= Solution
Illuminate a narrow entrance slit with the <mercury-vapor lamp>, use a collimating <lens> to illuminate two parallel narrow slits coherently, and insert a bandpass filter selecting the green line. Place a screen or scanning detector a measured distance $D$ beyond the double slit. The source slit and the two apertures should be parallel. The aperture widths should be appreciably smaller than their separation to provide an extended <diffraction> envelope; the green filter should reject the other strong mercury lines. The layout is drawn below.
For slit separation $d$ and observation angle $\theta$, the <optical path difference> in air is $\Delta=d\sin\theta$. At a bright <interference fringe>, $\Delta=m\lambda$. For a distant screen and small angles, $\sin\theta\simeq\tan\theta\simeq y/D$, giving
$$
y_m\simeq\frac{m\lambda D}{d},\qquad \boxed{\beta=y_{m+1}-y_m=\frac{\lambda D}{d}.}
$$
Measuring several <interference fringe> spacings and dividing by their number reduces relative position error. The derivation assumes the slits have equal <phase> on illumination and that the screen region lies within their common <diffraction> envelope.
\Image[/past-exam-of-the-physics-2-paper-of-the-university-of-oxford/2019-mercury-double-slit-fringes.png]
{title=Double-slit layout and two-colour fringes across the central thirty millimetres}
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