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 ion's motion towards the source of the radiation.
The Doppler effect is a phenomenon where the observed wavelength of a wave changes due to the relative motion between the source of the wave and the observer.
In this case, the ion is moving towards the source of the infrared radiation. We know that the absorption line for helium at rest occurs at a wavelength of 1083 nm (λ0). The infrared light being directed towards the ion has a wavelength of 2000 nm (λ).
According to the Doppler effect equation for light waves, the observed wavelength (λ') is related to the rest wavelength (λ0) by the formula:
λ' = λ0 * (1 + v/c),
where v is the velocity of the ion and c is the speed of light.
We can rearrange this equation to solve for the velocity of the ion:
v = c * (λ'/λ0 - 1).
Substituting the given values, we have:
v = 3.0 × 10^8 m/s * (2000 nm / 1083 nm - 1).
Now we can find the velocity of the ion.
v = 3.0 × 10^8 m/s * (1.846 - 1) = 5.538 × 10^7 m/s.
Since we know the ion is at rest initially, this velocity is the final velocity attained by the ion when it absorbs the infrared radiation.
The time taken for the ion to absorb the radiation can be calculated using the equation of motion:
v = u + at,
where v is the final velocity, u is the initial velocity (which is 0 in this case), a is the acceleration, and t is the time.
Since there are no radiative losses due to acceleration, we can assume the acceleration is constant and equal to the electric field E. Therefore, the equation becomes:
v = E * t.
Rearranging the equation to solve for time, we have:
t = v / E = (5.538 × 10^7 m/s) / (2 N/C).
Calculating this value, we find:
t ≈ 2.769 × 10^7 s.
Therefore, after approximately 2.769 × 10^7 seconds (or about 330 days), the helium ion will absorb the infrared radiation in the laboratory frame.