imagine you are a proton sitting at the center of a massive machine called a cyclotron. Your goal is to reach an incredible speed—one-sixth the speed of light!

To do this, the cyclotron uses two main components: a powerful magnetic field and a high-frequency alternating electric field. The magnetic field forces you to move in a circular path, while the electric field exists only in the narrow gap between two D-shaped hollow metal electrodes, aptly named "Dees".

Here is the beautiful part of the physics. The magnetic field does absolutely zero work on you. It only steers you. All your energy comes from the electric field in the gap.

Every time you cross this gap, the electric field gives you a precise kick of energy equal to your charge multiplied by the voltage, or $qV$. Because you are moving in a circle, you cross this gap exactly twice in one full revolution.

Therefore, the energy you gain per revolution is:

If you make $n$ full revolutions inside the cyclotron, the total energy you absorb from the radio frequency oscillator is simply $n$ times the energy gained in a single revolution.

This total energy absorbed doesn't just vanish; it manifests entirely as your final kinetic energy as you exit the machine. We know from classical mechanics that kinetic energy is $\frac{1}{2}m{v}^2$. Equating these two gives us our master equation:

Now, let's crunch the numbers. We are given the accelerating potential $V=12000\text{ V}$, the mass of a proton $m_p=1.67\times 10^{-27}\text{ kg}$, and its charge $q_p=1.6\times 10^{-19}\text{ C}$.

The target speed $v$ is one-sixth the speed of light $c$:

First, let's calculate the final kinetic energy required:

Next, let's calculate the energy gained in just one revolution:

To find the number of revolutions $n$, we divide the total kinetic energy by the energy gained per revolution:

Since the proton must complete full revolutions to be extracted at the correct phase at the edge of the Dee, we take the integer part. Thus, the proton makes 543 revolutions.

You might wonder, why stop at one-sixth the speed of light? Why not keep accelerating the proton until it reaches $0.99c$?

This is where Einstein's theory of relativity crashes the party. As the proton's speed approaches the speed of light, its relativistic mass begins to increase significantly.

Because the frequency of the cyclotron's oscillator is fixed and depends on a constant mass ($f=\frac{qB}{2\pi m}$), the heavier, faster proton starts to lag behind. It falls out of sync with the alternating voltage and stops accelerating. This is the fundamental speed limit of a standard cyclotron, and it is exactly why physicists had to invent the synchrotron!