Using power

The power on a particle is the velocity multiplied by the force, i.e.,

$$P = F v$$

$$= E q \cos (k z - \omega t) v.$$ (54)

Different order approximations of $v$ and $z$ can be used in Eq. (54) to evaluate the power. If the approximations $v \approx v^{(0)} = v_0$ and $z \approx z^{(0)} = z_0 + v_0 t$ are used, Eq. (54) is written as

$$P \approx E q \cos (k z_0 + \alpha t) v_0,$$ (55)

which is obviously zero when it is averaged over initial position $z_0$ in one wavelength. If the approximations and $v \approx v^{(1)}$ and $z \approx z^{(1)}$ are used, Eq. (54) is written as

$$P \approx \left\{ q E \cos (k z_0 + \alpha t) - \sin (k z_0 + \alpha t) k \... ...0 + \frac{q E}{m} \frac{\sin (k z_0 + \alpha t) - \sin (k z_0)}{\alpha} \right]$$

$$\approx E q \cos (k z_0 + \alpha t) v_0 + E q \cos (k z_0 + \alpha t) \fr... ...alpha t) + \cos (k z_0)}{\alpha^2} - \frac{\sin (k z_0)}{\alpha} t \right] v_0,$$ (56)

where, the term proportional to $E^3$ has been neglected. Equation (56) is identical with Eq. (24). Therefore the derivation after this point is the same as given in Sec. 2.2.