A helium ion is at rest in a laboratory when it is put in an electric field of E=2 N/C. An infrared light, of wavelength 2000 nm, is directed towards the ion. The ion is moving towards the source of radiation. After what time in the laboratory frame in seconds will the ion absorb the infrared radiation?
Details and assumptions
The first absorption line of helium at rest occurs at a wavelength of λ0=1083 nm.
The mass of the helium atom (approximately the same as of the Helium ion) is 6.65×10−27 kg.
Only one electron is taken from the helium atom to make it into an ion.
Neglect radiative losses due to acceleration.
You may neglect any relativistic effects in the acceleration of the ion, but not otherwise.
To determine the time it takes for the helium ion to absorb the infrared radiation, we need to consider the Doppler effect due to the motion of the ion towards the source of radiation.
The Doppler effect describes how the frequency and wavelength of a wave are affected by the relative motion between the source of the wave and the observer. In this case, the motion of the helium ion towards the source of radiation will cause a shift in the wavelength of the absorbed radiation.
The Doppler shift formula for the observed wavelength (λ_obs) is given by:
λ_obs = λ_source * (1 + v/c)
Where λ_source is the wavelength of the source radiation (2000 nm in this case), v is the velocity of the helium ion, and c is the speed of light.
We can rearrange the formula to solve for v:
v = c * ((λ_obs / λ_source) - 1)
Now, let's calculate the velocity of the helium ion using the known values:
v = c * ((λ_obs / λ_source) - 1)
v = 3 * 10^8 m/s * ((1083 nm / 2000 nm) - 1)
v = 3 * 10^8 m/s * (0.5415 - 1)
v = - 3 * 10^8 m/s * (0.4585)
v = - 1.375 * 10^8 m/s
Note that the negative sign indicates that the ion is moving towards the source of radiation, as stated in the question.
Now, we can determine the time it takes for the ion to absorb the infrared radiation using the formula:
t = Δx / v
Where Δx is a distance over which the absorption occurs. In this case, we can assume it is a distance of one wavelength of the radiation, which would be:
Δx = λ_obs = 2000 nm = 2 * 10^-6 m
t = (2 * 10^-6 m) / (- 1.375 * 10^8 m/s)
t ≈ -1.45 * 10^-14 s
Note that the negative sign indicates that the time is "negative" because it refers to the time in the past when the ion absorbed the radiation. The absolute value of the time (∣t∣) is approximately 1.45 * 10^-14 seconds.
Therefore, after approximately 1.45 * 10^-14 seconds in the laboratory frame, the helium ion will absorb the infrared radiation.