Lloyd's Single Mirror and Fresnel's Bi-prism

LLOOYD'S SINGLE MIRROR EXPEREIMENT:

Lloyd's single mirror consist of a glass plate having length about 30 cm and width of 6 to 8 cm and capable of reflecting light energy only from upper surface. Instead thin kind of plate, a kind of piece of polished metal can also be used to perform the experiment. Here, the light waves directly coming from the source and the waves reflected from the single mirror are interfering beam. The wave that reflect from the surface produces an extra path difference of \(\frac{\lambda}{2}\) or equivalent phase difference of \(\pi\). We know when light wave reflect from denser surface to rarer medium, there exists an extra path difference of \(\frac{\lambda}{2}\). Due to this extra path difference, the central point appears dark instead of being bright. In both side of the central dark band, alternative arrangement of bright and dark fringes are produced. In this experiment, the point s and it's virtual image s' are taken as the coherent sources. The position of bright and dark bands can be calculated by using the same mathematics as in the Young's double slit experiment. In this experiment, the bright fringes and dark fringes are evenly spaced.

$$i.e.\;\; \beta=\frac{\lambda D}{d}$$

FRESNEL'S BIPRISM EXPERIMENT:

Fresnel's used a biprism experiment to demonstrate the wave nature of light.. A biprism is made by combining two acute angle prism by placing base to base, then the biprism has obtuse angle of 17\(9^\circ\) and remaining acute angle on both side are \(\frac{1}{2}^\circ\) or 30'. The light wave incident on the upper surface refracts downwards and seems to be coming from the point source \(S_1\), which is virtual. Similarly, the light wave incident on the lower surface refracts upwards and seems to be coming from the point source \(S_2\), which is also virtual. The virtual point \(S_1\) and \(S_2\) are two coherent source and the light wave coming from the point source \(S_1\) and \(S_2\) are interfering on the screen. At the center of the screen, there is no path difference so that a bright image is produced. In both side of central image alternative arrangement of dark and bright fringes are observed. The position of image from central bright point can be observed by using the mathematics as in the Young's double slit experiment. The bright fringes and dark fringes are equally spaced, which is given by $$\beta=\frac{\lambda D}{d}$$

DETERMINATION OF WAVE LENGTH OF LIGHT BY USING FRESNEL'S BIPRISM:

In the case of Fresnel's Biprism, the bright fringes and dark fringes are equally spaced. The separation between two successive bright fringes or dark fringes is called fringe width and it is given by $$\beta=\frac{\lambda D}{d}$$

$$or,\;\;\;\lambda=\frac{\beta d}{D}\dotsm(1)$$

Where, \(\lambda\) = wave length

d = slit separation

D = distance between slit and screen

• Determination of D :

The distance between slit and screen is relatively large distance. So, it can be measure with the help of meter scale provided in the optical bench.

• Determination of \(\beta\) :

To determine the fringe width \(\beta\), A system of wire known as cross wire is inserted in the field of view and fringes are made on the cross-wire. Initially, a bright image is identified with the help of cross-wire and N number of bright fringe are displaced through the identified position by rotating external scale provided. Let, the external scale is displaced from \(x_\circ\) to \(x_N\) due to N number of fringes, we know, the fringes are of equal width.

Fringe width \(\beta\) = \(\frac{Total displacement}{Total number of fringes}\)

$$=\frac{X_N-X_\circ}{N}$$

• Determination of d :

The slit separation is obtained by displacement method. In this method, a convex lens is introduced in between the Biprism at two different position. Initially, the lens is placed near of the Biprism as shown in the figure below.

In this case, two light image of the point source \(S_1\) and \(S_2\) are observed at the separation d, so, we can write ,

$$\frac{d'}{d}=\frac{v}{u}\dotsm(1)$$

Where; v = image distance

And u = object distance.

Secondly, the convex lens is displaced towards the screen so that v' = u and u' = v as shown in figure.

FIGURE HERE:

In this case also, two line images are produced on the screen at the separation \(d_2\),\. So,

$$\frac{d_2}{d}=\frac{u}{v}\dotsm(2)$$

Now, from equation (1) and (2),

$$d=\sqrt{d_1 d_2}\dotsm(3)$$

The equation (3) is used to calculate the slit separation.

Alternatively,

Let \(\alpha\) = angle of prism

S = deviation produced by each ray

Then from geometry of the figure as in the experimental arrangement.

$$d=2\alpha(\mu-1)y_1$$

Where, \(y_1\) = displacement between the slit and Biprism

\(\mu\) = Refractive index of Biprism.

References:

Adhikari, P.B, Daya Nidhi Chhatkuli and Iswar Prasad Koirala. A Textbook of Physics. Vol. II. Kathmandu: Sukunda Pustak Bhawan, 2012.

Jenkins, F.A and H.E White. Fundamental of optics. New York (USA): McGraw-Hill Book Co, 1976.

wood, R.W. Physical Optics. New York (USA): Dover Publication , 1934.