Examples of omega-3 in the following topics:
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- $\displaystyle x_0 \left ( -\omega^2 + i \tau \omega^3 + \omega_0^2 \right )= \frac{q E_0}{m}$
- $\displaystyle x_0 = -\frac{e E_0}{m} \frac{1}{\omega^2-\omega_0^2 - i \tau \omega_0^3} $
- $\displaystyle \tan \delta = \frac{\omega^3\tau}{\omega^2-\omega_0^2}~ \text{ and }~|x_0| = \frac{e E_0}{m} \left [ \left ( \omega^2-\omega_0^2 \right )^2 + \omega_0^6 \tau^2 \right ]^{-1/2}.$
- $\displaystyle P = \frac{q^2 |x_0|^2 \omega^4}{3 c^3} = \frac{q^4 E_0^2}{3m^2 c^3} \frac{\omega^4}{\left (\omega^2-\omega_0^2\right )^2+\left(\omega_0^3 \tau\right)^2}$
- $\omega^2 - \omega_0^2 = (\omega - \omega_0) (\omega + \omega_0) \approx 2 \omega_0 (\omega-\omega_0)$
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- $\displaystyle \alpha = \pm i \omega_0 \sqrt{1 - \omega_0^2 \tau^2} - \frac{1}{2} \omega_0^2 \tau.$
- $\displaystyle x(t) = x_0 e^{-\Gamma t/2} \cos \omega_0 t~\text{ where }~\Gamma \equiv \omega_0^2 \tau = \frac{2 q^2 \omega_0^2}{3mc^3}$
- $\displaystyle {\hat x}(t) = \frac{x_0}{4\pi} \left [ \frac{1}{\Gamma/2 - i\left(\omega+\omega_0\right)} + \frac{1}{\Gamma/2 - i\left(\omega-\omega_0\right)} \right ]$
- $\displaystyle \frac{d W}{d\omega} = \frac{8 \pi \omega^4}{3 c^3}|{\hat d}(\omega)|^2 = \frac{8 \pi \omega^4}{3 c^3} \frac{q^2 x_0^2}{(4\pi)^2} \frac{1}{(\omega - \omega_0)^2 + \left(\Gamma/2\right)^2} \\ \displaystyle = \left ( \frac{1}{2} k x_0^2 \right ) \frac{\Gamma}{2\pi}\frac{1}{(\omega - \omega_0)^2 + \left(\Gamma/2\right)^2} $
- $\displaystyle \frac{\Delta \lambda}{\lambda} = \frac{\Delta \omega}{\omega}, \Delta \lambda = \frac{2e^2 \omega_0^2}{3 m c^3} \frac{\lambda}{\omega_0} = \frac{4 \pi e^2 }{3 m c^2} = \frac{4\pi}{3} r_0 = 1.2 \times 10^{-12}~{cm}$
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- $\displaystyle \gamma\theta \approx 2 \gamma \left ( \gamma^2 \omega_B \sin \alpha \right) t \propto \omega_c t$
- $\displaystyle {\hat E}(\omega) = \frac{1}{2\pi} \int_{-\infty}^\infty g(\omega_c t) e^{i\omega t} dt. = \frac{1}{2\pi} \int_{-\infty}^\infty g(\xi) e^{i\xi \omega/\omega_c} \frac{d\xi}{\omega_c}. = h(\omega/\omega_c)$
- so the average power per unit frequency is a function of $\omega/\omega_c$,
- $\displaystyle P = \frac{2}{3} r_0^2 c \beta^2_\perp \gamma^2 B^2 = C_1 \int_0^\infty F\left(\frac{\omega}{\omega_c}\right) d\omega = \omega_c C_1 \int_0^\infty F(x) dx$
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- Figure 1.10 shows two masses ( $m_1$ and $m_2$ ) connected to fixed walls with springs $k_1$ and $k_3$ and connected to one another by a spring $k_2$ .
- $\displaystyle{ m_2 {\ddot{x}_2} = -k_3 x_2 - k_2 (x_2 - x_1). }$
- $\displaystyle{ 2 \omega _0 ^2 - \omega ^2 = \omega _0 ^2 \Rightarrow \omega ^2 = \omega _0 ^2 . }$
- $\displaystyle{ 2 \omega _0 ^2 - \omega ^2 = - \omega _0 ^2 \Rightarrow \omega ^2 = 3 \omega _0 ^2 . }$
- We will refer to the frequency $\omega ^2 = 3 \omega_0 ^2 $$\omega ^2 = \omega_0 ^2$ as "fast" and $\omega ^2 = \omega_0 ^2$ as "slow".
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- $\displaystyle {W}{\omega d\Omega} \equiv {W_\|}{\omega d\Omega} + {W_\perp}{\omega d\Omega} \\ {W_\perp}{\omega d\Omega} = \frac{q^2 \omega^2}{4\pi^2 c} \left | \int \frac{ct'}{a} \exp \left [ \frac{i\omega}{2\gamma^2} \left ( \theta^2_\gamma t' + \frac{c^2 \gamma^2 t'^3}{3a^2} \right ) \right ] dt' \right |^2\\ {W_\|}{\omega d\Omega} = \frac{q^2 \omega^2 \theta^2}{4\pi^2 c} \left | \int \exp \left [ \frac{i\omega}{2\gamma^2} \left ( \theta^2_\gamma t' + \frac{c^2 \gamma^2 t'^3}{3a^2} \right ) \right ] dt' \right |^2$
- $\displaystyle y \equiv \gamma \frac{c t'}{a \theta_\gamma} ~\mbox{and}~ \eta \equiv \frac{\omega a \theta_\gamma^3}{3 c \gamma^3} \approx \frac{\omega}{2\omega_c}$
- $\displaystyle {W_\perp}{\omega d\Omega} = \frac{q^2 \omega^2}{4\pi^2 c} \left ( \frac{a \theta^2_\gamma}{\gamma^2 c} \right )^2 \left | \int_{-\infty}^\infty y \exp \left [ \frac{3}{2} i \eta \left ( y + \frac{1}{3} y^3 \right ) \right ] dt' \right |^2\\ {W_\|}{\omega d\Omega} = \frac{q^2 \omega^2 \theta^2}{4\pi^2 c} \left ( \frac{a \theta_\gamma}{\gamma c} \right )^2 \left | \int_{-\infty}^\infty y \exp \left [ \frac{3}{2} i \eta \left ( y + \frac{1}{3} y^3 \right ) \right ] dt' \right |^2\\ $
- $\displaystyle \frac{W_\perp}{\omega d\Omega} = \frac{q^2 \omega^2}{4\pi^2 c} \left ( \frac{a \theta^2_\gamma}{\gamma^2 c} \right )^2 K^2_\frac{2}{3} (\eta)\\ \frac{W_\|}{\omega d\Omega} = \frac{q^2 \omega^2 \theta^2}{4\pi^2 c} \left ( \frac{a \theta_\gamma}{\gamma c} \right )^2 K^2_\frac{1}{3} (\eta)$
- $\displaystyle {W_\perp}{\omega} = \frac{2 q^2 \omega^2 a^2 \sin\alpha}{3\pi c^3 \gamma^4} \int_{-\infty}^\infty \theta_\gamma^4 K^2_\frac{2}{3} (\eta) d\theta \\ {W_\|}{\omega} = \frac{2 q^2 \omega^2 a^2 \sin\alpha}{3\pi c^3 \gamma^4} \int_{-\infty}^\infty \theta_\gamma^2 \theta^2 K^2_\frac{1}{3} (\eta) d\theta $
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- 1.3.
- So $(\omega^2 - \omega_ 0 ^2)^2= \omega_ 0 ^2 \gamma ^2 + 1/2 \omega_0 \gamma^3 + \gamma^4 /16$ .
- Similarly $\gamma ^2 \omega^2 = \omega_ 0 ^2 \gamma ^2 + \omega_0 \gamma^3 + \gamma^4 / 4$ .
- $\displaystyle{\rho^{-2}(\omega_0 + \gamma/2) = 2 \omega_ 0 ^2 \gamma ^2 + \frac{3}{2} \omega_0 \gamma^3 + \frac{5}{16} \gamma^4 . }$
- $\displaystyle{\frac{\gamma ^3 \omega _0}{\gamma^2 \omega_0 ^2} = \frac{\gamma}{\omega_0} = \frac{1}{Q}. }$
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- $(\omega _0 ^2 - \omega^2) x_0 = \frac{qB}{m} i \omega y_0$
- $(\omega _0 ^2 - \omega^2) y_0 = - \frac{qB}{m} i \omega x_0 .$
- $(\omega _0 ^2 - \omega^2) y_0 = - \frac{ \left(\frac{qB}{m} i \omega \right) ^2 }{\omega _0 ^2 - \omega^2} y_0 .$
- $\displaystyle{ \left(\omega _0 ^2 - \omega^2\right)^2 = \left(\frac{qB}{m} \omega \right)^2 }$
- $\displaystyle{ \omega _0 ^2 - \omega^2 = \pm \frac{qB}{m} \omega . }$
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- Fig.3.2 illustrates how for $v>c/\sqrt{\epsilon}$ each point yields a unique retarded time denoted by the circles.
- $\displaystyle \frac{d W}{d\omega d\Omega} = \frac{q^2 \omega^2 \epsilon^{1/2}}{4\pi^2 c^3} \left | \int_{-\infty}^\infty {\bf n} \times ({\bf n} \times \beta) e^{i \omega \left ( t'- {\bf n} \cdot {\bf r}_0 (t') \epsilon^{1/2} / c \right )} dt' \right |^2.$
- $\displaystyle \frac{d W}{d\omega d\Omega} = \frac{q^2 \epsilon^{1/2}}{c^3} |{\bf n} \times {\bf v}|^2 \left | \frac{\omega}{2\pi} \int_{-\infty}^\infty e^{i \omega t' \left ( 1 - {\bf n} \cdot {\bf v} \epsilon^{1/2} / c \right )} dt' \right |^2.$
- $\displaystyle \frac{\omega}{2\pi} \int_{-T}^T e^{i \omega t' \left ( 1 - {\bf n} \cdot {\bf v} \epsilon^{1/2} / c \right )} dt' = \frac{\omega T}{\pi} \frac{\sin \left [ \omega T \left ( 1 - \epsilon^{1/2} \beta \cos \theta \right ) \right ]}{\left [ \omega T \left ( 1 - \epsilon^{1/2} \beta \cos \theta \right ) \right ]}.$
- $\displaystyle \frac{d W}{d\omega} = \frac{q^2 \omega}{c^2} \sin^2 \theta_c \left (2 c \beta T \right ) = \frac{q^2 \omega}{c^2} \left [ 1 - \frac{1}{\beta^2 \epsilon(\omega)} \right ] \left ( 2 c \beta T \right )$
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- $\displaystyle {\hat E} (\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} E(t) e^{i\omega t} d t.$
- $\displaystyle \frac{dW}{d\Omega d\omega} = R^2 \frac{dW}{dA d\omega}= c |R \hat E(\omega)|^2$
- $\displaystyle R {\hat E} (\omega) = \frac{q}{2\pi c} \int_{-\infty}^{\infty} \left [ \frac{\bf n}{\kappa^3} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]\right ]_\mathrm{ret} e^{i\omega t} d t.$
- $\displaystyle \frac{d W}{d\Omega d\omega} = \frac{q^2}{4\pi^2 c} \left | \int_{-\infty}^{\infty} \frac{{\bf n} \times \left [ ({\bf n} - \beta ) \times \dot{\beta} \right ]}{\left ( 1-\beta\cdot {\bf n} \right )^2} e^{i\omega (t'-{\bf n}\cdot {\bf r}(t')/c)} d t' \right |.$
- $\displaystyle \frac{d W}{d\omega d\Omega} = \frac{q^2 \omega^2}{4\pi^2 c^3} \left | \int_{-\infty}^\infty {\bf n} \times ({\bf n} \times \beta) e^{i \omega \left ( t'-{\bf n} \cdot {\bf r}_0 (t') / c \right )} dt' \right |^2
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- Even before calculating the form of $F(\omega/\omega_c)$, we can determine some interesting properties of the radiation spectrum.
- $\displaystyle P_\mbox{tot} (\omega) = C \int_{\gamma_1}^{\gamma_2} P(\omega) \gamma^{-p} d\gamma \propto \int_{\gamma_1}^{\gamma_2} F\left(\frac{\omega}{\omega_c}\right) \gamma^{-p}d\gamma.$
- Let's change variables to $x\equiv \omega/\omega_c$.
- Remember that $\omega_c = A \gamma^2$ so $\gamma^2 \propto \omega/x$, we get
- This power-law spectrum is valid essentially between $\omega_c(\gamma_1)$ and $\omega_c(\gamma_2)$.