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A singly charged ion means that the ion has acquired or lost a single fundamental charge, which is equal to the charge of an electron. In the case of a singly charged ion, like our Li isotope ion, it carries an elementary charge denoted as \( e \), which is approximately \( 1.602 \times 10^{-19} \; \text{C} \).
This charge is crucial, as it determines how the ion interacts with electric fields and magnetic fields.
In this example, the Li isotope has a charge resulting from either losing an electron (positively charged) or gaining one (negatively charged). Since the exercise involves ions, we assume the ion is positive, making it "+1" charged. Understanding the charge of the ion helps explain its behavior once it encounters a magnetic field, as we'll delve into in subsequent sections.

These two forms of energy are fundamental in understanding how the ion moves before entering the magnetic field.
When the ion is accelerated through a potential difference of 220 V, it gains kinetic energy, which is the energy due to its motion.
This gain in kinetic energy originates from the decrease in electrical potential energy thanks to the voltage.
The kinetic energy \( KE \) can be expressed as \( \frac{1}{2}mv^2 \), where \( m \) is the mass of the ion and \( v \) is its velocity.
Meanwhile, the potential energy \( U \) associated with the 220 V is given by the equation \( qV \), with \( q \) being the charge of the ion and \( V \) the potential difference.

By equating kinetic and potential energies, we can find the ion's velocity as:

• \( v = \sqrt{\frac{2qV}{m}} \)

This relationship is important because it bridges between static and dynamic states of the ion, allowing us to calculate how fast the ion moves as it enters the magnetic field.

When the ion enters the magnetic field, it is subjected to a magnetic force that acts perpendicularly to its velocity; this leads to a centripetal force, making it travel in a circular path.
The magnetic force \( F_{B} \) experienced by a moving charge in a magnetic field is described with \( F_{B} = qvB \), where \( B \) is the magnetic field strength.
For circular motion, the centripetal force \( F_{c} \) is given by \( F_{c} = \frac{mv^2}{r} \), where \( r \) represents the radius of the path.

Equating these forces allows us to solve for radius \( r \), which is key to understanding how the path curve is determined in the field:

• \( r = \frac{mv}{qB} \)

By substituting the known values from the exercise, the precise path radius is computed, showing how these forces dynamically maintain the ion's circular motion within the magnetic field.

The path followed by our \(^7\)Li isotope ion is greatly influenced by the balance between the kinetic forces and the magnetic field.
This balance results in a circular trajectory, whose radius depends on the mass of the ion, its velocity, its charge, and the magnetic field it encounters.
For the Li isotope with a given charge and mass, the step-by-step calculations show that its radius in a magnetic field of 0.874 T is approximately 0.0241 meters.
This path calculation is essential to various applications, such as mass spectrometry in identifying isotypes. The unique paths based on different masses allow accurate identification of isotopes and ions, thus showcasing the fundamental principles of physics and their practical applications.