91影视

Equation for the first minimum of the dark fringes:

$\mathbf{a}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{胃}\mathbf{=}\mathbf{尘位}$

Intensity is given by

$\mathbf{I}\mathbf{=}{\mathbf{I}}_{\mathbf{0}}\left({\cos }^2\frac{蠒}{2}\right){\left(\frac{\sin {\displaystyle }\frac{尾}{2}}{\frac{尾}{2}}\right)}^{\mathbf{2}}$

Where

$\mathbf{尾}\mathbf{=}\frac{\mathbf{2}\mathbf{蟺补}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{胃}}{\mathbf{位}}$

Since the bright fringe contains5 fringes, so the minimum fringe that makes an interference of total dark is for $\mathit{m}\mathbf{=}\mathbf{卤}\mathbf{3}$.

The equation is given by

$\mathit{a}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{胃}\mathbf{=}\mathit{m}\mathit{位}$

Hence,

$\mathit{a}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{胃}\mathbf{=}\mathit{位}$ (1)

While it is the third bright fringe in the double-slit pattern,

$\mathit{d}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{胃}\mathbf{=}\mathbf{3}\mathit{位}$ (2)

Now, the intensity is given by

$\mathbf{I}\mathbf{=}{\mathbf{I}}_{\mathbf{0}}\mathbf{(}{\mathbf{cos}}^{\mathbf{2}}\mathbf{}\frac{\mathbf{蠒}}{\mathbf{2}}\mathbf{)}{\mathbf{\left(}\frac{\mathbf{sin}{\displaystyle }\frac{\mathbf{尾}}{\mathbf{2}}}{\frac{\mathbf{尾}}{\mathbf{2}}}\mathbf{\right)}}^{\mathbf{2}}$

Where

$\mathit{尾}\mathbf{=}\frac{\mathbf{2}\mathbf{蟺补}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{胃}}{\mathbf{位}}$

Thus,

$\mathbf{I}\mathbf{=}{\mathbf{I}}_{\mathbf{0}}\mathbf{(}{\mathbf{cos}}^{\mathbf{2}}\mathbf{}\frac{\mathbf{蠒}}{\mathbf{2}}\mathbf{)}{\mathbf{\left(}\frac{\mathbf{sin}{\displaystyle }\frac{\mathbf{蟺补}\mathbf{}\mathbf{蝉颈苍胃}}{\mathbf{位}}}{\frac{\mathbf{蟺补}\mathbf{}\mathbf{蝉颈苍胃}}{\mathbf{位}}}\mathbf{\right)}}^{\mathbf{2}}$

Now, for $\mathit{m}\mathbf{=}\mathbf{卤}\mathbf{3}$, the intensity is zero.

$$\begin{gathered} \mathbf{0}\mathbf{=}{\mathit{I}}_{\mathbf{0}}\left(co{s}^2\frac{蠒}{2}\right){\left(\frac{sin\frac{蟺asin胃}{位}}{\frac{蟺asin胃}{位}}\right)}^{\mathbf{2}} \\ \mathbf{鈬�}\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\frac{\mathbf{蟺}\mathbf{a}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{胃}}{\mathbf{位}}}{\frac{\mathbf{蟺}\mathbf{a}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{胃}}{\mathbf{位}}}\mathbf{=}\mathbf{0} \\ \mathbf{鈬�}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\frac{\mathbf{蟺}\mathbf{a}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{胃}}{\mathbf{位}}\mathbf{=}\mathbf{0} \end{gathered}$$

Plug the value,

$\begin{array}{c}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\left(\frac{3蟺asin胃}{dsin胃}\right)\mathbf{=}\mathbf{0}\\ \mathbf{鈬�}\frac{\mathbf{3}\mathbf{a}}{\mathbf{d}}\mathbf{=}\mathbf{1}\\ \mathbf{鈬�}\mathbf{d}\mathbf{/}\mathbf{a}\mathbf{=}\mathbf{3}\mathbf{.0}\end{array}$

Now, the fringe for the central maximum fringe are for

$\mathit{m}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{卤}\mathbf{1}\mathbf{,}\mathbf{卤}\mathbf{2}$

So,

$\mathit{a}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{胃}\mathbf{=}\mathbf{2}\mathit{位}$ (3)

And

$\mathit{d}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{胃}\mathbf{=}\mathit{m}\mathit{位}$ (4)

Divide (4) by (3) and plug d /a,

$$\begin{gathered} \frac{\mathbf{d}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{胃}}{\mathbf{a}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{胃}}\mathbf{=}\frac{\mathbf{m}\mathbf{位}}{\mathbf{2}\mathbf{位}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{鈬�}\frac{\mathbf{m}}{\mathbf{2}}\mathbf{=}\mathbf{3} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{鈬�}\mathit{m}\mathbf{=}\mathbf{卤}\mathbf{6} \end{gathered}$$

This means that the second bright fringe envelope is contained between$\mathit{m}\mathbf{=}\mathbf{卤}\mathbf{3}$ and $\mathit{m}\mathbf{=}\mathbf{卤}\mathbf{6}$.

Implies that this envelope only contains two bright fringes which are for $\mathit{m}\mathbf{=}\mathbf{卤}\mathbf{4}\mathbf{,}\mathbf{卤}\mathbf{5}$.

Thus, the ratio d /a must be 3.0. There are3.0 fringes contained with the first diffraction maximum on one side of the central maximum.