An ion with charge 1.6 ×10^-19 C moves at speed 1.0 ×10^6 m / s into and perpendicular to the 0.

step 1 So we have a charged ion with a speed moving into a magnetic field, and it creates a circular pa

An ion with charge 1.6 ×10^-19 C moves at speed 1.0 ×10^6 m...

An ion with charge $1.6 \times 10^{-19} \mathrm{C}$ moves at speed $1.0 \times 10^{6} \mathrm{m} / \mathrm{s}$ into and perpendicular to the $0.30-\mathrm{T} \mathrm{mag}-$ netic field of a mass spectrometer. After entering the field, the ion moves in a circular path of radius 0.31 $\mathrm{m} .$ What physical quantities describing the field and the ion can you determine using this information? Determine them.

An ion with charge $1.6 \times 10^{-19} \mathrm{C}$ moves at speed $1.0 \times 10^{6} \mathrm{m} / \mathrm{s}$ into and perpendicular to the $0.30-\mathrm{T} \mathrm{mag}-$
netic field of a mass spectrometer. After entering the field, the
ion moves in a circular path of radius 0.31 $\mathrm{m} .$ What physical
quantities describing the field and the ion can you determine
using this information? Determine them.

Key Concepts

Centripetal Motion

In the presence of a constant perpendicular force, an object will execute uniform circular motion. The magnetic (Lorentz) force in this context serves as the centripetal force required to keep the ion moving in a circle, and its relation is expressed by m*v²/r = q*v*B. This provides a way to relate the mass, velocity, magnetic field strength, and radius of the circular path.

Cyclotron Frequency

This concept refers to the frequency at which a charged particle orbits in a uniform magnetic field. It is determined by the ratio ? = qB/m, and the period T = 2?/?. Calculating the cyclotron frequency (or period) is important in understanding the dynamics of charged particles in magnetic fields and is a key parameter in the operation of devices such as mass spectrometers.

Lorentz Force

This concept involves the force exerted on a charged particle moving through a magnetic field. When the ion’s velocity is perpendicular to the magnetic field, the force is given by F = qvB, which acts at right angles to the velocity. This force is responsible for bending the path of the charged particle, making it undergo circular motion.

Mass-to-Charge Ratio Determination

By using the relationship between the magnetic force and the centripetal force, one can determine the mass-to-charge ratio (m/q) of the ion. Rearranging the equation r = m*v/(q*B) allows for calculating either the ion’s mass (if the charge is known) or the charge (if the mass is known), which is a fundamental principle in mass spectrometry.

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