Question

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 657 nm laser pointer to shine through the hole and project a circular diffraction pattern on a wall 22.2 m away. She then uses her ruler to measure that the central bright circle is 12.2 cm in diameter. What diameter does Irina calculate for the needle?

Answer 1

To solve this problem we will first determine the position of the first minimum and then use the Rayleigh for resolution criterion. The criterion basically says that the diameter is equivalent to 1.22 the product of the wavelength by the circular diffraction pattern at the rate of the first minimum. The first minimum for its part is the middle of the central bright circle. So, position of first minima,

$y = \frac{12.2}{2} = 6.1 cm$

Diameter of needle (formula from Rayleigh's criteria for resolution),

$d = 1.22 \frac{\lambda D}{y}$

$d = 1.22 \frac{(657*10^{-9}) (22.2)}{ 6.1*10^{2}}$

$d = 291.7\mu m$

Therefore the diameter that Irina calculate for the needle is $291.7\mu m$