Answer :
Using the single-slit diffraction formula, Irina can calculate the diameter of the needle to be approximately 0.12 mm by applying the values λ = 657 nm, L = 22.2 m, and D = 12.2 cm to the formula (λL) / D.
Irina is attempting to calculate the diameter of a needle that she tested with a laser to create a diffraction pattern, a topic related to Physics and more specifically to the study of wave optics. The setup she uses is akin to the single-slit diffraction experiment. In this phenomenon, light waves pass through a small aperture and spread out to form a pattern of bright and dark regions on a screen.
Using the formula for single-slit diffraction:
Diameter of needle (d) can be estimated by d =
(λL) / D, where
λ is the wavelength of the laser light, L is the distance to the screen, and D is the diameter of the central bright fringe. In this case, λ = 657 nm (or 657 x 10^-9 m), L = 22.2 m, and D is 12.2 cm (or 0.122 m).
Calculating the diameter of the needle:
d = (λL) / D
d = (657 x 10^-9 m × 22.2 m) / 0.122 m
d = (657 x 10^-9 m × 22.2 m) / 0.122 m
d ≈ 0.12 mm,
assuming that the hole size is a good approximation for the needle's diameter.
It's worth noting that this is an idealized calculation and real-world factors such as the finite width of the laser beam and imperfections in the needle and the paper could affect the measurement.
To determine the diameter of the needle, Irina can apply the formula for the first dark ring in a diffraction pattern and use the measured diameter of the central bright spot to calculate the needle's diameter, resulting in approximately 264 micrometers.
To calculate the diameter of the needle used by Irina, we must understand the diffraction pattern created when laser light passes through a small aperture. This is an application of the principle of diffraction and interference. The diameter of the central bright circle, known as the Airy disc, is related to the diameter of the aperture through which the light is shining.
Let's begin the calculation by using the formula for the radius of the first dark ring in the diffraction pattern, which is given by:
r = [tex]1.22 \times \lambda \times L / D[/tex]
Where:
r is the radius of the central bright spot (Airy disc)
[tex]\lambda[/tex] is the wavelength of the laser light
L is the distance to the screen
D is the diameter of the aperture (the needle in this case)
We are given:
[tex]\lambda[/tex] = 657 nm (converted to meters, this is 6.57 x [tex]10^{-7}[/tex] m)
L = 22.2 m
The diameter of the central bright spot is 12.2 cm (converted to radius and meters, this is 0.061 m)
By rearranging the formula to solve for D:
D = [tex]1.22 \times \lambda \times L / r[/tex]
We get:
D = [tex]1.22 \times 6.57 \times 10^{-7} m \times 22.2 m / 0.061 m[/tex]
By performing the calculation, we find that:
[tex]D \approx 2.64 \times 10^{-4} m[/tex], or 264 micrometers.
This is the calculated diameter of the needle.
