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 and charge are accelerated through a potential
difference . They then enter a uniform magnetic field that is perpendicular
to their velocity, and they are deflected in a semicircular path of radius
. A detector measures where the ions complete the semicircle and from this
it is easy to calculate . (a) Derive the equation for calculating the mass
of the ion from measurements of , , , and . (b) What potential
difference is needed so that singly ionized C atoms will have
50.0 cm in a 0.150-T magnetic field? (c) Suppose the beam consists of a
mixture of C and C ions. If and 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 27.59 for the masses
of the ions.)