- #1
mbigras
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- 2
Homework Statement
According to classical electromagnetic theory an accelerated electron radiates energy at the rate [itex]Ke^{2}a^{2}/c^{3}[/itex], where [itex]K = 6*10^{9} Nm^{2}/C^{2}[/itex], [itex]e = [/itex] electronic charge, [itex]a = [/itex] instananeous acceleration, and [itex]c = [/itex] speed of light.
a) If an electron were oscillating along a straight line with frequency [itex]v[/itex] (Hz) and amplitude [itex]A[/itex], how much energy would it radiate away during 1 cycle? (Assume that the motion is described adequately by [itex]x = A\sin{2 \pi v t}[/itex] during anyone cycle.)
b) What is the [itex]Q[/itex] of this oscillator?
c) How many periods of oscillation would elapse before the energy of the motion was down to half the initial value?
d) Putting for [itex]v[/itex] a typical optical frequency(i.e., for visible light) estimate numerically the approximate Q and "half-life" of the radiating system.
Homework Equations
[tex]Q = \frac{\omega_{0}}{\gamma}[/tex]
The Attempt at a Solution
For part a, I took the integral of the rate that the energy radiates from 0 to [itex]\frac{1}{2v}[/itex]. So the energy radiated during 1 cycle is [itex]\frac{8 \pi^{4} v^{3} A^{2} K e^{2}}{c^{3}}[/itex] J
I feel confused about part b. I'm given the rate the energy radiates and from that I think I should find [itex]\omega_{0}[/itex] and [itex]\gamma[/itex] which will tell me about [itex]Q[/itex]. By knowing how much energy is being lost I can imagine how that tells you about the damping but right now I don't see how they're related. Something I was thinking was to integrate the given rate:
[tex]\int \frac{dE}{dt} dt = \int \frac{K e^{2}}{c^{3}} \frac{d^{2} x}{d t^{2}} dt[/tex]
[tex]E = \frac{1}{2} \frac{K e^{2}}{c^{3}} \left( \frac{dx}{dt} \right)^{2} + constant[/tex]
Now it's starting to look like a familiar differential equation...but really, I'm not sure what going on here. I think my main question is: How is the quality of an oscillatory system related to the rate that it losses energy due to damping?