The equation for missing order in the double-slit diffraction pattern→
The nature of the diffraction pattern due to the double slits depends upon the relative values of $e$ and $d$. If, however, $e$ is kept constant and $d$ is varied, then certain orders of interference maxima will be missing.
We know that, the direction of interference maxima
$(e+d)\:sin\theta=\pm n\lambda \qquad(1)$
The direction of diffraction minima
$e \: sin\theta=\pm m\lambda \qquad(2)$
Divide the equation $(1)$ by equation $(2)$
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$\frac{(e+d)}{e}=\frac{n}{m}$
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Case (I)→
If $e=d$ then
So for $m=1,2,3,....$
The $n=2,4,6,....$
Thus, the $2_{nd}, 4^{th}, 6^{th}, ...$ order interference maxima will be missing.
Case (II) →
If $e=\frac{d}{2}$ then
So for $m=1,2,3,....$
The $n=3,6,9,....$
Thus, the $3_{rd}, 6^{th}, 9^{th}, ...$ order interference maxima will be missing.
