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
The charge of He ion = e
The aceleration of ion in the electric field
ma=eE
a=eE/m
KE=mv²/2= m(at)²/2=
= (eEt)²/2m
ℇ=KE
ℇ = hc/λ
hc/λ=(eEt)²/2m
t={sqrt(2mhc/λ)}/eE=
={sqrt(2•6.65•10⁻²⁷•6.63•10⁻³⁴•3•10⁸/1083•10⁻⁹)}/1.6•10⁻¹⁹•2=
=1.5•10⁻⁴ s
wrong
To determine the time it takes for the helium ion to absorb the infrared radiation, we need to consider the Doppler effect and the motion of the ion.
First, let's find the speed of the helium ion. We can use the equation for the Doppler effect:
Δλ/λ = v/c
where Δλ is the change in wavelength, λ is the initial wavelength, v is the velocity of the source (the ion), and c is the speed of light.
Given that the initial wavelength λ0 is 1083 nm and the final wavelength λ is 2000 nm, we can rearrange the equation to solve for v:
v/c = (λ - λ0) / λ
v/c = (2000 nm - 1083 nm) / 1083 nm
v/c ≈ 0.8504
Since the ion is moving towards the source of radiation, the velocity v is negative. Therefore, v = -0.8504c.
Next, we can calculate the acceleration of the ion using Newton's second law:
F = ma
where F is the force on the ion and a is its acceleration. In this case, the force is given by the electric field:
F = qE
where q is the charge of the helium ion and E is the electric field. The charge of the helium ion is the charge of a single electron, e = -1.602 × 10^-19 C.
F = (-1.602 × 10^-19 C) × (2 N/C)
F = -3.204 × 10^-19 N
Now, we can use Newton's second law to find the acceleration:
-3.204 × 10^-19 N = (6.65 × 10^-27 kg) × a
a ≈ -4.822 × 10^7 m/s^2
Finally, we can use the equation of motion to determine the time it takes for the ion to absorb the infrared radiation. The equation of motion is:
Δx = v0t + (1/2)at^2
where Δx is the change in position (which is the distance the ion travels), v0 is the initial velocity (which is zero since the ion is at rest), a is the acceleration, and t is the time.
We want to solve for t, so rearrange the equation:
Δx = (1/2)at^2
Δx / [(1/2)a] = t^2
Taking the square root of both sides:
t = √[Δx / ((1/2)a)]
The change in position Δx is equal to the speed of light times the time it takes for the helium ion to absorb the radiation:
Δx = c * t
Substituting this into the previous equation:
t = √[(c * t) / ((1/2)a)]
Simplifying:
t^2 = (c * t) / ((1/2)a)
t^2 = 2(c / a) * t
t = 2(c / a)
Substituting the values:
t = 2[(3.00 × 10^8 m/s) / (-4.822 × 10^7 m/s^2)]
t ≈ 12.42 s
Therefore, in the laboratory frame, the helium ion will absorb the infrared radiation after approximately 12.42 seconds.