The diameters of fine fibers can be accurately measured using interference patterns. Two optically flat pieces of glass that each have a length L are arranged with the fiber between them, as shown below. The setup is illuminated by monochromatic light, and the resulting interference fringes are observed. Suppose that L is 20.0 cm and that yellow sodium light (590 nm) is used for illumination. If 19 bright fringes are seen along this 20-cm distance, what are the limits on the diameter of the fiber? Hint: The nineteenth fringe might not be right at the end, but you do not see a twentieth fringe at all.
Bright Fringes at distances=n*lambda. n is an integer.
The Attempt at a Solution
So I took each fringe to represent a distance of 1/2 wavelengths, as it has to reach the other side and return, and took the minimum and maximum distance to be the distances constrained by what 19 fringes dictates, and what 20 fringes dictates. For nineteen fringes I took 590nm*9.5=5.605µm (answer to be in µm), and the maximum diameter to be 590nm*10=5.9µm. These did not work, and so I tried half for radius, double in case I for some reason had found the radius, yet none of these three combinations worked. What am I doing wrong?