To tackle your question about the helium-neon laser emitting monochromatic light, we can break it down into three parts: calculating the energy and momentum of each photon, determining the number of photons emitted per second, and finding the speed of a hydrogen atom that would match the momentum of a photon. Let’s dive into each part step by step.
Calculating Energy and Momentum of Each Photon
The energy of a photon can be calculated using the formula:
E = h * f
Where:
- E is the energy of the photon.
- h is Planck's constant, approximately 6.626 x 10^-34 J·s.
- f is the frequency of the light.
First, we need to find the frequency of the light using the relationship between wavelength and frequency:
f = c / λ
Where:
- c is the speed of light, approximately 3.00 x 10^8 m/s.
- λ is the wavelength, which is given as 632.8 nm (or 632.8 x 10^-9 m).
Now, substituting the values:
f = (3.00 x 10^8 m/s) / (632.8 x 10^-9 m) ≈ 4.74 x 10^14 Hz
Next, we can find the energy of each photon:
E = (6.626 x 10^-34 J·s) * (4.74 x 10^14 Hz) ≈ 3.14 x 10^-19 J
Now, for the momentum of a photon, we use the formula:
p = E / c
Substituting the energy we just calculated:
p = (3.14 x 10^-19 J) / (3.00 x 10^8 m/s) ≈ 1.05 x 10^-27 kg·m/s
Determining the Number of Photons Emitted per Second
To find the number of photons emitted per second, we can use the power of the laser:
Power (P) = Energy per photon (E) * Number of photons per second (N)
Rearranging gives us:
N = P / E
Substituting the values:
N = (9.42 x 10^-3 W) / (3.14 x 10^-19 J) ≈ 3.00 x 10^16 photons/s
Finding the Speed of a Hydrogen Atom with Equivalent Momentum
To find the speed of a hydrogen atom that has the same momentum as the photon, we can use the momentum formula:
p = m * v
Where:
- m is the mass of the hydrogen atom, approximately 1.67 x 10^-27 kg.
- v is the speed we want to find.
Rearranging gives us:
v = p / m
Substituting the values:
v = (1.05 x 10^-27 kg·m/s) / (1.67 x 10^-27 kg) ≈ 0.63 m/s
Summary of Findings
In summary, we found that:
- The energy of each photon is approximately 3.14 x 10^-19 J.
- The momentum of each photon is about 1.05 x 10^-27 kg·m/s.
- The laser emits around 3.00 x 10^16 photons per second.
- A hydrogen atom would need to travel at approximately 0.63 m/s to match the momentum of a photon.
This analysis illustrates the fascinating interplay between light and matter, showcasing how fundamental physics principles apply to real-world scenarios like laser technology.
