- #1
sayebms
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consider energy for a damped electric oscillator . ("[itex]f[/itex]" indicates the dipole moment of the oscillator)
in the absence of the damping force
[itex]U= \frac{1}{2}kx^2 +1/2 (\frac{d^2x}{dt^2}) ^2 [/itex]
and the energy conservation tells us [itex]dU=0[/itex].
but if there is damping force we get the following using larmor formula and energy of a dipole in an electric field, for the conservation of energy:
[itex] \int_t^τ ( \frac{dU}{dt} + \frac{2}{3c^2} (\frac{d^2f}{dt^2})^2 -E \frac{df}{dt} ) [/itex]
and here is i don't understand: using the fact that [itex] \frac{4π^2 v_0}{3c^3 L} = σ [/itex]
and the above conseravtion of energy formula we get to
[itex] Kf+L \frac{d^2f}{dt^2} -2/(3c^3) \frac{d^3f}{dt^3}=E [/itex]
i don't really know how we got to this last formula using the above equations. any help is appreciated. and for those who have access to the book The question is from page 184 of the book "Planck's Columbia Lectures".
in the absence of the damping force
[itex]U= \frac{1}{2}kx^2 +1/2 (\frac{d^2x}{dt^2}) ^2 [/itex]
and the energy conservation tells us [itex]dU=0[/itex].
but if there is damping force we get the following using larmor formula and energy of a dipole in an electric field, for the conservation of energy:
[itex] \int_t^τ ( \frac{dU}{dt} + \frac{2}{3c^2} (\frac{d^2f}{dt^2})^2 -E \frac{df}{dt} ) [/itex]
and here is i don't understand: using the fact that [itex] \frac{4π^2 v_0}{3c^3 L} = σ [/itex]
and the above conseravtion of energy formula we get to
[itex] Kf+L \frac{d^2f}{dt^2} -2/(3c^3) \frac{d^3f}{dt^3}=E [/itex]
i don't really know how we got to this last formula using the above equations. any help is appreciated. and for those who have access to the book The question is from page 184 of the book "Planck's Columbia Lectures".