Problem 20 - University Physics with Modern Physics

Question

Cyclotrons are widely used in nuclear medicine for producing short-lived radioactive isotopes. These cyclotrons typically accelerate H\(^-\) (the \(hydride\) ion, which has one proton and two electrons) to an energy of 5 MeV to 20 MeV. This ion has a mass very close to that of a proton because the electron mass is negligible-about \\(\frac{1}{2000}\\) of the proton's mass. A typical magnetic field in such cyclotrons is 1.9 T. (a) What is the speed of a 5.0-MeV H\(^-\)? (b) If the H\(^-\) has energy 5.0 MeV and \(B =\) 1.9 T, what is the radius of this ion's circular orbit?

Step-by-Step Solution

Answer

Convert MeV to Joules to find speed, then calculate orbit radius using magnetic field formula.

Step 1: Understand the Conversion from MeV to Joules

Given the energy of the H\(^-\) ion as 5.0 MeV, we first need to convert this energy into joules. Recall that 1 MeV = \(1.602 \times 10^{-13}\) J. Thus, 5.0 MeV = \(5.0 \times 1.602 \times 10^{-13}\) J = \(8.01 \times 10^{-13}\) J.

Step 2: Calculate the Speed of the Ion

Use the energy formula for kinetic energy, \(E = \frac{1}{2}mv^2\), where \(E\) is the kinetic energy, \(m\) is the mass of the ion, and \(v\) is the speed.Rearranging the formula gives: \(v = \sqrt{\frac{2E}{m}}\).The mass of an H\(^-\) ion is approximately the mass of a proton, \(m \approx 1.67 \times 10^{-27}\) kg.Substitute \(E\) and \(m\) into the formula: \(v = \sqrt{\frac{2 \times 8.01 \times 10^{-13}}{1.67 \times 10^{-27}}}\) m/s.Calculate \(v\) to find the speed of the ion.

Step 3: Calculate the Radius of the Orbit

In a magnetic field, the radius \(r\) of the cyclotron orbit is given by the formula \(r = \frac{mv}{qB}\), where \(m\) is the ion's mass, \(v\) is the velocity, \(q\) is the charge of the ion, and \(B\) is the magnetic field strength. The charge \(q\) of an H\(^-\) ion is the same as that of a proton, \(q = 1.602 \times 10^{-19}\) C. Using the calculated speed from Step 2, substitute \(m\), \(v\), \(q\), and \(B = 1.9\) T:\(r = \frac{1.67 \times 10^{-27} \times v}{1.602 \times 10^{-19} \times 1.9}\).Calculate \(r\) to find the radius of the orbit.

Key Concepts

Nuclear Medicine

Nuclear medicine is a specialized area of medicine that uses small amounts of radioactive materials, known as radiotracers, to diagnose and treat diseases. These radiotracers are introduced into the body, where they emit gamma rays that can be detected by special cameras to create detailed images of the organs and tissues. This imaging technique provides valuable information that might not be achievable through other methods.

One vital application of nuclear medicine is in the detection of cancerous cells. Since cancer cells absorb sugars at a high rate, radioactive tracers bonded to glucose molecules can help reveal the presence and spread of cancer by highlighting areas with excessive uptake.

Another important aspect is treating conditions by concentrating radioactive isotopes in specific areas of the body to destroy diseased cells, such as hyperthyroidism therapy using radioactive iodine. This approach offers high precision, minimizing damage to surrounding healthy tissue.

Cyclotrons play a critical role in nuclear medicine by producing short-lived radioactive isotopes. These isotopes decay quickly, necessitating their production on-site or near the location of their intended medical application.

Magnetic Field

The magnetic field in a cyclotron serves the purpose of bending particles into circular orbits, allowing them to be accelerated over and over as they gain energy. Cyclotrons employ a uniform magnetic field that makes charged particles like ions travel in a spiral path.

For the cyclotron in question, the magnetic field is given as 1.9 Tesla (T), which is a measure of the strength of the magnetic field in which the ions move. The direction of the magnetic field is perpendicular to the plane of the particle's motion, causing the charged particles to exhibit circular motion.

The radius of this spiral path is dependent on several factors such as the mass of the ion, the velocity, the charge it carries, and the strength of the magnetic field. In the formula \( r = \frac{mv}{qB} \), the radius \( r \) is determined by these variables, indicating how tightly the ion path curves within the magnetic field.

Overall, the magnetic field in a cyclotron is crucial for guiding the acceleration progression and dictating the precise path of the ions.

Radioactive Isotopes

Radioactive isotopes are unstable atoms that release excess energy in the form of radiation until they reach a stable form. These isotopes are instrumental in various fields, especially in nuclear medicine, where they allow for precise imaging and targeted therapies.

The production of these radioactive isotopes often involves cyclotrons which bombard a target with particles, such as protons or deuterons. This bombardment produces isotopes through nuclear reactions, which can either add to or split the nucleus of the atoms in the target material.

The short-lived nature of many isotopes, such as fluorine-18 used in PET scans, means they must be produced close to the location where they are used.

Radioactive isotopes in medical applications are chosen for their ability to target specific physiological pathways, allowing for clear imaging of various organs or treatment of diseases at the cellular level. For example, iodine-131 accumulates in the thyroid gland, which facilitates imaging or targeted therapy for thyroid disorders.

In summary, the strategic use of radioactive isotopes in medicine enhances the ability to diagnose and treat conditions with high precision.