In a Michelson interferometer, in order to shift the pattern by half a fringe, one of the mirrors at the end of an arm must be moved by:
A)one wavelength
B)half a wavelength
C)one-quarter wavelength
D)It depends on the wavelength.
E)It depends on which mirror is moved.

Question

Quizplus · Accepted Answer

In a Michelson interferometer, the phase difference between the two beams is given by:

$\Delta\varphi = \frac{4\pi}{\lambda}(x_2-x_1)$

where $\lambda$ is the wavelength of the light, $x_2$ is the distance traveled by the light in the arm with the movable mirror, and $x_1$ is the distance traveled by the light in the arm with the fixed mirror.

When the movable mirror is moved by a distance $\Delta x$, the phase difference changes by:

$\Delta\varphi = \frac{4\pi}{\lambda}(\Delta x)$

To shift the pattern by half a fringe, we need to change the phase difference by $\frac{\pi}{\lambda}$. Therefore:

$\frac{\pi}{\lambda} = \frac{4\pi}{\lambda}(\Delta x)$

$\Delta x = \frac{1}{4}\lambda$

This means that to shift the pattern by half a fringe, we need to move the mirror by a distance equal to one-quarter of the wavelength. Therefore, the correct answer is (C).

In a Michelson Interferometer, in Order to Shift the Pattern