All Problems Interference of Light
Problem 5.94
In Michelson's interferometer the yellow sodium line composed of two wavelengths \(\lambda_{1}=589.0 \mathrm{nm}\) and \(\lambda_{2}=589.6 \mathrm{nm}\) was used. In the process of translational displacement of one of the mirrors the interference pattern vanished periodically (why?). Find the displacement of the mirror between two successive appearances of the sharpest pattern.
5.94. The transition from one sharp pattern to another occurs if the following condition is met: \[ (k+1) \lambda_{1}=k \lambda_{2} \] where \(k\) is a certain integer. The corresponding displacement \(\Delta h\) of the mirror is determined from the equation \(2 \Delta h=k \hat{\lambda}_{2} .\) From these two equations we get \[ \Delta h=\frac{\lambda_{1} \lambda_{2}}{2\left(\lambda_{2}-\lambda_{1}\right)} \approx \frac{\lambda^{2}}{2 \Delta \lambda}=0.3 \mathrm{~mm} \]