Diffraction by a Single Slit: Locating the Minima — Question 8

Question 8

Thermal widening moves a diffraction minimum toward the axis

A slit width obeys $a(T)=a_0[1+\alpha\Delta T]$, with $\alpha=2.0\times10^{-5}\,\mathrm{K^{-1}}$. A fixed-order minimum initially lies at $\theta=30^\circ$. The wavelength remains constant and the slit warms by $10\,\mathrm K$. Use first-order changes.

1. Differentiate the minimum condition to find the angular shift.

2. If that minimum initially lies at $y=0.500\,\mathrm m$ on a fixed screen, find its transverse shift without making a small-angle approximation.

Question 8 - Solution

Thermal widening moves a diffraction minimum toward the axis

(a) Method and calculation

$$a\sin\theta=m\lambda\Longrightarrow\sin\theta\,da+a\cos\theta\,d\theta=0.$$$$\boxed{\Delta\theta\simeq-\alpha\Delta T\tan\theta=-1.15470\times10^{-4}\,\mathrm{rad}}.$$

(b) Interpretation and further calculation

$$y=L\tan\theta\Longrightarrow\Delta y=L\sec^2\theta\Delta\theta=-\alpha\Delta T\,y\sec^2\theta.$$$$\boxed{\Delta y=-0.133333\,\mathrm{mm}}.$$

Physical check

A wider opening narrows the pattern. Retaining the secant-squared factor matters when the observed angle is not small.