The accompanied figure shows an arrangement for measuring...

The accompanied figure shows an arrangement for measuring mass of ions by an instrument called the mass spectrometer. An ion of mass $m$ and charge $+q$ is produced essentially at rest in source $S$, a chamber in which a gas discharge is taking place. The ion is accelerated by a potential difference $V_{\mathrm{acc}}$ and allowed to enter a region of constant magnetic field $\overrightarrow{\mathbf{B}}$ o. In the uniform magnetic field region, the ion moves in a semicircular path striking a photographic plate at a distance $x$ from the entry point. Derive a formula for mass $\mathrm{m}$ in terms of $B_{0}, \quad q, V_{\mathrm{acc}}$ and $x$.

Key Concepts

Geometric Interpretation of the Ion Trajectory

The physical setup, where the ion follows a semicircular path, allows for geometric analysis to connect the measurable displacement on a detector to the radius of the curvature. Knowing the relationship between the radius and the displacement x on the photographic plate facilitates the derivation of the ion mass in terms of the measured distance, magnetic field strength, charge, and accelerating voltage.

Motion of Charged Particles in Magnetic Fields

When a charged particle moves within a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field, known as the Lorentz force. This force causes the ion to follow a circular or semicircular path if the velocity is perpendicular to the field. The radius of this path is linked to the ion’s momentum and the strength of the magnetic field by the relationship p = m*v = qB*r.

Mass Spectrometry

Mass spectrometry is an analytical technique used to determine the mass-to-charge ratio of ions. In this context, the instrument accelerates ions and then employs magnetic fields to cause their deflection; by measuring the radius of curvature of the ionic path, one can deduce the mass of the ion. This method underpins the precise identification and quantification of different substances based on their ionized masses.

Ion Acceleration by Electric Potential

Ions in a mass spectrometer are accelerated by an applied electric potential difference, which converts their potential energy into kinetic energy. This acceleration is described by the energy conservation principle where the electric potential energy (qV) is transformed into kinetic energy ((1/2)mv²), allowing one to relate the ion’s velocity directly to the applied voltage.

Kinetic Energy and Velocity Relationship

The relationship between kinetic energy and velocity is central to determining the speed of an ion after acceleration. Using the expression (1/2)mv² = qV, one can solve for the velocity v of the ion. This relation is crucial as the velocity impacts the curvature of the ion's trajectory in a magnetic field.

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