CBSE Class 12 Physics Optics Exam Notes

CBSE Class 12 Physics Optics Exam Notes. Please refer to the examination notes which you can use for preparing and revising for exams. These notes will help you to revise the concepts quickly and get good marks.

OPTICS

Optics – exam notes

REFLECTION

Laws of reflection

1. The incident ray, the reflected ray and the normal at the point of incidence lie in the same plane.
2. The angle of incidence is equal to the angle of reflection, i.e. the incident ray and the reflected ray make the same angle with the normal at the point of incidence.

Characteristics of images formed by a plane mirror

1. The image of a real object formed by reflection from a plane mirror is virtual, erect and laterally inverted.
2. The object and the image are on opposite sides of the mirror, and their distances from the mirror are the same.
3. The size of the image is equal to the size of the object.
4. If the plane mirror is rotated through a certain angle, the reflected ray for any fixed incident ray is rotated through double the angle. If the mirror is moved through a certain distance towards or away from the object, the image moves through double the distance respectively.
5. When two plane mirrors are inclined at an angle \(\theta\) and an object is placed between them, multiple images are formed due to successive reflections. (i) If \(\dfrac{360^\circ}{\theta}\) is even, the number of images is \(n=\dfrac{360^\circ}{\theta}-1\). (ii) If \(\dfrac{360^\circ}{\theta}\) is odd and the object lies unsymmetrically, \(n=\dfrac{360^\circ}{\theta}\). (iii) If the object is placed at the angle bisector of the two mirrors, the number of images reduces by one in both cases (i) and (ii). (iv) If the two mirrors are parallel to each other, an infinite number of images are formed, of diminishing intensity due to absorption of light in the multiple reflections.
6. The velocity of the object with respect to the mirror is the negative of the velocity of the image with respect to the mirror.

SPHERICAL MIRRORS

A spherical mirror is a smooth reflecting surface which forms a part of a spherical surface. If reflection takes place from the inner reflecting surface, the mirror is called a concave mirror. If the reflection takes place from the outer surface, it is called a convex mirror.

The reflection of light from a concave or convex mirror is shown in the figure.

Parameters associated with a mirror. The pole (vertex) P is the geometrical centre of the reflecting surface. The centre of curvature C is the centre of the sphere of which the mirror forms a part. The radius of curvature R is the distance between the pole and the centre of curvature. The principal axis is the line CP passing through the pole and the centre of curvature. If a ray of light is incident from an object at infinity, so that the beam is parallel to the principal axis, the image is formed at the principal focus F after reflection. The focal length f is the distance between the pole P and the focus F along the principal axis. When a beam of light is incident parallel to the principal axis, the reflected rays converge to the point F for a concave mirror and appear to diverge from a point F for a convex mirror. The aperture of a spherical mirror is the effective diameter MM′ of the light-reflecting area of the mirror. For a mirror of small aperture, the focal length is half the radius of curvature.

Sign convention (based on coordinate geometry)

1. The rays of light are considered to travel from left to right.
2. All distances are measured from the pole. Distances in the direction of light, to the right of the pole, are taken as positive; distances measured opposite to the direction of light, to the left of the pole, are taken as negative.
3. Transverse distances above the principal axis are taken as positive and those below the principal axis as negative.
4. If condition (1) is followed, this sign convention follows the right-handed Cartesian coordinate system.

Rules for image formation. These rules are used for locating the image of an object by considering the reflection of the following three rays, based on the laws of reflection:

1. A ray incident parallel to the principal axis passes through the principal focus after reflection in the case of a concave mirror, and appears to pass through the focus in the case of a convex mirror.
2. A ray which passes through the principal focus of a concave mirror, or which appears to go towards the principal focus of a convex mirror, goes parallel to the principal axis after reflection.
3. A ray passing through the centre of curvature of a concave mirror, or appearing to go towards the centre of curvature of a convex mirror, is reflected from the mirror along the same path.

Image formation by a concave mirror (for real objects)

Ray diagrams for the table below (Diagram column, rows 1 to 6):

Image formation by a convex mirror. The image is formed between the pole and the focus for all positions of the real object, except in the case of an object at infinity, where the image is formed at F in the focal plane. The image formed is virtual, erect and diminished.

Mirror formula. If an object is placed at a distance \(u\) from the pole of a mirror, its image is formed at a distance \(v\) from the pole, and its focal length is \(f\), then

\[\dfrac1u+\dfrac1v=\dfrac1f,\qquad f=\dfrac R2\]

As per the sign convention mentioned above, \(f\) and \(R\) are negative for a concave mirror and positive for a convex mirror. The power of a mirror, in dioptres, is

\[P=-\dfrac1f\ (f\text{ in metres})=-\dfrac{100}{f}\ (f\text{ in centimetres})\]

Magnification by a mirror. If an object of linear size \(O\) is situated vertically on the axis of a concave or convex mirror at a distance \(u\) from the pole and its image of size \(I\) is formed at a distance \(v\) from the pole, the lateral (transverse) magnification is

\[m=\dfrac IO=-\dfrac vu\]

If the magnification is negative, the image is inverted with respect to the object; a positive magnification means the image is erect.

REFRACTION OF LIGHT

Laws of refraction

1. The incident ray, the refracted ray and the normal to the surface separating the two media lie in the same plane.
2. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for any two media for a light beam of a specific wavelength, and is equal to the refractive index of the second medium with respect to the first medium. This is Snell’s law:

\[\dfrac{\sin i}{\sin r}={}_1\mu_2=\dfrac{{}_a\mu_2}{{}_a\mu_1}\quad\text{or}\quad{}_a\mu_1\sin i={}_a\mu_2\sin r\]

where \({}_a\mu_1\) is the refractive index of the first medium with respect to air, \({}_a\mu_2\) is that of the second medium with respect to air, and \({}_1\mu_2\) is the refractive index of the second medium with respect to the first.

The refraction of light is produced due to the different velocity of light in different media. If \(c\) is the velocity of light in vacuum and \(c'\) is the velocity of light in a medium, the absolute refractive index is \(\mu=\dfrac c{c'}\). If \(\mu_1\) and \(\mu_2\) are the absolute refractive indices of the first and second media, \(\mu_1=\dfrac c{c_1}\) and \(\mu_2=\dfrac c{c_2}\), so \({}_1\mu_2=\dfrac{\mu_2}{\mu_1}=\dfrac{c_1}{c_2}\), where \(c_1\) and \(c_2\) are the velocities of light in the two media.

The frequency \(f\) of light remains the same as light goes from one medium to another, whereas the wavelength changes from \(\lambda_1\) to \(\lambda_2\). Then

\[\dfrac{c_2}{c_1}=\dfrac{f\lambda_2}{f\lambda_1}=\dfrac{\lambda_2}{\lambda_1},\qquad{}_2\mu_1=\dfrac{\mu_1}{\mu_2}=\dfrac{c_2}{c_1}=\dfrac{\lambda_2}{\lambda_1}\]

The refractive index of a medium is different for rays of different wavelengths.

Refraction through a transparent sheet. Let a ray be incident at face AB of a sheet of index \(\mu_2\) placed between media of indices \(\mu_1\) (above) and \(\mu_3\) (below). Applying Snell’s law at faces AB and CD:

\[\mu_1\sin i_1=\mu_2\sin i_2\ \ldots(i),\qquad \mu_2\sin i_2=\mu_3\sin i_3\ \ldots(ii)\]

(the angle of incidence for face CD is \(i_2\)). From (i) and (ii), \(\mu_1\sin i_1=\mu_3\sin i_3\), or \(\sin i_3=\dfrac{\mu_1}{\mu_3}\sin i_1\) …(iii). For \(\mu_1=\mu_3\), \(i_1=i_3\), i.e. the incident ray and the emergent ray are parallel. This shows that the deviation of the ray is not affected by the refractive index of the sheet; it is decided by \(\mu_1\) and \(\mu_3\). \(\mu_2\) only causes a lateral displacement, which is given by

\[x=\dfrac{t\sin(i_1-i_2)}{\cos i_2}\]

Apparent depth. If an object O is placed in a denser medium like glass or water of thickness \(t\), it appears to be raised when viewed from a rarer medium like air, to a point I. When it is viewed normally, \(OP=t\) is the real depth and PI is the apparent depth, given by

\[\mu=\dfrac{\sin i}{\sin r}=\dfrac{\text{real depth}}{\text{apparent depth}}\]

\[\text{Apparent shift}=OI=OP-IP=t-\dfrac t\mu=t\left(1-\dfrac1\mu\right)\]

When an object is placed in a rarer medium and viewed normally from a denser medium, \(\mu=\dfrac{\text{apparent depth}}{\text{real depth}}\), where \(\mu\) is the refractive index of the denser medium with respect to the rarer medium.

Critical angle and total internal reflection. When a ray of light is incident from a denser medium into a rarer medium, it bends away from the normal, so that the angle of incidence is less than the angle of refraction. The angle of incidence in the denser medium for which the angle of refraction is \(90^\circ\) is called the critical angle \(C\) for the two given media. If \({}_a\mu_1\) and \({}_a\mu_2\) are the refractive indices with respect to air of the denser and the rarer media,

\[{}_1\mu_2=\dfrac{{}_a\mu_2}{{}_a\mu_1}=\dfrac{\sin C}{\sin90^\circ}=\sin C\quad\Rightarrow\quad{}_2\mu_1=\dfrac1{\sin C}\]

If the rarer medium is air (\({}_a\mu_2=1\)), the refractive index of the denser medium with respect to air is \(\mu=\dfrac1{\sin C}\). If the angle of incidence is more than the critical angle, no refraction takes place and the ray is completely reflected back into the denser medium. This phenomenon is called total internal reflection.

REFRACTION AT SPHERICAL SURFACES

Refraction at a single spherical surface. If the boundary between two transparent media is curved, either convex or concave, an object O in a medium of refractive index \(\mu_1\) forms an image I in the medium of refractive index \(\mu_2\). When the ray from O is incident at an angle \(i\) in the medium \(\mu_1\), the refracted ray forms the image I in the medium \(\mu_2\) at angle \(r\), following Snell’s law, \(\mu_1\sin i=\mu_2\sin r\). The relation between \(u=OP\) and \(v=IP\) is

\[\dfrac{\mu_2}{v}-\dfrac{\mu_1}{u}=\dfrac{\mu_2-\mu_1}{R}\]

This relation is valid for single spherical or plane refracting surfaces, and the sign convention for spherical mirrors and spherical refracting surfaces is the same.

REFRACTION THROUGH LENSES

A lens is made of a transparent material with two refracting surfaces such that at least one of them is curved. A convex lens is thicker in the middle and a concave lens is thinner in the middle. Plano-convex and plano-concave lenses have one plane surface and the other surface convex or concave respectively.

Parameters associated with a lens. The centre of curvature C is the centre of the sphere of which the lens surface forms a part; the radius of curvature of either surface is the radius of that sphere. The optical centre O is a point through which a ray goes undeviated. The principal axis is the line passing through the centres of curvature and the optical centre. If the object is at infinity, the image is formed at the principal focus, and vice versa. The focal length is the distance between the optical centre of the lens and the point at which a parallel beam of light converges or appears to converge. The aperture of a lens is the effective diameter of its light-transmitting area. The intensity of the image is directly proportional to the square of the aperture.

Rules of image formation

1. A ray passing through the optical centre travels undeviated through the lens.
2. A ray incident parallel to the principal axis, after refracting through the lens, either passes through the focus or appears to pass through the focus after extrapolation.
3. A ray passing through the focus, or directed towards it, becomes parallel to the principal axis after refraction.

The rays from a point of the object intersect or appear to intersect after refraction through the lens and form an image. If the rays actually intersect, a real image is formed; if they only appear to intersect, a virtual image is formed.

Image formation by a convex lens

Ray diagrams for the table below (Diagram column, rows 1 to 6):

Image formation by a concave lens. The image formed by a concave lens of a real object placed beyond F and the optical centre is virtual, erect and diminished. The image is formed at F if the object is at infinity.

Lens formulae. If an object is lying in a medium of refractive index \(\mu_1\) at a distance \(u\) from the optical centre of a lens having radii of curvature \(R_1\) and \(R_2\) and refractive index \(\mu_2\), the image is formed at a distance \(v\) from the optical centre, where

\[\dfrac1v-\dfrac1u=\left(\dfrac{\mu_2}{\mu_1}-1\right)\left(\dfrac1{R_1}-\dfrac1{R_2}\right)\]

If \(f\) is the focal length of the lens,

\[\dfrac1f=\dfrac1v-\dfrac1u=\left(\dfrac{\mu_2}{\mu_1}-1\right)\left(\dfrac1{R_1}-\dfrac1{R_2}\right)\]

If \(\mu\) is the refractive index of the material of the lens with respect to the surrounding medium (air),

\[\dfrac1f=\dfrac1v-\dfrac1u=(\mu-1)\left(\dfrac1{R_1}-\dfrac1{R_2}\right)\]

If \(I\) and \(O\) are the lateral sizes of the image and the object, the magnification is \(m=\dfrac IO=\dfrac vu\). The power of the lens is

\[P=\dfrac1{f(\text{m})}=\dfrac{100}{f(\text{cm})}\ \text{dioptre}\]

Focal length by the displacement method. When the distance between an object and a screen is more than four times the focal length of a converging lens, real images of the object are formed on the screen for two positions of the lens: for the first position the object lies at a distance between \(f\) and \(2f\) (image \(I_1\)), and for the second position the object lies at a distance greater than \(2f\) (image \(I_2\)). If \(D\) is the distance between the object and the screen and \(d\) is the distance between the two positions of the lens, the focal length of the lens is

\[f=\dfrac{D^2-d^2}{4D}\]

If the magnification for the first position of the lens is \(m\), the magnification for the second position is \(\dfrac1m\). The size of the object is \(O=\sqrt{I_1I_2}\), where \(I_1\) and \(I_2\) are the lengths of the images for the two lens positions.

Lenses in contact. If two thin lenses of focal lengths \(f_1\) and \(f_2\) are placed in contact, the equivalent focal length \(F\) is given by

\[\dfrac1F=\dfrac1{f_1}+\dfrac1{f_2}\]

If one surface of the lens is silvered, the effective focal length \(F\) of the combination is given by

\[\dfrac1F=\dfrac2{f_l}+\dfrac1{f_m}\]

where \(f_l\) is the focal length of the lens and \(f_m\) is the focal length of the silvered surface. When two lenses of focal lengths \(f_1\) and \(f_2\) are kept a distance \(d\) apart, the focal length of the combination is given by

\[\dfrac1F=\dfrac1{f_1}+\dfrac1{f_2}-\dfrac d{f_1f_2}\]

If there is a medium of refractive index \(\mu\) between the lenses, the equivalent focal length \(F\) is given by

\[\dfrac1F=\dfrac1{f_1}+\dfrac1{f_2}-\dfrac{d/\mu}{f_1f_2}\]

DEFECTS OF IMAGES

The relations developed for the formation of images by lenses and mirrors are approximate and do not produce perfectly focused images of objects. The imperfections or defects in such images are called aberrations. If the defect in the image is due to such approximations, it gives rise to spherical aberration, apart from coma, distortion, astigmatism, etc. If the defect is due to the dispersion of white light into its constituent colours, similar to that in a prism, it is called chromatic aberration, producing a blurred, coloured image of the object.

(a) Spherical aberration. This aberration is produced due to the spherical nature of the lens or mirror. The rays of light from a point object that are incident near the principal axis (paraxial rays) are focused at a larger distance, at the point \(I_M\), and the rays incident near the periphery (margin) of the lens are focused nearer the lens, at \(I_P\). It gives rise to a defocused, elongated image of a point object. Spherical aberration can be reduced by any one of the following methods:

1. using two thin convex lenses of focal lengths \(f_1\) and \(f_2\) separated by a distance \(d=f_1-f_2\);
2. using a plano-convex lens with its convex surface facing the incident or the emergent light, whichever is more parallel;
3. using either a lens of large focal length, or a specially designed aplanatic lens, a crossed lens or a parabolic reflector.

(b) Chromatic aberration. As the refractive index is different for the different wavelengths (colours) of white or any composite light, the image of an object illuminated by white light is coloured. Such a defect can be removed by using an achromatic doublet, comprising a convex and a concave lens of focal lengths \(f_1\) and \(f_2\) and dispersive powers \(\omega_1\) and \(\omega_2\), such that the ratio of their focal lengths is equal to the ratio of their dispersive powers:

\[\dfrac{\omega_1}{f_1}+\dfrac{\omega_2}{f_2}=0\quad\text{or}\quad\dfrac{\omega_1}{\omega_2}=-\dfrac{f_1}{f_2}\]

where the dispersive power is \(\omega=\dfrac{\mu_v-\mu_r}{\mu-1}\), and \(\mu_v\), \(\mu_r\) and \(\mu\) are the refractive indices for violet, red and mean yellow light respectively.

REFRACTION THROUGH A PRISM

If a ray of light is incident at an angle \(i\) on a glass prism of refracting angle \(A\), the ray is refracted at angle \(r\) at the first face, is incident at angle \(r'\) on the second face and emerges in air at angle \(i'\). When the incident ray is extrapolated forward and the emergent ray backward, the angle between them is the angle of deviation \(\delta\). It can be shown that

\[A+\delta=i+i',\qquad A=r+r'\]

If the angle of incidence is increased continuously from a small value, the angle of deviation first decreases until it reaches a minimum value, and then it increases. When the angle of deviation is minimum through an isosceles prism, the ray passes symmetrically through the prism, parallel to its base. Then \(r=r'\) and \(i=i'\):

\[r=\dfrac A2,\qquad i=\dfrac{A+\delta_{min}}2\]

The refractive index \(\mu\) of the prism is given by

\[\mu=\dfrac{\sin\left(\dfrac{A+\delta_{min}}2\right)}{\sin\left(\dfrac A2\right)}\]

If the angle of the prism \(A\) is small and the angle of deviation \(\delta\) is small (for a small angle of incidence), then \(\sin\theta\approx\theta\) and

\[\mu=\dfrac{\dfrac{A+\delta}2}{\dfrac A2}\ \Rightarrow\ \delta=(\mu-1)A\]

DISPERSION OF LIGHT

When a beam of white light, consisting of a spectrum of wavelengths from long wavelengths (red) to short wavelengths (violet), passes through a prism, it is split up into its constituent colours. This phenomenon of splitting a beam of composite light into its constituent colours is called dispersion. Dispersion takes place because the refractive index \(\mu\) of a medium depends on the wavelength \(\lambda\) of light, as given by Cauchy’s relation

\[\mu\approx A+\dfrac B{\lambda^2}\]

\(\mu\) is maximum for violet and minimum for red light.

Angle of dispersion. It is the angle between the emerging violet and red rays: \(\alpha=\delta_V-\delta_R=A(\mu_V-\mu_R)\).

Dispersive power. The ratio of the angle of dispersion to the deviation \(\delta\) of the mean yellow ray produced by a prism is called the dispersive power \(\omega\) of the material of the prism:

\[\omega=\dfrac{\delta_V-\delta_R}{\delta}=\dfrac{\mu_V-\mu_R}{\mu-1}\]

where \(\mu_V\), \(\mu_R\) and \(\mu\) are the refractive indices for violet, red and mean yellow light respectively.

Deviation without dispersion. If two suitable prisms of small angle, made from two different transparent materials, are combined and placed opposite to each other, the net dispersion can be zero; such a pair is called an achromatic pair of prisms, which produces deviation of the ray without any dispersion. For two prisms of angles \(A\) and \(A'\) and refractive indices \(\mu_V,\mu_R\) and \(\mu_V',\mu_R'\) for violet and red light, the net dispersion is

\[(\mu_V-\mu_R)A+(\mu_V'-\mu_R')A'=0\ \Rightarrow\ A'=-\dfrac{(\mu_V-\mu_R)A}{(\mu_V'-\mu_R')}\]

If \(\omega\) and \(\omega'\) are the dispersive powers of the prisms and \(\delta\) and \(\delta'\) their mean deviations, then \(\dfrac{\mu_V-\mu_R}{\mu-1}(\mu-1)A+\dfrac{\mu_V'-\mu_R'}{\mu'-1}(\mu'-1)A'=0\), i.e.

\[\omega\delta+\omega'\delta'=0\]

Dispersion without deviation. If the deviation produced for the mean ray by the first prism is equal and opposite to that produced by the second prism, the combination of the two prisms produces dispersion without any deviation. For zero net deviation, \(\delta+\delta'=0\):

\[(\mu-1)A+(\mu'-1)A'=0\ \Rightarrow\ A'=-\dfrac{(\mu-1)A}{(\mu'-1)}\]

(the negative sign shows that the second prism is inverted), where \(\mu\) and \(\mu'\) are the refractive indices for the mean colour for the prisms of angles \(A\) and \(A'\) respectively.

WAVE OPTICS – INTERFERENCE

When two waves of the same frequency moving along the same direction in a medium superimpose, they give rise to a phenomenon called interference. Points of constructive interference have maximum intensity, while points of destructive interference have minimum intensity.

If two waves \(y_1=A_1\sin\omega t\) and \(y_2=A_2\sin(\omega t+\phi)\) superimpose, the resultant wave is \(y=R\sin(\omega t+\theta)\), where

\[R^2=A_1^2+A_2^2+2A_1A_2\cos\phi\quad\text{or}\quad I=I_1+I_2+2\sqrt{I_1}\sqrt{I_2}\cos\phi\quad(\because I\propto A^2)\]

\[\tan\theta=\dfrac{A_2\sin\phi}{A_1+A_2\cos\phi}\]

For maxima, \(\cos\phi=1\), \(\phi=2n\pi\), \(n=0,1,2,\dots\) For minima, \(\cos\phi=-1\), \(\phi=(2n-1)\pi\), \(n=1,2,\dots\) If \(I_1=I_2=I_0\), then \(I=4I_0\cos^2\dfrac\phi2\), and

\[\dfrac{I_{max}}{I_{min}}=\dfrac{(A_1+A_2)^2}{(A_1-A_2)^2}=\dfrac{(\sqrt{I_1}+\sqrt{I_2})^2}{(\sqrt{I_1}-\sqrt{I_2})^2}\]

The following conditions are required to produce a sustained interference pattern:

1. The sources should be coherent, i.e. the phase difference between them should be constant. For this the frequency of the sources should be the same.
2. The amplitudes of both waves should be equal or nearly equal, so as to obtain bright and dark fringes of maximum contrast.
3. The two sources should be very close to each other.
4. The two sources or slits should be very narrow, otherwise a broad source will be equivalent to a number of narrow sources emitting their own overlapping wavelets.

If the two sources are obtained from a single parent source by splitting the light into two narrow sources, they form coherent sources which produce a sustained interference pattern, due to the constant phase difference between the waves from such coherent sources. A simple type of coherent source is described in the following experiment.

YOUNG’S DOUBLE SLIT EXPERIMENT

A parallel beam of monochromatic light from a slit S is incident on two narrow pinholes or slits \(S_1\) and \(S_2\), separated by a small distance \(d\). The wavelets emitted from these sources superpose at the screen placed in front of the slits and produce an alternate dark and bright fringe pattern, depending upon whether the waves reach a point with a phase difference \(\phi=(2n-1)\pi\) (destructive interference) or \(\phi=2n\pi\) (constructive interference). If the screen is placed at a perpendicular distance \(D\) from the middle point of the slits, the point O on the screen lies on the perpendicular bisector of \(S_1S_2\) and is equidistant from \(S_1\) and \(S_2\). The path difference between the two waves reaching O from \(S_1\) and \(S_2\) is zero, so the intensity at O is maximum.

Consider a point P located at a distance \(x_n\) from O on the screen. The path difference of the waves reaching P from \(S_2\) and \(S_1\) is

\[S_2P-S_1P=\dfrac{x_nd}{D}\]

The point P is bright (of maximum intensity) when the path difference is an integral multiple of the wavelength \(\lambda\), i.e. \(\phi=2n\pi\) corresponds to a path difference \(n\lambda\):

\[S_2P-S_1P=\dfrac{x_nd}{D}=n\lambda\]

The bright fringes are thus observed at distances

\[x_1=\dfrac{\lambda D}d,\ x_2=\dfrac{2\lambda D}d,\ x_3=\dfrac{3\lambda D}d,\ \dots,\ x_n=\dfrac{n\lambda D}d\]

The distance between consecutive bright fringes is \(\beta=\dfrac{n\lambda D}d-(n-1)\dfrac{\lambda D}d=\dfrac{\lambda D}d\).

The point P will be dark (of minimum intensity) when the path difference is an odd multiple of half the wavelength, i.e. \(\phi=(2n-1)\pi\):

\[S_2P-S_1P=\dfrac{x_nd}{D}=(2n-1)\dfrac\lambda2\ \Rightarrow\ x_n=(2n-1)\dfrac{\lambda D}{2d}\]

where \(n\) is an integer. The dark fringes are observed at distances \(x_1'=\dfrac{\lambda D}{2d}\), \(x_2'=\dfrac{3\lambda D}{2d}\), \(x_3'=\dfrac{5\lambda D}{2d}\), …, \(x_n'=\dfrac{(2n-1)\lambda D}{2d}\). The distance between two consecutive dark fringes is \(\dfrac{(2n-1)\lambda D}{2d}-\dfrac{(2n-3)\lambda D}{2d}=\dfrac{\lambda D}d\).

The spacing between any two consecutive bright fringes or two dark fringes is equal and is called the fringe width \(\beta\):

\[\beta=\dfrac{\lambda D}{d}\]

Effect of a thin transparent plate. If a thin transparent plate of thickness \(t\) and refractive index \(\mu\) is introduced in the path \(S_1P\) of one of the interfering waves, the entire fringe pattern is shifted through a constant distance. The air path \(S_1P\) is increased to an equivalent air path \(S_1P+(\mu-1)t\), so the path difference is

\[\delta=S_2P-[S_1P+(\mu-1)t]=S_2P-S_1P-(\mu-1)t=\dfrac{x_nd}{D}-(\mu-1)t\]

For the \(n\)th maximum at a distance \(x_n\), \(\dfrac{x_nd}{D}-(\mu-1)t=n\lambda\), so

\[x_n=\dfrac Dd\left[n\lambda+(\mu-1)t\right]\]

Thus, when a thin transparent plate of thickness \(t\) and refractive index \(\mu\) is introduced in one of the paths of the waves, the path difference changes by \(\dfrac{xd}{D}=(\mu-1)t\), i.e. \(x=\dfrac Dd(\mu-1)t\). The central maximum shifts by a distance equal to

\[\dfrac{D(\mu-1)t}{d}\]

INTERFERENCE BY A THIN FILM

If a ray is incident on a thin film of thickness \(t\) and refractive index \(\mu_2\) which has media of refractive indices \(\mu_1\) and \(\mu_3\) on its two sides, there is partial reflection and refraction at both interfaces of the film.

For interference in reflected rays. When \(\mu_2\gt\mu_1,\mu_3\), the conditions are:

1. Maxima: \(2t\mu_2\cos r=\left(n-\dfrac12\right)\lambda,\quad n=1,2,\dots\)
2. Minima: \(2t\mu_2\cos r=n\lambda,\quad n=1,2,\dots\)

When \(\mu_2\) is between \(\mu_1\) and \(\mu_3\), the conditions are:

1. Maxima: \(2\mu_2t\cos r=n\lambda,\quad n=1,2,\dots\)
2. Minima: \(2\mu_2t\cos r=\left(n-\dfrac12\right)\lambda,\quad n=1,2,\dots\)

For interference in the transmitted rays 3 and 4, the conditions for maxima and minima are interchanged in both cases of \(\mu_2\).

6. Velocity of object w.r.t. mirror = – velocity of image w.r.t. mirror.