I'm having trouble understanding the explanation of the Michelson Morley experiment given in Physics for Scientists and Engineers by Raymond Serway.
Serway computes the time difference between the two beams of light arriving at the telescope as follows,
$$ \Delta t = \frac{2L}{c} \left[ \left(1-\frac{v^2}{c^2}\right)^{-1}-\left(1-\frac{v^2}{c^2}\right)^{-1/2}\right]$$
And using the binomial expansion $ (1-x)^n\approx 1-nx$ for $x<<1$
He gets
$$\Delta t \approx \frac{Lv^2}{c^3}$$
I understand up to this part just fine.
The next part I don't understand:
A shift in the interference pattern should be detected when the interferometer is rotated through 90° in a horizontal plane so that the two beams exchange
roles. This rotation results in a time difference twice that given by $\Delta t \approx \frac{Lv^2}{c^3}$.
Therefore, the path difference that corresponds to this time difference is $\Delta d = c(2 \Delta t) = \frac{2Lv^2}{c^2}$
Why is the time difference twice that given by $\Delta t \approx \frac{Lv^2}{c^3}$ when the interferometer is rotated through 90 degrees?
Wouldn't the time difference be the same? By rotating the apparatus by 90 degrees, the light beam that was parallel to the ether wind is now perpendicular to it, and vice versa. I do not understand why you need to double the time difference.
This doubling seems crucial because the formula for calculating the fringe shift is
Shift = $\frac{2Lv^2}{\lambda c^2}$
and there is that factor of 2 on the numerator. But I do not understand why.
Best Answer
They've implicitly chosen a sign convention that makes $Δt>0$ in the original configuration. When you rotate the apparatus and the arms change places, the same convention gives $Δt=-\frac{Lv^2}{c^3}$. The difference between the $Δt$s is $\frac{2Lv^2}{c^3}$.
The original derivation assumed that the aether wind is parallel to one of the arms. Realistically, you wouldn't know that, so you would need to measure the fringe shift at many angles to find the amplitude of the sinusoid. (And at different times of day, if you can only rotate it in a plane.)