Answer to Question #87758 in Optics for Nicholas Legault

Question

Your teacher gives you a piece of cardboard with two pinholes on it, saying that they are separated by 205 μm ± 3%. In order to test this, you set up a double slit apparatus with the screen at a distance of 3.00 m away from the cardboard. You use a helium-neon laser (wavelength 638.2 nm) to generate an interference pattern. You measure the distance from the first to the seventh nodal line to be 56.9mm. Did your teacher give you accurate value for the separation of the pinholes?

Answer 1

The angles of the nodal lines are determined by the relation $d \sin \theta = n \lambda$, where $d$ is the distance between the pinholes, $\lambda$ is the wavelength of light, and $n = 0, 1, 2, \ldots$ For expected small angles, we can use the approximation $\sin \theta \approx \theta$. Then, the angular distance between the first and the seventh nodal line is $\theta = 6 \lambda / d$, and the corresponding distance on the screen will be $L = \theta D = 6 \lambda D / d$, where $D$ is the distance to the screen. The distance between the pinholes is measured with 3% precision, so the distance $L$, being inversely proportional to $d$, will also be determined with about 3% precision. Calculating this quantity, we have

$$L = \frac{6 \lambda D}{d} = \frac{6 \cdot \left( 638.2 \times 10^{-9}\, \text{m} \right) \cdot 3\, \text{m}}{205 \times 10^{-6}\, \text{m}} = 0.056\, \text{m} = 56\, \text{mm} \, .$$

The difference between this number and the measured value is 0.9 mm, which is 0.9/56.9 = 0.016, or 1.6% of the measured value, well within the 3% error. Thus, the teacher gave a fairly accurate value for the separation between the pinholes.

Answer: yes.