A free electron is located in the field of a monochromatic light wave. The intensity of light is `I = 150 W//m^(2)`, its frequency is `omega = 3.4 . 10^(15) s^(-1)`. Find:
(a) the electron's oscillation amplitude and its velocity amplitude,
(b) the ratio `F_(m)//F_(e)`, where `F_(m)` and `F_(e)` are the amplitudes of forces with which the magnetic and electric components of the ligth wave field act on the electron, demonstrate that the ratio is equal to `(1)/(2)v//c`, where `v` is the electron's velocity amplitude and `c` is the velocity of light.
Instruction. The action of the magnetic field component can be disregarded in the equation of motion of the electron since the calculations shown it to be negligible.

Explanation

In a travelling plane electromagnetic wave the intensity is simply the time averaged magnitude of the Polynting vector:-
` I = lt|oversetrarr(E) xx oversetrarr(H)|gt = lt sqrt((epsilon_(0))/(mu_(0)))E^(2) gt = lt c epsilon_(0)E^(2)gt`
on using `c = (1)/(sqrt(epsilon_(0)mu_(0))), E sqrt(epsilon_(0)) = H sqrt(mu_(0))`.
Now time averaged value of `E^(2)` is `E_(0)^(2)//2` so
`I = (1)/(2)c epsilon_(0) E_(0)^(2)` or `E_(0) = sqrt((2I)/(c epsilon_(0)))`,
(a) Represent the electric field at any point by `E = E_(0) sin omegat`. Then for the electron we have the equation.
`m ddotx = eE_(0)sin omegat`
so `x =- (eE_(0))/(m omega^(2)) sin omegat`
The amplitude of the forced oscillation is
`(eE_(0))/(m omega^(2)) = (e)/(m omega^(2)) sqrt((2I)/(c epsilon_(0))) = 5.1 xx 10^(-16) cm`
The velocity amplitude is clearly
`(eE_(0))/(m omega) = 5.1 xx 10^(-16) xx 3.4 xx 10^(15) = 1.73 cm//sec`
(b) For the electric force
`F_(e) =` amplitude of the electric force
`= eE_(0)`
For the magnetic force (which we have neglected above), it is
`(evB) = (evmu_(0)H)`
`= evE sqrt(epsilon_(0)mu_(0)) = ev(E)/(c)`
writing `v=- v_(0) cos omega t`
where `v_(0) = (eE_(0))/(m omega)`
we see that the magnetic force is apart from a sign
`(ev_(0)E_(0))/(2c)sin 2 omegat`
Hence `(F_(m))/(F_(e)) =` Ratio of amplitudes of the two forces
`= (v_(0))/(2c) = 2.9 xx 10^(-11)`
This is negligible and justifies the neglect of magnetic field of the electromagnetic wave in calculating `v_(0)`.