Mass Spectrograph A mass spectrograph is used to measure the masses of ions, or to separate ions of different masses (see Section 27.5). In one design for such an instrument, ions with mass m and charge q are accelerated through a potential difference V. They then enter a uniform magnetic field that is perpendicular to their velocity, and they are deflected in a semicircular path of radius R. A detector measures where the ions complete the semicircle and from this it is easy to calculate R.
(a) Derive the equation for calculating the mass of the ion from measurements of B, V, R, and q.
(b) What potential difference V is needed so that singly ionized 12c atoms will have R = 50.0 cm in a 0.150-T magnetic field?
(c) Suppose the beam consists of a mixture of 12c and 14C ions. If V and B have the same values as in part (b), calculate the separation of these two isotopes at the detector. Do you think that this beam separation is sufficient for the two ions to be distinguished? (Make the assumption described in Problem 27.65 for the masses of the ions.)