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Your forensic chemistry group, working closely with local law enforcement agencies, has acquired a mass spectrometer. It employs a uniform magnetic field that has a magnitude of 0.75 T. To calibrate the mass spectrometer, you decide to measure the masses of various carbon isotopes by measuring the position of impact of the various singly ionized carbon ions that have entered the spectrometer with a kinetic energy of 25 keV. A wire chamber with position sensitivity of 0.50 mm is part of the apparatus. What will be the limit on its mass resolution for ions in this mass range, that is, those whose mass is on the order of that of a carbon atom? SOLUTION ββ To determine the limit on mass resolution, we need the relationship πΉβπ΅ = ππ£β × π΅ The way I have defined the axis, the velocity is in the z direction, and the magnetic field is in the ββ |π πππ = ππ£π΅ y direction. So the cross product of the two is in the x direction. |πΉβ | = π|π£β||π΅ The particles will form a circle, so ππ£π΅ = two equations together gives us π = πππ΅ π£ ππ£ 2 π 2πΎπΈ and we also know that π£ = √ π π = πππ΅√2πΎπΈ which means that √π = putting these πππ΅ √2πΎπΈ . The question, however, doesn’t ask about mass, but rather the mass resolution – so we’d like to be able to state our information in terms of σm and σr. Therefore, take the derivative of the equation for mass above. 1 1 2 √π ππ = ππ΅ √2πΎπΈ ππ 2ππ΅√π Therefore ππ = ππ √2πΎπΈ So, what is dr here? If the resolution in the diameter is 0.5 mm, then half that division is a good approximation, so dr=0.25mm. q=1.6e-19C and B=0.75T. Finally, E=25keVx1.6x1019 J/eVx1000=4e-15J and m=1.99e-26 kg. Plugging in the numbers yields a mass resolution of dm=9.5e-29kg=0.06mp=104me