You find an unlabeled box of fine needles and want to determ

You find an unlabeled box of fine needles, and want to determine how thick they are. A standard ruler won\\\'t do the job, since all you can tell is that each needle is less than a millimeter thick. So to find the thickness, you use the needle to poke a hole in a piece of brown construction paper. Then you arrange your 657 nm laser pointer to shine through the hole, and a circular diffraction pattern, consisting of a central bright circle surrounded by alternating dark and bright rings, appears on the wall 24.2 m away. Now you can use your ruler to measure that the central bright circle is 12.7 cm in diameter. What is the diameter of the needle?

Solution

let needle diameters is d.

distance of the pattern from the hole is D=24.2 m

distance of central bright from center =y=12.7 cm/2=0.0635 m

wavelength=lambda=657 nm

then using the formula:

y=m*lambda*D/d

and noting that here m=1.635 for first maximum intensity

d=657*10^(-9)*24.2*1.635/(0.0635)=4.0937*10^(-4) m

hence diameter of the needle is 4.0937*10^(-4) m