The picture above shows part of a mass spectrometer that can be used to measure

Question

The picture above shows part of a mass spectrometer that can be used to measure molecular charge-to-mass ratios. A charged molecule (orange circle) is ionized and accelerated through an electric potential difference into a region with a uniform magnetic field. Here the magnetic field points out of the screen. The field makes the positively charged molecules undergo circular motion as shown. By adjusting the voltage difference between the plates, one can change the radius of curvature of the charged particles until they reach the detector. For a magnetic field of B = 9 mT and a spacing between the entrance and exit holes of d = 8 cm, what is the potential V (in mV) required to detect O2+?

(You may assume that the mass of the O2+ ion is 2.656e-26 kg.)

Explanation / Answer

from the conservation of energy

q V = 1/2 m v^2

v= sqrt 2 qV/m

the radius of path is

r= mv/ qB

d/2 = m / qB * sqrt 2 qV/m

d^2/4 = m^2 / q^2B^2 * 2 qV/m

V= d^2 q B^2/ 8m

= (0.08)^2 ( 1.602 * 10^-19) ( 9 * 10^-3 )^2/ 8 ( 2.65 6 * 10^-26 kg)

=0.390 V

or 390 mV

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