Rotating mirror method

Functional principle and quantities of the rotating mirror method required for the calculation

Simpler scheme of the Fizeau-Foucault apparatus and the improvement by Michelson

The rotating mirror method is a method developed by Léon Foucault in 1850/51 to measure the speed of light . He was thus able to determine the speed of light relatively precisely at 298,000 km / s.

functionality

A light source is arranged behind a projection screen with a passage opening in such a way that its light falls on a rotating mirror . From this it is directed onto a fixed mirror, from which it is reflected back onto the rotating mirror. Since the rotating mirror has continued to rotate in the meantime and is therefore at a different angle to the light beam, the light beam is no longer reflected back to the starting point (the light source opening) , but to a point next to it on the projection screen. A prerequisite is a sufficient angular speed of the mirror; so this has to rotate fairly quickly so that there is a measurable difference at all. ${\displaystyle P}$

By measuring the distance between the reflection point and the light beam at , it is possible to use the formulas listed below to determine the speed of light in the laboratory with a known rotational frequency of the mirror and known distances between the fixed mirror and the rotating mirror ( ) and the screen from the rotating mirror ( ). ${\displaystyle X}$ ${\displaystyle P}$ ${\displaystyle D}$ ${\displaystyle S}$ ${\displaystyle L}$

calculation

The light that is reflected from the rotating mirror onto the fixed mirror and from there back onto the rotating mirror covers the distance twice in that time . So: ${\displaystyle \Delta t}$ ${\displaystyle S}$

${\displaystyle 2S=c\cdot \Delta t}$

During the transit time of the light, the rotating mirror rotating with the rotation frequency has rotated through the angle : ${\displaystyle \Delta t}$ ${\displaystyle f}$ ${\displaystyle \beta }$

${\displaystyle \beta =2\pi \cdot f\cdot \Delta t}$

If you solve the first equation and insert it into the second equation, the result is the speed of light ${\displaystyle \Delta t}$ ${\displaystyle c}$

${\displaystyle c={\frac {4\pi fS}{\beta }}}$

Since the rotation of the reflected beam is caused by the rotation of the plumb bob (mirror) and the change of the angle of incidence by , the reflected light beam rotates by twice the angle according to the law of reflection . The distance between the projection screen surface and the rotating mirror also applies ${\displaystyle \beta }$ ${\displaystyle \beta }$ ${\displaystyle \alpha =2\beta }$ ${\displaystyle L}$

${\displaystyle X=L\cdot \tan(2\beta )}$

In the case of a projection screen with a cylindrical surface, the center of which lies in the axis of the rotating mirror, the following applies to small angles

${\displaystyle X=L\cdot \tan(2\beta )\approx L\cdot 2\beta }$ ,

provided the angle is given in radians . If you put the result in the above equation for , you get: ${\displaystyle \beta }$ ${\displaystyle \beta \approx {\frac {X}{2L}}}$ ${\displaystyle c}$

${\displaystyle c={\frac {8\pi \cdot f\cdot L\cdot S}{X}}}$

Further measurements and improvements

In 1879, measurements by Albert A. Michelson using the rotating mirror method showed a speed of light of 299,910 ± 50 km / s. After further improving the experimental setup, Michelson published a value of 299,853 ± 60 km / s in 1883. This value comes very close to the speed of light in a vacuum of 299,792.458 km / s.

Web links

• Speed ​​of Light Demonstration by the Foucault Method , article by Kevin McFarland on the University of Rochester website
• Speed ​​of light: Measurement using the rotating mirror method ( Memento from November 13, 2013 in the Internet Archive ), collection of lectures at Ulm University , January 18, 2002