Question

A beam of yellow laser light \((590 \mathrm{nm})\) passes through a circular aperture of diameter \(7.0 \mathrm{mm}\). What is the angular width of the central diffraction maximum formed on a screen?

Short answer

Question: Calculate the angular width of the central diffraction maximum formed on a screen when a light beam with a wavelength of \(590\) nm passes through a circular aperture with a diameter of \(7.0\) mm. Answer: The angular width of the central diffraction maximum is approximately \(5.85 \times 10^{-3}\) degrees.

Step-by-step solution

Identify the given values

Wavelength of the light (\(\lambda\)) = \(590\) nm = \(590 \times 10^{-9}\) m (as we should convert the unit to meters) Diameter of the aperture (\(D\)) = \(7.0\) mm = \(7.0 \times 10^{-3}\) m (as we should convert the unit to meters)

Calculate the angular width of the central maximum

Using the formula, \(\theta = 1.22\frac{\lambda}{D}\), we can plug in the given values: \(\theta = 1.22 \frac{(590 \times 10^{-9} \mathrm{m})}{(7.0 \times 10^{-3} \mathrm{m})}\)

Solve for the angular width

Calculating the value, we get: \(\theta = 1.22 \frac{(590 \times 10^{-9})}{(7.0 \times 10^{-3})} = 1.02 \times 10^{-4} \mathrm{radians}\) Since the answer should be in degrees, we'll convert radians to degrees: \(\theta = 1.02 \times 10^{-4} \cdot \frac{180}{\pi} \approx 5.85 \times 10^{-3} \mathrm{degrees}\)

Write the final answer

The angular width of the central diffraction maximum formed on the screen is approximately \(5.85 \times 10^{-3}\) degrees.