# 8.4: A Non-Sinusoidal Time Dependence

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Nothing in the calculation of the radiation fields required the time variation of the dipole moment to be sinusoidal. If a charge undergoes an acceleration ~a at the retarded time t_{R} = t − R/c then the Poynting vector at time t on a surface of radius R will have the radial component

\[S_{\mathrm{r}}=\frac{1}{c \mu_{0}}\left(\frac{q a \sin \theta}{4 \pi \epsilon_{0} c^{2} R}\right)^{2}, \nonumber \]

(see Equations (7.4.5)). This expression can be written

\[S_{\mathrm{r}}=\frac{c}{4 \pi} \frac{1}{4 \pi \epsilon_{0}}\left(\frac{\mathrm{q}^{2} \mathrm{a}^{2} \sin ^{2} \theta}{\mathrm{c}^{4} \mathrm{R}^{2}}\right)_{\mathrm{tr}}, \label{8.23}\]

and the power integrated over a sphere of radius R is given by

\[\mathrm{P}_{\mathrm{q}}=\frac{2}{3} \frac{1}{4 \pi \epsilon_{0}}\left(\frac{\mathrm{q}^{2} \mathrm{a}^{2}}{\mathrm{c}^{3}}\right)_{\mathrm{t}_{\mathrm{R}}} \quad \text { Watts }, \label{8.24}\]

where a(t_{R}) means that the acceleration is measured at the retarded time (t − R/c) if the power is measured at the time t. Eqn.(\ref{8.24}) can be used to understand the production of the continuous X-ray spectrum, refer to Figure (8.4.4). The conversion efficiency for X-ray production is rather small; approximately 10^{−7} of the incident power is converted to continuous spectrum X-ray energy.