Mass Spectrograph. A mass spectrograph is used to measure the masses of ions, or to...

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 27.59 for the masses of the ions.)

27.59 .. Suppose the electric field between the plates in Fig. 27.24 is 1.88 × 10⁴ V/m and the magnetic field in both regions is 0.682 T. If the source contains the three isotopes of krypton, ⁸²Kr, ⁸⁴Kr, and ⁸⁶Kr, and the ions are singly charged, find the distance between the lines formed by the three isotopes on the particle detector. Assume the atomic masses of the isotopes (in atomic mass units) are equal to their mass numbers, 82, 84, and 86. (One atomic mass unit = 1 u = 1.66 × 10^-27 kg.)