In a mass spectrograph, ions with a mass m and charge q are accelerated through
ID: 2179205 • Letter: I
Question
In a mass spectrograph, ions with a mass m and charge q are accelerated through a potential difference V. They then enter a uniform magnetic field B that is perpendicular to their velocity and are deflected into a semicircular orbit of radius R. A detector measures where the particles exit the semicircle, thereby providing a measurement of R.(a) Derive and equation for calculating the mass m of the ion from the measurements of B, V, R, and q.
(b) What potential V would be required so that singly ionized C(12) ions (i.e. with charge q=e) will execute orbits of radius R=50cm in a magnetic field of B=0.150T? (take the mass of a C(12) ion to be m(12)=1.993*10^-26kg)
(c) Suppose that the ion beam consists of a mixture of C(12) and C(14) ions. If V and B have the same values as in (b), calculate the spatial separation of these two isotopes at the detector. (Take the mass of a C(14) ion to be m(14)=2.325*10^-26kg)
Please help/explain!!
Explanation / Answer
(a) Derive and equation for calculating the mass m of the ion from the measurements of B, V, R, and q.
q V = m v2/R
===> v = (RqV/m)
m v2/R = q v B
===> m v = R q B
===> m (RqV/m) = R q B
===> m = B (Rq/V)
===> m = RqB2/V
(b) What potential V would be required so that singly ionized C(12) ions (i.e. with charge q=e) will execute orbits of radius R=50cm in a magnetic field of B=0.150T? (take the mass of a C(12) ion to be m(12)=1.993*10^-26kg)
m = RqB2/V
====> V = RqB2/m = 0.50*1.6e-19*0.150*0.150/1.993e-26 = 9.03e4 V
(c) Suppose that the ion beam consists of a mixture of C(12) and C(14) ions. If V and B have the same values as in (b), calculate the spatial separation of these two isotopes at the detector. (Take the mass of a C(14) ion to be m(14)=2.325*10^-26kg)
R = m (V/qB2) = (2.325e-26-1.993e-26)*(90316/(1.6e-19*0.150*0.150)) = 0.0833 m
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