Young's Interference Experiment — Question 10

Question 10

A uniform medium and a changed slit spacing can preserve a fringe pattern

An experiment in air uses wavelength $\lambda_0=600\,\mathrm{nm}$, slit spacing $d_1=0.40\,\mathrm{mm}$, and screen distance $L_1=2.0\,\mathrm m$. It is rebuilt entirely in a uniform medium of index $n=1.50$ with $L_2=3.0\,\mathrm m$. Neglect dispersion.

1. Find the new slit spacing needed to preserve the paraxial fringe spacing.

2. If instead the old slit spacing is retained, find the spacing ratio and determine whether the full angular bright-fringe pattern is also preserved.

Question 10 - Solution

A uniform medium and a changed slit spacing can preserve a fringe pattern

(a) Method and calculation

$$\Delta y_1=\lambda_0L_1/d_1,\qquad\Delta y_2=\lambda_0L_2/(nd_2).$$$$\Delta y_2=\Delta y_1\Longrightarrow\boxed{d_2=d_1L_2/(nL_1)=0.40\,\mathrm{mm}}.$$

(b) Interpretation and further calculation

$$d_2=d_1:\quad\boxed{\Delta y_2/\Delta y_1=L_2/(nL_1)=1};\quad\Delta y_1=3.00\,\mathrm{mm}.$$$$\sin\theta_{m,2}=\frac{m\lambda_0}{nd_1}=\frac{\sin\theta_{m,1}}{n}\Rightarrow\boxed{\text{angular pattern contracts}}.$$

Physical check

Equal paraxial screen spacing can conceal unequal angular spacing; exact large-angle screen positions would not remain identical.