91Ó°ÊÓ

Two slits of width $a$ and slit separation$d$ .

The intensity at angle $\mathrm{θ}$from light of wavelength$\mathrm{λ}$passing through two slits of width$a$ and separation$d$is given by

role="math" localid="1663144245716" $\mathit{I}\mathbf{=}{\mathit{I}}_{\mathbf{m}}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathbf{\left(}\frac{\mathbf{Ï€»å²õ¾\pm ²Ôθ}}{\mathbf{λ}}\mathbf{\right)}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{\left(}\frac{\mathbf{Ï€}\mathbf{a}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{θ}}{\mathbf{λ}}\mathbf{\right)}}{{\mathbf{\left(}\frac{\mathbf{Ï€}\mathbf{d}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{θ}}{\mathbf{λ}}\mathbf{\right)}}^{\mathbf{2}}}$ …(¾±)

The intensity at angle $\mathrm{θ}$ from light of wavelength$\mathrm{λ}$passing through a single slit of width$a$ is given by

$\mathit{I}\mathbf{=}{\mathit{I}}_{\mathbf{m}}\frac{{\mathbf{sin}}^{\mathbf{2}}\mathbf{\left(}\frac{\mathbf{Ï€²¹²õ¾\pm ²Ôθ}}{\mathbf{λ}}\mathbf{\right)}}{{\mathbf{\left(}\frac{\mathbf{Ï€»å²õ¾\pm ²Ôθ}}{\mathbf{λ}}\mathbf{\right)}}^{\mathbf{2}}}$ …(¾±¾±)

Here, $I_m$is the intensity of the central maxima.

For $d=a$equation (i) becomes

$$\begin{gathered} I=I_m{\cos }^2\left(\frac{\mathrm{Ï€²¹²õ¾\pm ²Ôθ}}{\mathrm{λ}}\right)\frac{{\sin }^2\left({\displaystyle \frac{\mathrm{Ï€²¹²õ¾\pm ²Ôθ}}{\mathrm{λ}}}\right)}{{\left(\frac{\mathrm{Ï€²¹²õ¾\pm ²Ôθ}}{\mathrm{λ}}\right)}^2} \\ =I_{m}4{\cos }^2\left(\frac{\mathrm{Ï€²¹²õ¾\pm ²Ôθ}}{\mathrm{λ}}\right)\frac{{\sin }^2\left({\displaystyle \frac{\mathrm{Ï€²¹²õ¾\pm ²Ôθ}}{\mathrm{λ}}}\right)}{4{\left(\frac{\mathrm{Ï€²¹²õ¾\pm ²Ôθ}}{\mathrm{λ}}\right)}^2} \\ =I_m\frac{{\sin }^2\left({\displaystyle \frac{\mathrm{Ï€}2\mathrm{²¹²õ¾\pm ²Ôθ}}{\mathrm{λ}}}\right)}{{\left(\frac{\mathrm{Ï€}2\mathrm{²¹²õ¾\pm ²Ôθ}}{\mathrm{λ}}\right)}^2} \end{gathered}$$

This is equal to equation (ii) with$a→2a$.

Thus the double slit diffraction pattern reduces to a single slit diffraction pattern of slit width $2a$ .