Question

In proton accelerators used to treat cancer patients, protons are accelerated to \(0.61 c .\) Determine the energy of each proton, expressing your answer in mega-electron-volts (MeV).

Short answer

Answer: The energy of each proton is approximately 195.9 MeV.

Step-by-step solution

Determine Lorentz Factor (Gamma), \(\gamma\)

First, we need to determine the Lorentz factor (gamma), which is given by the equation: \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\), where \(v\) is the velocity of the proton and \(c\) is the speed of light. We are given that \(v=0.61c\), so let's plug in the values and calculate \(\gamma\):$$\gamma = \frac{1}{\sqrt{1-\frac{(0.61c)^2}{c^2}}}$$.

Simplify and Calculate Gamma Value

To simplify, divide \(0.61^2c^2\) by \(c^2\) which gives:$$\gamma = \frac{1}{\sqrt{1-0.3721}}$$. Now, subtract 0.3721 from 1 and then take the square root: $$\gamma = \frac{1}{\sqrt{0.6279}}$$After taking the square root, we get:$$\gamma \approx 1.25$$.

Calculate the Energy of a Proton using the Relativistic Energy Equation

Now that we have the value for gamma, we can use the relativistic energy equation to find the energy of each proton:$$E = \gamma m_pc^2$$, where \(m_p\) is the mass of a proton, which is approximately \(1.67\times10^{-27} kg\), and \(c\) is the speed of light in a vacuum, which is approximately \(3.00\times10^8 m/s\). Now, we plug in our values:$$E = (1.25)(1.67\times10^{-27}kg)(3.00\times10^8 m/s)^2$$.

Calculate the Energy in Joules

To calculate the energy of a proton in Joules, multiply the numbers and square c:$$E \approx (1.25)(1.67\times10^{-27}kg)(9\times10^{16} m^2/s^2)$$Using a calculator to multiply these values, we get: $$E \approx 3.14\times10^{-10}J$$

Convert the Energy from Joules to Mega-electron-volts (MeV)

To convert the energy from Joules to Mega-electron-volts, we can use the conversion factor:$$1eV = 1.6\times10^{-19}J$$. Then, we can calculate the energy in electron-volts and then divide by \(10^6\) to get MeV:$$E \approx \frac{3.14\times10^{-10}J}{1.6\times10^{-19}J/eV}\times \frac{1 MeV}{10^6 eV}$$After performing the calculation, we get: $$E\approx 195.9 MeV$$. So, the energy of each proton is approximately \(195.9 MeV\).

Key concepts

Lorentz Factor

The Lorentz factor is a key element in the realm of special relativity, representing how much time, length, and relativistic mass change for an object while it is moving. The factor is expressed by the symbol \(\gamma\), and it is calculated using the formula \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\), where \(v\) represents the object's velocity, and \(c\) stands for the speed of light in a vacuum.

Understanding the Lorentz factor is crucial because it tells us how much energetically 'heavier' and time-dilated an object becomes as it moves close to the speed of light. For a proton in a proton accelerator, as its velocity \(v\) approaches the speed of light, this factor grows, meaning that the proton's inertial mass increases – a phenomenon that must be accounted for when calculating the proton's total energy as part of cancer treatment therapies.

Proton Accelerators in Cancer Treatment

Proton accelerators are a frontier technology in cancer treatments known as proton therapy. Unlike traditional radiation therapy that uses X-rays, proton therapy targets tumors with high-energy proton beams. These beams can be controlled with millimeter precision, ensuring that the maximum energy is delivered directly to the tumor while sparing healthy surrounding tissue.

Proton accelerators must speed protons up to a significant fraction of the speed of light, which is where relativistic physics come into play. By knowing the relativistic energy of the protons, oncologists and medical physicists can accurately determine the dose delivered to the cancer cells, optimizing treatment effectiveness and minimizing side effects.

Relativistic Energy Equation

The relativistic energy equation is crucial for calculating the total energy of particles moving at speeds close to the speed of light. This equation is given by \(E = \gamma m c^2\), which shows that the energy (\(E\)) of a particle is the product of its rest mass (\(m\)), the speed of light squared (\(c^2\)), and the Lorentz factor (\(\gamma\)).

For protons in a cancer-treating accelerator, this equation allows one to compute the energy required to reach and treat a tumor adequately. The equation underscores the fact that as the velocity of an object increases, so does its energy – rapidly so when approaching the speed of light due to the increasing Lorentz factor, which amplifies the energy significantly.

Energy Conversion Joules to MeV

In physics, especially when dealing with particles like protons, energy is often measured in electron-volts (eV), with one electron-volt being the amount of kinetic energy gained or lost by a single electron accelerating through an electric potential difference of one volt. In proton therapy, the energy of protons is typically expressed in mega-electron-volts (MeV), where one MeV equals one million electron-volts.

The conversion from Joules to electron-volts is carried out using the relation \(1 eV = 1.6\times10^{-19} J\). Therefore, to convert the energy calculated in Joules to MeV, you'll divide the energy in Joules by \(1.6\times10^{-19}\) and then divide by \(10^6\) to convert electron-volts to mega-electron-volts. This conversion is fundamental when clinicians are determining the correct dosage in proton therapy, ensuring patients receive the most accurate treatment dose.