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# 8.4: A Non-Sinusoidal Time Dependence

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 tR = 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

$S_{\mathrm{r}}=\frac{c}{4 \pi} \frac{1}{4 \pi \epsilon_{0}}\left(\frac{\mathrm{q}^{2} \mathrm{a}^{2} \sin ^{2} \theta}{\mathrm{c}^{4} \mathrm{R}^{2}}\right)_{\mathrm{tr}}, \label{8.23}$ Figure $$\PageIndex{4}$$: A schematic diagram of an X-ray tube illustrating the production of the white X-ray spectrum. The electrons undergo a de-acceleration upon striking the metal anode. This de-acceleration is of order a= 1024 m/sec2 for a typical 20 keV potential drop between the anode and the cathode: this assumes an electron stopping distance of 35×10−10 m. During a brief period, ∼ 10−16 seconds, the electron radiates at the rate of ∼ 5.7 × 10−6 Watts, therefore each electron emits a pulse of radiation containing ∼ 5.7 × 10−22 Joules. The number of electrons that impinge on the anode per second for a beam of 1 mAmp is 6.25 × 1015. The average power in the X-ray beam will be (6.25 × 1015)(5.7 × 10−22) = 3.6 × 10−6 Watts. This energy is distributed over a range of frequencies from zero to 4.8×1018 Hz (hνmax =| e | V). This calculation does not include the energy contained in the characteristic X-ray spectrum emitted from the target. Figure $$\PageIndex{5}$$: Schematic diagram of an atom in a time varying electric field. The atom develops a time varying dipole moment that scatters the incident radiation.

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(tR) 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.