Wiggling Charges Revisited
I'm not sure I understand what you mean by "wiggling." How does
the Bohr model for an atom relate to the ideas we were
talking about earlier? You compared "wiggling charges" to
oscillating springs; how does that fit in with Bohr's idea of
jumps between orbits?
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Let's step back from the Bohr model for a moment and look at an
earlier model for the atom--don't worry, you've seen this before.
Remember when you were playing around with orbits, and I said
you'd created a simplified version of a hydrogen atom?
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That was just an electron orbiting around a proton.
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That's right; in this model (which is sometimes called the
"Rutherford model" after Ernest Rutherford, the scientist who
first envisioned it), an atom is made up of electrons orbiting a
nucleus in the same way that planets orbit around the sun. The
electrons are held in their orbits by the electric force, just as
the planets are kept in theirs by gravity, and the entire atom
resembles a miniature solar system. This is known as a
"classical" model, meaning that it doesn't use any quantum ideas,
just Newtonian mechanics applied to the electric
force. |
 Rutherford
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So does the Rutherford model relate to the "charge on a spring"
idea?
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It certainly does. An
electron orbiting a nucleus moves periodically, just like the
charge on a spring you saw
earlier.
Both are
undergoing simple harmonic motion; if you project
the orbiting electron's motion into one dimension,
it will look just like a mass oscillating on a
spring. |
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So if an electron is orbiting, you could say it's "wiggling" at a
certain frequency, which depends on the radius of its orbit. That
makes sense. But when we discussed the "charge on a spring," you
said that if an electron wiggles at a particular
frequency, it must produce electromagnetic
radiation at that same frequency. So if you had,
say, a hydrogen atom with an electron orbiting at
some fixed radius, it would always be
giving off radiation--that can't be right.
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Yes, Rutherford's picture of the atom has a couple of
fundamental problems. First of all, there's no apparent reason
why an electron's orbit couldn't have just any old radius, and
thus any old frequency. That flatly contradicts
the experimental evidence of spectral
lines. |
Well, that's why Bohr came up with the idea of electrons' being
restricted to certain special orbits.
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All right, that's a partial solution. Let's assume that an
electron's orbit, for some mysterious reason, can only have
certain discrete radii. Now suppose that an electron is happily
cruising along in one of these legal orbits. As you've said, the
electron is "wiggling," so
it ought to
be producing radiation. So, let's say our electron
emits an electromagnetic wave of the proper frequency.
That's all well and good until you start to think about the
energy contained in that wave...
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Waves have energy?
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Click the image to jump to the applet.
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Take another look at that wiggling charge.
Notice how when the wave reaches the negative charge on the left,
that charge starts bouncing up and down? The
original + charge never touched that negative
charge; only the wave did.
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If the wave made that second charge move, then it must have
carried energy from the + charge to the - charge.
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So the moving wave contains energy...and energy is conserved.
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Then in order to emit the wave, the electron has to give up some
of
its
own energy.
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Exactly; the electron would have to slow down, which would
decrease its kinetic energy. But if it did that, it wouldn't be
able to remain at a fixed radius; it would be
pulled in closer to the nucleus. (Note: this
actually does happen for macroscopic
objects in orbit; stars in a binary
system, for example, spiral slowly in toward
one another, because as they orbit they give off
energy in the form of gravitational waves.)
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And meanwhile the electron would still be wiggling, so it would
give off another wave (at a different frequency), which would
make its orbit decay even more...when would this ever
end?
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It wouldn't. Our hapless electron would keep right on spiraling
inward until it crashed into the nucleus, at which point there
would be no more atom. So if this classical picture were correct,
atoms would be highly unstable, and nothing made of atoms could
possibly exist for more than a fraction of a second. You and I
couldn't be having this conversation if it weren't for quantum
mechanics.
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Okay, so it sounds like you're saying that the Bohr model and the
Rutherford model actually have nothing to do with each other.
Rutherford's classical picture was just completely wrong, so Bohr
had to come
up with something entirely new.
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Well...yes and no. This classical model is inaccurate, but
there are cases in which the correspondence principle
applies.
Just as Newtonian mechanics is a good approximation to relativity
at low velocities, Rutherford's model is a good approximation to
the Bohr model for closely packed energy levels.
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