Question

The single-slit diffraction pattern shown in the figure was produced with light from a laser. The screen on which the pattern was projected is located a distance of \(3.17 \mathrm{~m}\) from the slit. The slit has a width of \(0.555 \mathrm{~mm}\). The width of the central maximum is \(w=5.81 \mathrm{~mm}\). What is the wavelength of the laser light?

Short answer

Answer: The wavelength of the laser light is approximately \(737 \mathrm{~nm}\).

Step-by-step solution

Note down the given values

We are given the following values: - Distance between the screen and the slit, \(L = 3.17 \mathrm{~m}\) - Width of the slit, \(a = 0.555 \mathrm{~mm} = 0.555 \times 10^{-3} \mathrm{~m}\) - Width of the central maximum, \(w = 5.81 \mathrm{~mm} = 5.81 \times10^{-3} \mathrm{~m}\)

Find the angular width of the central maximum

First, we need to find the angle \(\theta\) that corresponds to the width of the central maximum. To do this, we can use the relationship between \(\theta\), \(L\), and \(w\): $$\tan\theta=\frac{w}{2L}$$ Solving for \(\theta\), we get: $$\theta=\tan^{-1}\left(\frac{w}{2L}\right)$$ Now, we can plug in the given values to find \(\theta\): $$\theta=\tan^{-1}\left(\frac{5.81\times 10^{-3}}{2\times3.17}\right)$$

Calculate the wavelength using the formula for the angular width of the central maximum

The formula for the angular width of the central maximum in single-slit diffraction is given by: $$a\sin\theta=m\lambda$$ where \(m=1\) for the central maximum. We have already found the angle \(\theta\) in step 2, so we can plug in the known values and solve for \(\lambda\): $$0.555 \times 10^{-3}\sin\left(\tan^{-1}\left(\frac{5.81\times 10^{-3}}{2\times3.17}\right)\right)=\lambda$$

Solve for the wavelength

Now we just need to plug in the values and solve for \(\lambda\): $$\lambda = \frac{0.555 \times 10^{-3}\sin\left(\tan^{-1}\left(\frac{5.81\times 10^{-3}}{2\times3.17}\right)\right)}{1}$$ Computing the values, we get \(\lambda \approx 7.37 \times 10^{-7} \mathrm{~m}\) or \(\lambda \approx 737 \mathrm{~nm}\). The wavelength of the laser light is approximately \(737 \mathrm{~nm}\).