User:MichaelK/Measuring c

The speed at which radiation propagates through a vacuum is exactly 299,792,458 m/s. We measure the speed of light in this experiment by determining how much a mirror that is rotating at a known speed has rotated during the time it takes a beam of light to travel out to a known distance and back. We use an apparatus specifically manufactured for the purpose of measuring the speed of light. This device consists of a motor that spins a mirror at a very high rate, a laser, and several mirrors and lenses to focus and direct the beam. A laser beam is shown through a lens L₁ which focuses it at a point at position s, then through a beam-splitter which deflects some of the radiation into a measuring microscope. The rest of the radiation is reflected by a rotating mirror R_m to a fixed mirror F_m 13 meters away. This mirror reflects the beam back to the rotating mirror which, during the time the beam has propagated from the fixed mirror, has rotated a certain amount. This amount depends on the time it takes for the beam of light to reach the fixed mirror and return, which is given by the equation <math>t=\frac{D}{c}</math>, where t is the elapsed time, D is the distance between the fixed mirror and the rotating mirror, and c is the speed of light. The beam is then deflected by the beam-splitter 90 degrees to a measuring microscope at point s’. The beam splitter is located at the same distance from points s and s’. When the laser beam strikes the rotating mirror, it is deflected at an angle which is equal to 2θ, where θ is the angle of deflection of the mirror. Point S₁ is where the beam is reflected onto the fixed mirror F_m, and S is the point where the virtual image of the beam appears to have been reflected onto the virtual image of the fixed mirror after the rotating mirror has rotated during the time it took the beam to travel from the rotating mirror to the fixed mirror and back again. The distance between S₁ and S is ∆S. Due to the focusing of the beam by lens L1 and the spherical shape of the fixed mirror, the apparent displacement of the beam due to rotation when viewed from points s or s’ will be (-i/o)∆S, where i is the distance of the lens from the image and o is the distance of the lens from the object. The distance of the lens from the image is the distance between lens L₂ and lens L₁ minus the focal length of L₁, and the distance of the lens from the object is the distance between the rotating mirror (R_m) and the fixed mirror (F_m) added to the distance between lens L₂ and the rotating mirror. Therefore, <math>-i/o=\frac{A}{D+B}</math>. So the distance between location at which the beam actually shines onto the fixed mirror and the location of the point the beam appears to shine onto the fixed mirror in the virtual image reflected from the rotating mirror when viewed from point s or point s’ is <math>\Delta s=\Delta s'=\frac{A}{D+B}\Delta s</math>. This distance is determined by measuring the difference in the position of the beam at point s’ when the mirror is rotating clockwise and when the mirror is rotating counter-clockwise. Since a beam of radiation would undergo a displacement of 2D∆θ on the fixed mirror if the rotating mirror were rotated by an angle of θ, <math>\Delta s'=sD\Delta\theta \frac{A}{D+B}</math>. Since ∆θ = 2ωt if ω is constant, and <math>t=\frac{D}{c}</math>, then <math>\Delta\theta=\frac{2D\omega}{c}</math>. So <math>c=\frac{4AD^2\omega}{(D+B)\Delta s'}</math>. The device measures the rotational speed in revolutions per second, so that value must be multiplied by 2π before being entered into the equation. The reason that the beam appears as a dot in the microscope instead of as a line is because of the spherical shape of the mirror that is thirteen meters away.