Speed of Light

# Michelson’s Method to Determine the Speed of Light

An American physicist Albert A. Michelson determined the speed of light for which he received a Nobel Prize in 1907. Michelson’s method to determine the speed of light uses mirror having various faces rotating in very high speed.

The simplified version of his experiment is shown in the following figure.

Let $M_1$ be an octagonal steel mirror which can be rotated about a vertical axis through its centre with the help of a motor of variable speed. Light from a source $S$ passes through a slit $S_1$ which falls on the face $1$ of the mirror $M_1$ at an angle of $45˚$. The mirror $M_1$ reflects it to a distant concave mirror $M_2$ at a point $A$. The beam is parallel to the axis of the mirror $M_2$. Then, the mirror $M_2$ reflects it to a plane mirror $M_3$ which is at the focus of the mirror $M_2$.

The mirror $M_3$ again reflects it to a point $B$ of the mirror $M_2$ from where it is reflected parallel to the axis of the mirror $M_2$. If the mirror $M_1$ is at rest, the beam falls to the face $3$ and it will be reflected towards the observer who can see the image of the source through the eyepiece $E$.

The mirror is set into rotation with a low speed. In this case, the beam of light returning from the mirror $M_2$ to the mirror $M_1$ will not be incident at the angle of $45˚$. As a result, the image will not enter the eyepiece.

When the speed of the mirror is gradually increased and suitably adjusted, then the image will again appear in the eyepiece. This would happen when the speed of rotation is such that the adjacent face $2$ of the mirror $M_1$ is in position formerly occupied by face $1$, during which light travels from the mirror $M_1$ to mirror $M_2$ and back to mirror $M_1$.

## Mathematical Measurement of Speed of Light by Michelson’s Method

Let the distance between the mirrors $M_1$ and $M_2$ be $d$, then, the time taken by the light to travel from $M_1$ to $M_2$ and back to $M_1$ is,$t=\frac{2d}{c}\text{ __(1)}$

During this time period, each mirror of octagonal is rotated through an angle $θ$. So, the total angle rotated by all the eight mirrors will be $8θ$ which will be equal to $2π$ i.e. $8θ=2π$$θ=\frac{π}{4} \text{ __(2)}$

If $f$ is the frequency of the rotation of the mirror, then, $ω=\frac{θ}{t}$ $t=\frac{θ}{ω}$ $t=\frac{θ}{2πf} \text{ __(3)}$ [Circular Motion]

From equations $(2)$ and $(3)$, $t=\frac{π/4}{2πf}$ $t=\frac{1}{8f} \text{ __(4)}$

From $(1)$ and $(4)$, $\frac{2d}{c}=\frac{1}{8f}$ $c=16fd$

This can be used to determine the speed of light.

Michelson mounted the revolving mirror on Mount Wilson and placed a reflector on Mount San Antorio, $35$ $\text{km}$ away. He rotated the mirror with an air jet at $528$ revolutions $\text{sec}^{-1}$. He measured the frequency by comparing the rotating mirror with an electrically driven tuning fork.

Therefore, putting the value of $f=528$ $\text{rev/sec}$ and $d=35\;\text{km}=35×10^3\;\text{m}$, $c$ can be calculated. Michelson’s result for the speed of light in vacuum was $2.99910×10^8±50$ $\text{ms}^{-1}$.

Modern methods of measuring the speed of light use highly coherent and unidirectional laser beams. Because of this, the speed of light can be measured with very high degree of accuracy. The accepted value of the speed of light in free space is $2.99774×10^8$ $\text{ms}^{-1}$.

1. The distance $(d)$ between two mirrors is very large (about $35$ $\text{km}$). So, it gives an accurate measure of speed of light.
2. The distance travelled by light and the rate of rotation of the mirror can be measured accurately.
3. Light cannot reach the eyepiece directly from the source.
4. It is a null deflection method so correction in the measurement of displacement is not required.
5. The appearance of the image of the slit is abrupt which helps to determine the speed of light accurately.