You find an unlabeled box of fine needles, and want to determine how thick they

Question

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 21.7 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?

Explanation / Answer

here by using the formula

d * sin(theta) = (m + 0.5 ) * lambda

d * ( y / L) = ( m + 0.5 ) * lambda

for the central fringe , m = 0 ,then

d * (a / 2*L) = lambda / 2

d = L * lambda / a

then put the values in this formula

d = ( 21.7 * 657 * 10^-9 ) / ( 12.7 * 10^-2)

d = 11.2259 * 10^-5 m

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