Mass-spectrometer (device that selects atomic particles of different masses and
ID: 2026818 • Letter: M
Question
Mass-spectrometer (device that selects atomic particles of different masses and charges) consists of cylindrical capacitor with inner radius R1=2.4 cm and outer radius R2=3 cm. Ions enter the mass-spectrometer through narrow slit S which is situated in between the capacitor plates as shown on the picture. Uniform magnetic field B=0.2 T is applied parallel to the capacitor’s axis, direction out of the plane. Find potential difference that should be applied between capacitance’s plates in order for the lithium ion Li+7 (such ions have charge +e, mass 7m, m=1.67*10-27 kg) to go around the middle section of the capacitor, i.e. on the circle of R=2.7 cm. Energy of incoming ions is 1000 eV. By how much the potential should be changed in order for lithium ions Li+6 to go around the same trajectory?
Explanation / Answer
The centripetal force necessary to make ions traverse that path is mv2/r. We need v but can get it from incoming energy: 1000 eV = 1.602 * 10-16 J = (1/2)mv2 so v = (2K/m)1/2
= ((2 * 1.602 * 10-16 J)/(7 * 1.67 * 10-27 kg))1/2 = 1.6553 * 105 m/s
so mv2/r = 1.19 * 10-14 N where r = 0.027 m
The amount of magnetic force is qvB = (1.602 * 10-19 C)(1.6553 * 105 m/s)(2 T) = 5.304 * 10-15 N which leaves 6.56 * 10-15 N to be accomplished by the E field.
Since F = qE E = F/q for E = (6.56 * 10-15N)/(1.602 * 10-19 C) = 4.10 * 104 V/m
If you use lithium 6 instead the mass changes to 6m, the velocity changes to 1.788 * 105 m/s, and the required voltage changes to 3.83 * 104 V/m.