Irina finds an unlabeled box of fine needles, and wants to determine how thick they are. A standard ruler will not do the job, as each needle is less than a millimeter thick. So, to find the thickness, she uses a needle to poke a hole in a piece of brown construction paper. Then, she positions a 640 nm laser pointer to shine through the hole and project a circular diffraction pattern on a wall 21.7 m away. She then uses her ruler to measure that the central bright circle is 14.2 cm in diameter. What diameter does Irina calculate for the needle?

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aristeus aristeus · Answer 1 · 2019-08-09T08:53:17+02:00

Answer:

[tex]d=\frac{1.22\times 640\times 10^{-9}\times 21.7}{7.35\times 10^{-2}}=2305.21\times 10^{-7}m[/tex]

Explanation:

The expression which represent the first diffraction minima by a circular aperture is given by [tex]d sin\Theta =1.22\lambda[/tex]--------eqn 1

The angle through which the first minima is diffracted is given by [tex]tan\Theta =\frac{y_1}{D}[/tex]---------eqn 2

As [tex]\Theta[/tex] is very small so we can write [tex]sin\Theta =tan\Theta[/tex]

So from eqn 1 and eqn 2 we can write

[tex]y_1=\frac{1.22\lambda D}{d}[/tex]--------eqn 3

Here [tex]y_1[/tex] is the position of first maxima D is the distance of screen from the circular aperture d is the diameter of aperture

It is given that diameter of circular aperture is 14.7 cm so [tex]y_1=\frac{14.7}{2}=7.35 \ cm[/tex]

Now putting all these value in eqn 3

[tex]d=\frac{1.22\lambda D}{y_1}[/tex]

[tex]d=\frac{1.22\times 640\times 10^{-9}\times 21.7}{7.35\times 10^{-2}}=2305.21\times 10^{-7}m[/tex]