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 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?

Verified step by step guidance

1. First, convert the energy from MeV to joules. Recall that 1 MeV is equal to $1.60218 \times 10^{-13}$ joules.
2. Use the relativistic energy formula $E = \gamma mc^2$, where $E$ is the total energy, $m$ is the rest mass of the proton (since the mass of the electron is negligible), $c$ is the speed of light, and $\gamma$ is the Lorentz factor $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ to solve for $v$, the speed of the H- ion.
3. For part (b), use the formula for the radius $r$ of the circular path of a charged particle in a magnetic field, $r = \frac{mv}{qB}$, where $m$ is the mass of the ion, $v$ is the velocity (calculated in step 2), $q$ is the charge of the ion, and $B$ is the magnetic field strength.
4. Substitute the values: mass of the proton (approximately $1.67 \times 10^{-27}$ kg), the charge of H- (which is the elementary charge $e$, approximately $1.6 \times 10^{-19}$ C), and the magnetic field strength (1.9 T) into the formula.
5. Calculate the radius $r$ using the values and formula from step 3 without solving for the numerical answer.

Key Concepts

Kinetic Energy and Relativistic Effects

Kinetic energy (KE) is the energy an object possesses due to its motion, calculated using the formula KE = (1/2)mv² for non-relativistic speeds. However, at high speeds, close to the speed of light, relativistic effects become significant, and the kinetic energy must be calculated using the relativistic formula KE = γmc² - mc², where γ (gamma) is the Lorentz factor. Understanding how to apply these concepts is crucial for determining the speed of the H- ion.

Magnetic Force and Circular Motion

When a charged particle moves in a magnetic field, it experiences a magnetic force that acts perpendicular to its velocity, causing it to move in a circular path. The radius of this circular motion can be determined using the formula r = mv/(qB), where m is the mass, v is the velocity, q is the charge, and B is the magnetic field strength. This relationship is essential for calculating the radius of the H- ion's orbit in the cyclotron.

Mass-Energy Equivalence

Mass-energy equivalence, expressed by Einstein's equation E=mc², indicates that mass can be converted into energy and vice versa. In the context of the H- ion, understanding how its mass relates to its energy is important, especially when considering the ion's kinetic energy at MeV levels. This concept helps in analyzing the behavior of particles in high-energy physics, such as those in cyclotrons.