Measurement of Velocity of Light

Velocity of Light

Many experiments were designed to measure the velocity of light. The first attempt was made by Galileo in 1600. Galileo attempted to measure the velocity of light by covering and uncovering a lantern at night and timing how long the light took to reach an observer a few miles away. As the velocity of light is very large, the time was taken to transverse the distance was too small to measured and thus this method failed to measure the velocity of light. The first successful attempt to measure the velocity of light was made by Romer, observing the eclipses on a satellite of Jupiter. However, the first measurement of the velocity of light was made by Fizeau in 1849.

Foucault ‘s Method for the Velocity of Light

The rays of light from a source S are allowed to fall on a convex lens L, after passing through the glass plate P. these rays are converged to the point I by the lens L. If a mirror M­₁ is placed at A as shown in figure, light after reflection from the mirror M₁, converges at the pole of the concave mirror C, whose distance from A is adjusted such equal to its radius of curvature. Light is reflected back from C along its original path and finally the image is formed on the source S. As there is a half silvered glass plate P inclined at an angle of 45⁰ C to the axis of the lens, the image if formed at B₁, which can be viewed with the help of micrometer eyepiece.

Suppose M₁ is rotated at the uniform angular speed about an axis passing through A and the rays after reflection from the concave mirror C, find the plane mirror displaced by an angle θ to a new position M₂. The image is now observed at B₂. The displacement B­₁B₂ of the image is measured and velocity of light is calculated

Theory

Consider a point E on a concave mirror from which light is reflected back to the rotating mirror. When the plane mirror is at the position M₁, the rays reflected from it to the lens L seem to come from I which is an image E in M₁. When it has rotated through an angle θ, the rays reflected by it to the lens L appears to come from I₁ which is the image of E in the new position M₂ of the mirror. The distance is set as AE = AI = a, where a is a radius of curvature of the concave mirror. Thus, we find that A acts as the centre of curvature of the concave mirror.

If the mirror M₁ is turned through an angle θ, the reflected ray is turned through an angle 2θ and hence I₁ AI = 2θ. So,

\begin{align*} \therefore II_1 = a \times 2\theta = 2a\theta \\ \text {As} \:S \: \text { and } S_1 \: \text {are conjugate points with respect to} \: I\: \text {and } \: I_1, \text {for the lens L} \\ \frac {S_1S}{l} = \frac {II_1}{(a + b)} \\ \text {or,} \: S_1S &= \frac {2a\theta \times l}{(a + b)} \\ \end{align*} where l is distance between the lens and source S and b is the distance between the lens and the mirror M.\begin{align*} \text {Suppose,} S_1S = BB_2 = y. \\ \text {Then,} \\ y &= \frac {2a\theta \times l}{(a + b)} \\ \text {or,} \: \theta &= \frac {y(a + b)}{2al} \dots (i) \\ \end{align*} If n be the number of revolutions made per second by the rotating mirror, then the time taken by the plane mirror to rotate through an angle\(\: \theta \: \text {is} \)\begin{align*} t = \frac {theta }{2\pi n} \dots (ii) \\ \text { If the velocity of light is c,} \\ \text {the time taken by the light to covered distance 2a,} \\ \end{align*}

\begin{align*}\text {i.e. from A to E and back to A is given by} \\ \therefore t = \frac {2a}{c} \dots (iii) \\ \text {Therefore from equations} \: (ii) \: \text {and} \: (iii) \: , \text {we have} \\ \frac {2a}{c} = \frac {theta }{2\pi n} \\ \text {or,} \: \theta = \frac {4\pi na}{c} \dots (iv) \\ \text {From equations} \: (ii) \: \text {and} \: (iii) \: , \text {we have}\\ \frac {y(a + b)}{2al} = \frac {4\pi na}{c} \\ \therefore c = \frac {8\pi na^2l}{y(a+b)} \\ \end{align*}

As n, a, l, y and b are measurable quantities, the speed of light c can be calculated. The value of light found by Foucault was 2.98×10⁸ m/s.

Advantage

1. It can be used to measure the velocity of light in a liquid by putting it in a tube that is placed between two mirrors.
2. Space required in this required in this experiment is quite small and hence it may be performed inside a laboratory.

Drawbacks

1. The shift in the image is very small.
2. The image obtained is not very bright due to reflection and refraction of light at various surfaces.

Michelson’s Method for the Velocity of Light

An octagonal mirror M₁ is mounted on the shaft of a variable speed motor. Light from a bright source S is focused at an angle of 45⁰ on one of the faces of mirror M₁ after passing through a slit S₁. The reflected light falls on a distant concave mirror M₂. In the figure, M₃ is a plane mirror and with the help of this mirror M₃ plane mirror and with the help of the mirror M₃ placed at the centre of curvature of mirror M₂ the beam of light is returned back and falls on face 3 of the octagonal mirror M₁ again at an angle of 45⁰. The light reflected by this face is then collected by a telescope T and the eye at the position. At the rest position of the mirror M+1+, an image of the light source can be observed in the telescope.

If the mirror M₁ is rotated, the light returning to it from the mirror M₂ will not be incident at an angle of 45⁰ , and hence will not enter the telescope. When the speed of rotation of mirror M₁ is so adjusted that the face 2 of mirror occupies exactly the same position as was occupied by face 3 earlier during the time travels from M₁ to M₂ and back to M₁, then the image of a source will reappear.

If d be the distance between the mirror M₁ and M₂ and c be the speed of light, then the time taken by the light to travel from M₁ to M₂ and back to M₁ is

\begin{align*} t = \frac {2d}{c} \\\end{align*} If f is the number of revolutions per second of mirror M₁and m is the number of faces of this mirror then the angle rotated by the mirror during the time t is \begin{align*}\theta = \frac {2\pi }{m} \\ t = \frac {\theta }{2 \pi f} = \frac {2\pi }{2\pi f m} = \frac {1}{mf} \\ \text {or,} \: \frac {2d}{c} = \frac {1}{mf} \\ \therefore c &= 2mfd \\ \end{align*}

The value obtained is 2.99775×10⁸ m/s.

Advantage of Michelson’s Method

1. The distance d between the two stations is very large.
2. This null method and so, their measurement of the image.
3. The image is very bright so the result is also accurate.

Disadvantage of Michelson’s Method

1. It is very difficult to keep the high speed of rotation of the mirror constant for a long time.
2. At high speed, the rotating may break. But speed can be reduced by increasing the number of faces of the mirror.

References

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.