W9_parallel_resonance.eps

208 Week 6: Moving Charges and Magnetic Force

point (whereris the radius of curvature of the species’ particular trajectory).

The molecular weight of the components of the sample is thus registered two ways. Typically
a “marker” species of known weight and concentration is introduced that permits the distances
from the entrance point to be calibrated and checked against a known mass, and each particular
components is likely to be present in single ionized form (with charge e.g. +e), doubly ionized form
(with charge +2e) etc. This appears as “similar” patterns of bands on the film or detector which
permits one to tell which pattern corresponds to a particular charge, for example +e. From this
combination it is straightforward to deduce the charge and infer the mass of the various chemical
components visible in the detector fingerprint.

We can easily understand the physics behind the mass spectrometer. A charged ion of chargeq
and massmproduced in the goop boiler is accelerated to a kinetic energy:

1

2

mv^2 =qV 0 (433)

in the beam entering the magnetic field. It therefore has a velocity^61 :

v=

√

2 qV 0

m

(434)

and experiences a centripetal magnetic force (that causes it to move in a circle of radiusr) of:

Fr=qvB 0 =mv

2

r

(435)

so as usual:
v
r

= q

m

B 0 (436)

If we solve for the radiusrof its half-orbit to the film/detector, we get:

r= v

B 0

m

q

(437)

Substituting forv:

r=

√

2 qV 0

m

B 0

m

q

=

√

2 mV 0

qB 02

=

√

m

q

√

2 V 0

B 0

(438)

Alternatively, since one measuresrand wishes to findm(given a good guess forq):

m=

r^2 B^20

2 V 0 q (439)

As one can see, the mass-to-charge ratio determinesr, creating similar “bands” of molecular signal
for different ionizations of the same collection of constituent masses. Once the charge on any given
band is guessed/determined (where the lowest charge, in positive multiples ofe, will have the largest
radius spectral pattern for each set ofm’s) one can transform a knowledge ofrandqdirectly into
m.

Most of this process can be automated and computerized, and mass spectrometers based on this
general principle are at this point commonplace in the laborator.

(^61) As before in the case of the Thompson apparatus, in reality the “boiler” would produce a Maxwell-Boltzmann
range of entering velocities, but we can insert a velocity selector stage to narrow the distribution to “precisely” the
desired/expectedv.