Proton-beam therapy is preferable to X rays for cancer treatment because protons deliver most of their energy to the tumor, with less damage to healthy tissue. A cyclotron used to accelerate protons for cancer treatment repeatedly passes the protons through a 15-kV potential difference. (a) How many passes are needed to bring the protons' kinetic energy to \(1.2 \times 10^{-11} \mathrm{J} ?\) (b) What's the resulting proton energy in electronvolts?

One of the remarkable tools in modern medicine and physics is the cyclotron, a type of particle accelerator that is widely used, notably in proton-beam therapy. Its primary function is to accelerate protons to high speeds, which can then be used to target and destroy cancer cells with precision.

How does a cyclotron achieve this? The device uses a combination of electric and magnetic fields to accelerate charged particles, like protons, in a spiral outward path. Inside the cyclotron, protons are subjected to a constant magnetic field which directs them to move in a circular path, while an alternating electric field boosts their speed with every pass through the gap between the cyclotron's two hollow 'D'-shaped electrodes, known as the dees. The alternating voltage across the gap gives the protons a 'push', increasing their kinetic energy.

To understand the relevance of the 15-kV potential difference mentioned in the exercise, it is this voltage that is applied to the dees to accelerate the protons. Each time a proton crosses the gap, it gains energy equal to the charge of the proton multiplied by the potential difference it moves through. The challenge lies in synchronizing the electric field with the protons' circular motion to ensure that they receive the acceleration boost at the right time with each pass. This is crucial in optimizing the energy delivery to the desired location with minimal damage to surrounding tissues.

Understanding the kinetic energy of protons is fundamental when it comes to their application in therapies for cancer treatment. The exercise you're tackling revolves around a central question: how do you calculate the kinetic energy of a proton after several passes in a cyclotron?

To calculate this, we can use the formula derived from the elementary principles of physics, where the kinetic energy (K.E.) of a particle is the energy it possesses due to its motion. The K.E. of a proton after n passes through the cyclotron is given by the sum of the energies gained through each pass. In other words, we take the energy acquired per pass and multiply it by the number of passes.

In the exercise, you were given the total kinetic energy a proton must reach and the energy gained per pass expressed in kilovolts. First, we converted kilovolts to joules, since kinetic energy is typically measured in joules. Then, you simply divide the total desired energy by the amount of energy gained per pass to determine the number of passes required. This allows us to understand, step-by-step, how the proton gains energy and reaches the sufficient level for therapeutic purposes.

The concept of energy conversion to electronvolts (eV) is pivotal in the context of proton-beam therapy and particle physics. An electronvolt is the amount of kinetic energy gained by a single electron when it accelerates through an electric potential difference of one volt. It's a convenient unit of energy used in the realm of atomic and particle physics because it's much more suited to the energy scales involved with particles.

In the step by step solution, we started by converting the energy from kilovolts to joules to deal with familiar units and calculations. However, for practical purposes, particularly in proton-beam therapy, it's beneficial to express this energy in electronvolts. This is because the energy ranges dealt with are much closer to the scale of electronvolts rather than joules, which are relatively large and thus cumbersome to use in this context.

To convert joules to electronvolts, we use the conversion factor where 1 eV is equivalent to 1.6 x 10^-19 joules. By dividing the total energy in joules by this amount, we can determine the energy in electronvolts. This ensures compatibility with conventional scientific terms and aids physicists and medical professionals in communicating and performing accurate calculations for treatment.