W9_parallel_resonance.eps

206 Week 6: Moving Charges and Magnetic Force

fromx(t). Substituting this into the last two equations, we find that as it emerges from between
the plates:

∆y 1 =

eE

2 m

t^21 =

eEL^2

2 mv 02

(425)

and
vy=eE
m

t 1 =eEL

mv 0

(426)

From our knowledge ofvxandvywhen the particle emerges, we can find:

tan(θ) =

vy

vx=

eEL

mv 02 (427)

which lets us easily determine:

∆y 2 =Dtan(θ) =

eELD

mv^20. (428)

Now we can relate themeasuredtotalydeflection to the known values ofL,D,E, and ourestimated
v 0 :

ytot = ∆y 1 + ∆y 2

= eEL

2

2 mv 02

+eELD

mv^20

= eEL

mv^20

(L

2

+D)

= e

m

EL

v^20

(L

2

+D) (429)

Inverting this last relation we find:

e

m

=

ytotv^20

EL(L 2 +D)

(430)

We know everything on the right (where wemeasureytot), so we have measurede/m!

Of course Thomson didn’t really knowv 0 – he had toestimateit from a mix of thermodynamics
and electrostatics in his first experiment. We, however, can see how the addition of a crossed
magnetic field permits him toprecisely determinev 0. With theEfield turned on, simply turn up
the magnetic fieldBuntil the particle’s deflection is once againzero. At that point, the apparatus
is functioning as a velocity selector, and we know from the argumentabove that:

v 0 =

E

B

(431)

this can be substituted into the expression above to obtain a much more accurate estimate fore/m,
one that doesn’t rely on a prior knowledge of the thermal distribution of electron energies before
they are accelerated by the first potential difference:

e

m

=

ytot

(E

B

) 2

EL(L 2 +D)

=

ytotE

B^2 L(L 2 +D)

(432)

This permits a measurement that is as accurate as one’s knowledge ofytotL,D,EandB; with
care within a few percent even using late 19th, early 20th century apparatus. Using it Thomson was
able to determine that therelativemass of the negative charge in a hydrogen atom compared to the
mass of the positive charge wasless than 0.1%! The electron wasextremely lightcompared to the
proton.

Of course, Thomson still did not know that the proton existed; theplum pudding model described
the positive mass as being an “amoebic blob” that somehow bound theelectron to the atom. It