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?
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 will take for the helium ion to absorb the infrared radiation, we can use the equation for the Doppler effect. The Doppler effect describes the change in frequency of a wave due to the motion of the source and/or observer.
First, let's calculate the initial velocity of the helium ion. Since it is at rest and moving towards the source of radiation, we can assume it has an initial velocity of zero.
Next, we need to calculate the final velocity of the helium ion. We can use the Doppler effect equation, which relates the change in wavelength (Δλ) to the initial wavelength (λ0), velocity of the source (vs), and the final velocity of the helium ion (vh):
Δλ/λ0 = vs / (speed of light) - vh/ (speed of light)
We know the initial wavelength (λ0) and the final wavelength (2000 nm), and we can assume the speed of light to be approximately 3x10^8 m/s.
Let's rearrange the equation to solve for vh:
vh = (vs / (speed of light)) - (Δλ/λ0) * (speed of light)
Substituting the given values, we have:
vh = 0 - ( (2 * 2000 nm) / 1083 nm) * (3x10^8 m/s)
Simplifying the equation:
vh = - (2000 / 1083) * (3x10^8) m/s
Now we can calculate the time it will take for the helium ion to absorb the radiation. Since we know the velocity of the ion, we can use the equation:
t = distance / velocity
The distance the ion needs to travel is not given, but we can assume it is relatively small. Therefore, we can neglect the distance and approximate the time as zero.
Therefore, in the laboratory frame, the ion will absorb the infrared radiation practically instantaneously.