Bainbridge's mass spectrometer, shown in Fig. 28-54,...

Bainbridge's mass spectrometer, shown in Fig. $28-54$, separates ions having the same velocity. The ions, after entering through slits, $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$, pass through a velocity selector composed of an electric field produced by the charged plates $\mathrm{P}$ and $\mathrm{P}^{\prime}$, and $\mathrm{a}$ magnetic field $\vec{B}$ perpendicular to the electric field and the ion path. The ions that then pass undeviated through the crossed $\vec{E}$ and $\vec{B}$ fields enter into a region where a second magnetic field $\vec{B}^{\prime}$ exists, where they are made to follow circular paths. A photographic plate (or a modern detector) registers their arrival. Show that, for the ions, $q / m=E / r B B^{\prime}$, where $r$ is the radius of the circular orbit.

Bainbridge's mass spectrometer, shown in Fig. $28-54$, separates ions having the same velocity. The ions, after entering through slits, $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$, pass through a velocity selector composed of an electric field produced by the charged plates $\mathrm{P}$ and $\mathrm{P}^{\prime}$, and $\mathrm{a}$ magnetic field $\vec{B}$ perpendicular to the electric field and the ion path. The ions that then pass undeviated through the crossed $\vec{E}$ and $\vec{B}$ fields enter into a region where a second
magnetic field $\vec{B}^{\prime}$ exists, where they are made to follow circular paths. A photographic plate (or a modern detector) registers their arrival. Show that, for the ions, $q / m=E / r B B^{\prime}$, where $r$ is the radius of the circular orbit.

Key Concepts

Velocity Selector

A velocity selector uses perpendicular electric and magnetic fields to filter particles by velocity. In this device, only particles whose electric force qE is exactly balanced by the magnetic force qvB will pass through undeviated. This condition leads to the relation v = E/B, ensuring that only particles with that specific velocity continue to the next stage of the mass spectrometer.

Lorentz Force

The Lorentz force is the net force experienced by a charged particle moving in electric and magnetic fields. It is given by the sum of the electric force (qE) and the magnetic force (qvB). In the context of a mass spectrometer, the appropriate selection (by crossing fields) and deflection (by applying a perpendicular magnetic field) are both determined by the Lorentz force acting on the charged particles.

Circular Motion in a Magnetic Field

When a charged particle enters a uniform magnetic field, it experiences a magnetic force that acts perpendicular to its velocity, causing it to move in a circular path. The balance between the magnetic force (qvB') and the centripetal force required for circular motion (mv^2/r) yields the relation r = mv/(qB'). This relation is central to determining the mass-to-charge ratio in a mass spectrometer once the velocity has been established.

Mass Spectrometry

Mass spectrometry is an analytical technique used to separate ions based on their mass-to-charge (m/q) ratios. In apparatus like Bainbridge's mass spectrometer, ions are first filtered by velocity and then bent into circular trajectories by a magnetic field. The radius of the path, together with the known fields, allows the determination of the mass-to-charge ratio, providing insights into the composition of the ion sample.

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