A mass spectrometer measures the mass of ions. An ion is accelerated to a high speed and injected into a region with a uniform magnetic field which is perpendicular to the ion's velocity. The ion will follow a curved trajectory and impact a photographic plate some distance away from the injection point. By measuring this distance, the ion's mass can be deduced.
[Figure: Diagram showing point O, magnetic field B, trajectory path, and point P on film, with ion having mass m, charge q, and velocity v]
Suppose singly charged (q=+e) ions of 39K19 and 41K19 are accelerated to an energy of 500 eV and injected into a magnetic field B=0.7 T. The velocity of the ions upon entering the magnetic field is perpendicular to the line OP.
(a) What speed does each of the ions have when they impact the photographic plate at a point P?
(b) What is the shape of the trajectory for the ions? Find the distance OP for each of the two ions.
(c) In reality, things are not quite so simple. Variations in the ions' velocities can cause problems with the resolution of the machine. If the initial energy of the ions cannot be held precisely at 500 eV, there will be a small variation in the distances OP. Suppose that the energy of the ions is 500±5 eV. Can the two ions be distinguished from one another when they hit the screen?
(d) Suppose that the velocity of the ions when they enter the