Answer
33 dark bands.
Work Step by Step
The wedge is now filled with water, so we take the hint given in the problem and use the wavelength in the water. Note that the index of refraction of water, like that of air, is less than that specified for the glass. Therefore, we can simply follow along with Example 24-10 because the phases changes at all interfaces are the same.
First, calculate the number of wavelengths at the location of the wire, for a round trip down and then up through the water.
$$\frac{2t}{\lambda_{water}}=\frac{2(7.35\times10^{-6}m)}{6.00\times10^{-7}m/1.33}=32.6\;wavelengths$$
There will be 33 dark bands in all, including the one at the point of contact. These correspond to path length differences of $0\lambda, 1\lambda,…,32\lambda$.
