A wire is supposed to be 0.10 millimeter in diameter. Laser light of wavelength 670 nm shining onto the wire creates a diffraction pattern on a wall that is 4.5 meters behind the wire. The central maximum of that diffraction pattern is 6.3 centimeters wide. What is the measured diameter of the wire?
To determine the measured diameter of the wire, we can use the concept of diffraction.
Diffraction refers to the bending or spreading of waves as they pass through an aperture or around an obstacle. In this case, the laser light passing through the wire creates a diffraction pattern on the wall.
The formula for the angular width (θ) of the central maximum in a single-slit diffraction pattern is given by:
θ = λ / (a)
where λ is the wavelength of the light and a is the size of the aperture (diameter of the wire in this case).
Given:
- Wavelength (λ) = 670 nm = 670 × 10⁻⁹ meters
- Angular width of the central maximum (θ) = 6.3 cm = 6.3 × 10⁻² meters
- Distance from the wire to the wall (L) = 4.5 meters
We can rearrange the formula to solve for the diameter of the wire (a):
a = λ / θ
Let's substitute the given values into the formula:
a = (670 × 10⁻⁹) / (6.3 × 10⁻²)
Now, let's calculate the value of a:
a = 670 × 10⁻⁹ / 6.3 × 10⁻²
a ≈ 10.63 × 10⁻⁷ meters
So, the diameter of the wire is approximately 10.63 × 10⁻⁷ meters, or 0.1063 micrometers.