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 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

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

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 (hν_max =| 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(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.