In a mass spectrometer used for measuring the masses of...

In a mass spectrometer used for measuring the masses of ions, the ions are initially accelerated by an electric potential $V$ and then made to describe semicircular paths of radius $R$ using a magnetic field B. If $V$ and $B$ are kept constant, the ratio $\left(\frac{\text { charge on the ion }}{\text { mass of the ion }}\right)$ will be proportional to (a) $1 / R^{2}$ (b) $R^{2}$ (c) $R$ (d) $1 / R$ (e) $R^{-3}$

In a mass spectrometer used for measuring the masses of ions, the ions are initially accelerated by an electric potential $V$ and then made to describe semicircular paths of radius $R$ using a magnetic field
B. If $V$ and $B$ are kept constant, the ratio $\left(\frac{\text { charge on the ion }}{\text { mass of the ion }}\right)$ will be proportional to
(a) $1 / R^{2}$
(b) $R^{2}$
(c) $R$
(d) $1 / R$
(e) $R^{-3}$

Key Concepts

Centripetal Force in Circular Motion

In a uniform magnetic field, the Lorentz force on a charged particle serves as the centripetal force necessary to maintain its circular trajectory. The balance between this force and the inertial tendency to move in a straight line yields a relationship between the particle's mass-to-charge ratio, its velocity, the magnetic field strength, and the curvature radius.

Mass Spectrometry and Charge-to-Mass Ratio Determination

Mass spectrometry uses the principles of electric acceleration and magnetic deflection to separate ions based on their charge-to-mass ratio. By measuring the radius of the ion's circular path under known electric and magnetic field conditions, one can deduce the particle's charge-to-mass ratio, which is essential for identifying and characterizing different ions.

Kinetic Energy Conversion

Charged particles accelerated through an electric potential gain kinetic energy equal to the product of their charge and the potential difference. This foundational principle is used in various instruments to convert electric potential energy into motion, allowing analysis of particle properties.

Motion of Charged Particles in Magnetic Fields

When a charged particle enters a magnetic field, it experiences a Lorentz force that acts perpendicular to its velocity. This force causes the particle to follow a curved, often circular, path, a phenomenon that is critical for differentiating particles based on their intrinsic properties.

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