Revision: Optics >> Ray Optics and Optical Instruments Physics (Theory) ISC (Science) ISC Class 12 CISCE

A branch of optics that describes light propagation in terms of rays is called ray optics.

A spherical mirror whose reflecting surface is curved outwards, is called a convex mirror.

OR

A convex mirror is one whose reflecting surface is away from the centre of the sphere of which the mirror is a part.

OR

The reflecting surface is on the outer side of the sphere (diverging mirror).

The radius of the sphere of which the reflecting surface of a spherical mirror forms a part is called the radius of curvature of the mirror. It is represented by the letter R.

OR

The radius of the sphere of which the mirror forms a part, is called the 'radius of curvature' of the mirror.

A straight line passing through the pole and the centre of curvature of a spherical mirror. This line is called the principal axis.

OR

The straight line joining the pole and the centre of curvature of the mirror and extended on both sides is called the 'principal axis' of the mirror.

The centre of the reflecting surface of a spherical mirror is a point called the pole. The pole is usually represented by the letter P.

OR

The central point of the reflecting surface of the mirror is called the 'pole' of the mirror.

“A mirror made by silvering the inner surface such that reflection takes place from the bulging surface” is called Convex Mirror.
The Centre of curvature is towards the silvered surface.

“A mirror made by silvering the outer or the bulging surface such that the reflection takes place from the concave surface.” Centre of curvature is towards the reflecting surface.

Pole “is the mid-point of the mirror”.

The centre of a hollow sphere of which the mirror forms a part is called the centre of curvature.

An imaginary line passing through the pole and the centre of curvature of a spherical mirror is called principal axis.

It is a point on the principal axis, where a beam of light, parallel to the principal axis, after reflection actually meet.

The linear distance between the pole and the center of curvature is called the radius of curvature.

The linear distance between the pole and the principal focus is called focal length.

Pole is the centre of the reflecting surface, in this case, a spherical mirror.

 Centre of curvature is the centre of the imaginary sphere to which the mirror belongs.

The distance between the pole and the principal focus is called the focal length (f) of a spherical mirror.

Principal focus of a spherical mirror is a point on the principal axis of the mirror, where all the rays travelling parallel to the principal axis and close to it after reflection from the mirror, converge to or appear to diverge from.

The focus of a concave mirror is a point on the principal axis of the mirror, where all the rays travelling parallel to the principal axis and close to it after reflection from the mirror converge to that point.

Normal to the surface of a mirror at any point is the straight line at the right angle to the tangent drawn at that point.

The reflecting surface of a spherical mirror forms a part of a sphere. This sphere has a centre. This point is called the centre of curvature of the spherical mirror. It is represented by the letter C.

OR

The centre of the sphere of which the mirror forms a part, is called the ‘centre of curvature' of the mirror.

“A mirror which is made from a part of a hollow sphere is called Spherical Mirror.

Aperture is the distance between the extreme points on the periphery of the mirror.

Mirrors whose reflecting surfaces are spherical are called spherical mirrors.

OR

A spherical mirror is a part of a hollow sphere, whose one side is silvered and coated with red oxide and the other side is the reflecting surface.

OR

A spherical mirror is a piece cut out of a spherical surface, which can be concave or convex.

A spherical mirror, whose reflecting surface is curved inwards, that is, faces towards the centre of the sphere, is called a concave mirror.

OR

A concave mirror is one whose reflecting surface is towards the centre of the sphere of which the mirror is a part.

OR

The reflecting surface is on the inner side of the sphere (converging mirror).

The ratio of the height of an image (h') to the height of an object (h) is known as linear magnification 

That is, 

`mh/h` 

where, h^' = height of image
            h =  height of object 

When travelling obliquely from one medium to another, the direction of propagation of light in the second medium changes. This phenomenon is known as refraction of light.

OR

Light changes its direction when going from one transparent medium to another transparent medium. This is called the refraction of light.

OR

The bending of the light ray from its path in passing from one medium to the other medium is called 'refraction' of light.

OR

When a ray of light impinges on a polished, smooth, shiny surface, the rebounding of light within the same medium is called reflection of light.

The ray that enters the second medium after crossing the boundary is called the refracted ray.

Refracted light is the part of light enters into the other medium and travels in a straight path but in a direction different from its initial direction and is called the refracted light.

1. Incident ray: The ray of light which falls on the mirror surface is called the incident ray.
2. Point of incidence: The point at which the incident ray falls on the mirror is called the point of incidence.
3. Normal: The normal is a line at right angles to the mirror surface at the point of incidence.
4. Reflected ray: The ray of light which is sent back by the mirror is called the reflected ray.
5. Angle of incidence: The angle of incidence is the angle made by the incident ray with the normal at the point of incidence.
6. Angle of reflection: The angle of reflection is the angle made by the reflected ray with the normal at the point of incidence.

Light rays that are parallel to the principal axis of a concave mirror converge at a specific point on its principal axis after reflecting from the mirror. This point is known as the principal focus of the concave mirror.

A normal is an imaginary line drawn perpendicular to the boundary at the point of incidence.

The change in the direction of the path of light when it passes from one transparent medium to another transparent medium is called refraction. The refraction of light is essentially a surface phenomenon.

or

When light passes from one transparent medium to another, its speed and direction change. This is called refraction.

The refractive index of second medium with respect to the first medium is defined as the ratio of the sine of the angle of incidence in the first medium to the sine of the angle of refraction in the second medium.

The perpendicular distance XY between the path of the emergent ray BC and the direction of the incident ray OD is called the lateral displacement.

It is defined as the ratio of the velocity of light in medium 1 to the velocity of light in medium 2.

Critical angle is the angle of incidence in the denser medium corresponding to which the angle of refraction in the rarer medium is 90°.

Total internal reflection is the complete reflection of light back into an optically denser medium when light travels from a denser medium to a rarer medium and the angle of incidence exceeds the critical angle.

When a ray of light propagates from a denser medium to a rarer medium, the angle of incidence for which the angle of refraction is 90° is called the critical angle.

The critical angle is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90 degrees.

A mirage is an optical illusion seen on hot roads or in deserts where distant objects appear reflected from water-like surfaces.

An optical fibre is a thin, transparent fibre of glass or plastic that transmits light signals using repeated total internal reflection.

The imaginary line passing through both centres of curvature is called the principal axis of the lens.

OR

The line joining the centres of curvature of the surfaces of the lens is called the 'principal axis' of the lens.

The distance between the optical centre and principal focus of a lens is called its focal length.

Principal focus (F) is the point on the principal axis at which light rays parallel to the principal axis converge after passing through a convex lens.

The radii (R₁ and R₂) of the spheres whose parts form surfaces of the lenses are called the radii of curvature of the lens.

The centres of spheres whose parts form surfaces of the lenses are called centres of curvatures of the lenses.

The point inside a lens on the principal axis, through which light rays pass without changing their path is called the optical centre of a lens.

OR

The point on the principal axis of a lens such that a ray of light directed towards it emerges from the lens in the same direction, without deviation.

A lens which is bent inwards in the middle is a concave lens. Such a lens diverges the light rays incident on it, so it is also called a diverging lens.

OR

This lens is thicker near the centre as compared to the edges. The lens with both surfaces spherical on the inside is called a concave or double concave lens.

OR

The lenses which are thinner in the middle and thicker at the edges, are called 'concave lenses'.

A lens which bulges out in the middle, is a convex lens. A light beam converges on passing through such a lens, so it is also called a converging lens.

OR

The lens which has two spherical surfaces which are puffed up outwards is called a convex or double convex lens.

OR

The lenses which are thicker in the middle and thinner at the edges, are called 'convex lenses'.

A lens is a transparent refracting medium bounded by either two spherical surfaces, or one spherical surface and the other surface plane.

OR

A lens is a transparent medium bound by two surfaces.

OR

A lens is a transparent medium (such as glass) bounded by two curved surfaces or one curved and one plane surface.

The SI unit of power of a lens is the dioptre.
One dioptre is the power of a lens whose focal length is 1 metre.

1D = 1m⁻¹

Power of a lens is defined as the ability of a lens to bend the rays of light. It is given by the reciprocal of focal length in metre.

The power of a lens is a measure of the deviation produced by it in the path of rays refracted through it.

The deviation of the incident light rays produced by a lens on refraction through it, is a measure of its power.

or

The power of a lens is defined as the reciprocal of its focal length. It is represented by the letter P.

OR

The power (P) of a thin lens is equal to the reciprocal of its focal length (f) measured in metres.

The focal length of a single lens that produces the same optical effect as the given combination of lenses kept in contact.

A prism is a transparent medium bounded by five plane surfaces with a triangular cross-section.

The angular separation between the two extreme colours (violet and red) in the spectrum (which is obtained by passing a beam of white light through a prism) is known as angular dispersion.

Scattering is the process of absorption and then re-emission of light energy by the dust particles and air molecules present in the atmosphere.

Angular magnification or magnifying power of an optical instrument is defined as the ratio of the visual angle made by the image formed by that optical instrument (β) to the visual angle subtended by the object when kept at the least distance of distinct vision (α).

For a normal, unaided human eye, D = 25 cm. If an object is brought closer than this, we cannot see it clearly. The minimum distance from the eye at which an object can be seen clearly is called the least distance of distinct vision.

OR

Due to the limitation of focusing the eye lens, it is not possible to take an object closer than a certain distance. This distance is called the least distance of distinct vision.

An optical instrument that uses objective and eye piece lenses to magnify tiny objects in detail is called a compound microscope.

The quantity μ sin ⁡θ, where μμ is the refractive index of the medium between the object and the objective, is called the numerical aperture (N.A.) of the objective of the microscope.

The ability of an optical instrument to produce distinctly separate images of two objects very close to each other is called the resolving power of the instrument.

The reciprocal of the least angular separation between the objects that are just resolved is called the resolving power of the telescope.

The reciprocal of the limit of resolution is called its resolving power.

The minimum distance of separation between two objects when they can be observed as separate by an optical instrument is called the limit of resolution of that instrument.

When sunlight passes through the Earth's atmosphere, much of the light is absorbed by the fine dust particles and air molecules in the atmosphere, which give out the absorbed light in some other direction. This is 'scattering of light'.

The angle between the emergent rays of any two colours is called ‘angular dispersion’ between those colours.

An astronomical telescope is an optical instrument used to observe distant heavenly objects by increasing the visual angle subtended at the eye.

A telescope that uses a concave mirror as the objective to collect and focus light from distant objects.

The power of changing the focal length of the eye lens to see objects clearly at different distances is called the power of accommodation of the eye.

The far point of a normal eye is the point at infinity, which can be seen distinctly when the eye is in a relaxed state.

The nearest distance up to which the eye can see clearly by applying maximum power of accommodation is called the least distance of distinct vision.

The near point of the eye is the nearest point at which an object can be seen distinctly.

A microscope is an optical instrument which forms a large image of a small and close object so that it subtends a large visual angle at the eye.

A simple microscope is a short-focus convex lens used to obtain a magnified, erect, and virtual image of a close object.

The magnifying power of a simple microscope is the ratio of the angle subtended by the image at the eye to the angle subtended by the object when placed at the least distance of distinct vision.

A compound microscope is an optical instrument which produces high magnification by using two converging lenses: an objective and an eyepiece.

The lens placed near the eye, of larger focal length and aperture, is called the eyepiece.

The lens placed near the object, of short focal length and small aperture, is called the objective lens.

When the velocity of light in a medium is compared with that in another medium, the parameter is called the relative refractive index.

“The perpendicular distance between the emergent ray and the direction of the incident ray is called the lateral shift.”

The plane passing through the focus of a lens and perpendicular to the principal axis is called the 'focal plane'.

The rays travelling parallel to the axis of the lens, after refraction through the lens, either go towards a fixed point on the axis or appear to come from a point. This point is called the 'second focus' or the 'principal focus' of the lens.

The distance of the second focus from the optical centre of the lens is called the 'second focal length' or the 'principal focal length' of the lens.

The rays starting from a fixed point on the principal axis of a lens, or appearing to go towards a fixed point on the axis, after refraction through the lens, become parallel to the principal axis. This point is called the 'first focus' of the lens.

The distance of the first focus from the optical centre of the lens is called the 'first focal-length' of the lens.

The critical angle for two given media is the angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90°.

A prism is a homogeneous, transparent medium bounded by two plane surfaces inclined to each other at an angle.

When a ray of light, travelling from a denser medium to a rarer medium, is incident at the interface of the two media at an angle greater than the critical angle for the two media, the ray is 'totally' reflected back into the denser medium.

A reflecting telescope that uses a paraboloidal primary mirror with a central hole and a convex secondary mirror, with the eyepiece placed behind the primary mirror.

A reflecting telescope in which a plane mirror inclined at 45° deflects light from a concave primary mirror to an eyepiece placed at the side.

The angle between the direction of the incident ray (produced forward) and the emergent ray (produced backward) is called the angle of deviation.

The coloured arcs seen in the sky when sunlight is dispersed by raindrops are called rainbows.

 The power of an optical instrument to produce distinctly separate images of two close objects is called the ‘resolving power' of that instrument.

The diameter of the periphery of the mirror is called the 'aperture' of the mirror.

The point on the principal axis at which light rays parallel to the principal axis, afterreflection from the mirror, actually meet or appear to come from, is called the 'principal focus' of the mirror.

The distance of the principal focus from the pole of the mirror is called the 'focal length' of the mirror.

The plane perpendicular to the principal axis and passing through the principal focus of the mirror is called the ‘focal plane' of the mirror.

Linear magnification produced by a spherical mirror is the ratio of the size of the image to the size of the object, both measured perpendicular to the principal axis.

The angle which an object subtends at our eye is called the 'visual angle’.

The magnifying power of an optical instrument is defined as the ratio of the visual angle subtended by the image formed by the instrument at the eye to the visual angle subtended by the object at the unaided eye.

The refractive index of a medium is the parameter that tells how much slower light travels in that medium compared to vacuum.

Mathematically,
n = \[\frac{\text{velocity of light in vacuum}}{\text{velocity of light in medium}}=\frac{c}{v}\]

where c = 3 × 10⁸ ms^-1.

The linear magnification produced by a spherical (convex or concave) lens is the ratio of the size of the image formed by the lens to the size of the object, both measured perpendicular to the principal axis.

When white light passes through a thin prism, the ratio of the angular dispersion between the violet and the red emergent rays and the deviation suffered by a mean ray (ray of yellow colour) is called the ‘dispersive power' of the material of the prism. It is denoted by ω.

Lens formula is: 

`1/v-1/u=1/f`

Now, the mirror formula is given by:

`1/v+1/u=1/f`

In both the formulas,

u = Object distance

v = Image distance

f = Focal length

In the mirror formula, a positive sign is present between the reciprocals of the image distance and object distance.

In lens formula, a negative sign is present between the reciprocals of image distance and object distance.

Magnification (m) = `-v/u`

The refractive index of medium 2 w.r.to medium 1:

\[n_{21} = \frac{\text{Speed of light in medium 1}}{\text{Speed of light in medium 2}} = \frac{v_1}{v_2}\]

The refractive index of medium 1 w.r.to medium 2:

\[n_{12}=\frac{\text{Speed of light in medium 2}}{\text{Speed of light in medium 1}}=\frac{v_{2}}{v_{1}}\]

The refractive index of the medium n_m  is given by:

\[n_{m} = \frac{\text{Speed of light in air}}{\text{Speed of light in the medium}} = \frac{c}{v}\]

OR

\[\frac {\text{sin i}}{\text{sin r}}\] = constant = n

n is called the refractive index of the second medium with respect to the first medium.

For light travelling from medium 1 to medium 2, where medium 1 is denser than medium 2:

Power of lens (in D) = \[\frac{1}{\text{focal length (in metre)}}\]

or

P = \[\frac {1}{f}\]

or

P = \[\frac {1}{f (m)}\]

Power of a Lens in a Medium:

P = (n₂ - n₁)\[\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)\] = \[\frac {n_1}{f}\]

\[\frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2}\]

For more than two thin lenses in contact:

M = m_o × M_e

R.P. = \[\frac {1}{\text {Limit of resolution}}\]

R.P. = \[\frac {1}{d}\] = \[\frac{2\mu\sin\theta}{\lambda}\]

where μ sin⁡ θ is the Numerical Aperture (N.A.) of the objective.

R.P. = \[\frac{1}{d\theta}=\frac{D}{1.22\lambda}\]

\[\frac {1}{v}\] + \[\frac {1}{u}\] = \[\frac {1}{f}\]

m = -\[\frac {v}{u}\]

OR

m = -\[\frac {v}{u}\] = \[\frac {f - v}{f}\] = \[\frac {f}{f - u}\].

\[\frac {1}{v}\] + \[\frac {1}{u}\] = \[\frac {1}{f}\]

n = \[\frac{\sin\frac{A+\delta_{m}}{2}}{\sin\frac{A}{2}}\]

\[\frac{n}{v}-\frac{1}{u}=\frac{n-1}{R}\]

m = \[\frac {v}{u}\]

OR

m = \[\frac {f}{f + u}\]

M = m₀ × m_e

Normal Adjustment, Image at D:

\[{M=\frac{L}{f_o}\left(1+\frac{D}{f_e}\right)}\]

Relaxed Eye, Image at Infinity:

\[{M=\frac{L}{f_o}\frac{D}{f_e}}\]

M = -\[\frac {f_o}{f_e}\]

• f_o​ = focal length of the concave (objective) mirror
• f_e​ = focal length of the eyepiece

Image at least distance of distinct vision:

M = 1 + \[\frac {D}{f}\]

Eye relaxed, image at infinity:

M = \[\frac {D}{f}\]

\[^1n_2=\frac{v_1}{v_2}=\frac{n_2}{n_1}\]

1. Both the Lenses are Convex:
   \[\frac {1}{f}\] = \[\frac {1}{f_1}\] + \[\frac {1}{f_2}\]
2. One Lens is Convex and the Other is Concave:
   \[\frac {1}{f}\] = \[\frac {1}{f_1}\] - \[\frac {1}{f_2}\]
   
3. Combined Power: 
   P = P₁ + P₂

Lateral shift (d) = t sin (i − r) sec r

\[\frac {1}{v}\] - \[\frac {1}{u}\] = \[\frac {1}{f}\]

\[^2n_3=\frac{n_3}{n_2}\]

\[_1n_2=\frac{1}{\sin C}\cdot\]

θ = (n_V - n_R) A

ω = \[\frac{n_{V}-n_{R}}{n_{Y}-1}\]

Internationally Accepted: 

ω = \[\frac{n_{F}-n_{C}}{n_{D}-1}\]

General Magnifying Power of a Telescope:

M = \[\frac {f_o}{u_e}\]

Final Image at the Least Distance of Distinct Vision:

M = \[\frac{f_o}{f_e}\left(1+\frac{D}{f_e}\right)\]

Normal Adjustment / Final Image at Infinity:

M = -\[\frac {f_o}{f_e}\]

The laws of refraction are fundamental for board examinations and objective tests.

First law

The incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane.

Second law

For a given pair of media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction remains constant.

\[\frac {\text {sin i}}{\text {sin r}}\] = constant

This constant is called the refractive index of the second medium with respect to the first medium.

1. First Law: Incident ray, refracted ray, and normal to the interface at the point of incidence all lie in the same plane.

2. Second Law (Snell's Law): \[\frac{\sin i}{\sin r}=\mathrm{constant}=\frac{\mu_2}{\mu_1}\]

Extended Snell's Law: \[\frac{\sin i}{\sin r}=\frac{\mu_2}{\mu_1}=_1\mu_2=\frac{v_1}{v_2}=\frac{\lambda_1}{\lambda_2}\]

• The incident ray, the refracted ray and the normal at the point of incidence, all lie in the same plane.
• The ratio of the sine of the angle of incidence i to the sine of the angle of refraction r is constant for the pair of given media. i.e., mathematically

\[\frac {\text{sin i}}{\text{sin r}}\] = constant = n

Where:

• i = angle of incidence
• r = angle of refraction
• n = refractive index of the second medium with respect to the first

This ratio is constant for a given pair of media and is known as the refractive index.

Note: If a light ray is incident along the normal (i = 0^∘), it passes undeviated into the second medium (r = 0^∘).

• The incident ray, the refracted ray and the normal to the interface of two transparent media at the point of incidence all lie in the same plane.
• The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant, for the light of a given colour and for the given pair of media. This law is also known as Snell’s law of refraction. (This is true for angle 0 < i < 90°)
  If i is the angle of incidence and r is the angle of refraction, then,
  \[\frac {\text{sin i}}{\text{sin r}}\] = constant

This constant value is called the refractive index of the second medium with respect to the first.

Ray 2 shows partially reflected ray.

Statement

When a light ray, after undergoing any number of reflections and refractions, has its direction reversed, it retraces its entire original path. This is called the principle of reversibility of light.

Explanation / Proof

Consider a light ray passing from medium 1 to medium 2 and suffering refraction at the boundary.
Let the angle of incidence be i and the angle of refraction be r.

By Snell’s law, the refractive index of medium 2 with respect to medium 1 is:

¹n₂ = \[\frac {sin ⁡i}{sin⁡ r}\]​

Now, suppose the refracted ray is reflected back and retraces the path in the reverse direction. In this case, the angle of incidence becomes r, and the angle of refraction becomes i.

Again, by Snell’s law, the refractive index of medium 1 with respect to medium 2 is:

²n₁ = \[\frac {sin ⁡r}{sin ⁡i}\]​

Multiplying the two equations:

¹n₂ × ²n₁ = 1

This shows that the ray follows the same path in the reverse direction, proving the reversibility of the light path.

Conclusion

Hence, a light ray always retraces its original path when its direction is reversed, even after multiple reflections and refractions. This establishes the principle of reversibility of light.

Rayleigh proved that the intensity of scattered light is inversely proportional to the fourth power of the wavelength; provided the scatterer is smaller in size than the wavelength of light :

Scattering ∝ \[\frac {1}{λ^4}\]

According to this law, the short waves of violet light (λ = 4000) are scattered about ten times more than the longer waves of red light (λ = 7000). The other colours are scattered by intermediate amounts.

• Pole (mirror) or optical centre (lens) is the origin; principal axis is the X-axis.
• Distances to the right are positive, to the left are negative; heights above the axis are positive, below are negative.
• Concave mirror: $f$ and R are negative; Convex mirror: $f$ and R are positive.
• Real images: image distance and magnification are negative; Virtual images: both are positive.
• Lenses are always negative; they are positive for real images and negative for virtual images; they are positive for convex lenses and negative for concave lenses.

• Lenses are widely used in daily life, such as in spectacles, peepholes, magnifiers, and telescopes.
• Light passing through a lens undergoes refraction twice: once on entering and once on exiting the lens.
• The shape of a lens affects the direction of light; convex lenses converge light, while concave lenses diverge it.
• Most lenses have surfaces that are parts of spheres, with common types including biconvex, biconcave, plano-convex, and meniscus lenses.

• Shorter wavelengths (violet and blue) are scattered more than longer wavelengths (red).
• The intensity of scattering follows Rayleigh’s law:
  I ∝ \[\frac {1}{λ^4}\]​
• Very small particles (smaller than the wavelength of light) scatter light more effectively than larger particles.
• The sky appears blue because blue light is scattered more due to its shorter wavelength.
• Red light scatters least, causing red/orange sunsets and making sunlight near Earth richer in red light.

• Refraction occurs due to a change in the speed of light when it passes from one medium to another.
• The greater the change in speed, the greater is the bending of light at the boundary of the two media.
• According to Snell’s law:
  If v₁ > v₂​, the ray bends towards the normal (rarer to denser medium).
  If v₁ < v₂​, the ray bends away from the normal (denser to rarer medium).

• Refractive index indicates the direction of bending of light at a boundary (towards or away from the normal).
• It gives the ratio of the speeds of light in vacuum and in the medium:
  n = \[\frac {c}{v}\]So, a higher refractive index means a lower speed of light in the medium.
• The frequency of light remains unchanged during refraction, but the wavelength changes; hence, the refractive index also gives information about the wavelength of light in a medium.

• Mirage is caused by total internal reflection in hot air layers, making objects appear inverted, as in water reflections.
• Diamonds sparkle because light undergoes repeated total internal reflections due to their small critical angle.
• Totally reflecting prisms use total internal reflection to reflect light efficiently.
• Right-angled prisms can turn light by 90° or 180° using total internal reflection.
• Prisms are better than mirrors because they reflect almost all light and produce clear images.
• Optical fibres guide light by total internal reflection and are used in communication and medical imaging.

• An object in a denser medium appears raised when viewed from a rarer medium due to refraction.
• Real depth is the actual depth of the object; apparent depth is the depth at which it appears.
• Refractive index is given by:
  n = \[\frac{\text{Real depth}}{\text{Apparent depth}}\]​
• Normal displacement is the difference between real and apparent depths:
  d = Real depth − Apparent depth
• For a medium of thickness t:
  d = t (1 − \[\frac {1}{n}\])

• The optical centre of the lens is taken as the origin; the principal axis is the X-axis and the perpendicular line through the optical centre is the Y-axis.
• Distances to the right of the optical centre are positive and to the left are negative; heights above the principal axis are positive and below are negative.

• Convexo-convex (Bi-convex): Both surfaces are convex; radii of curvature may be equal or different.
• Plano-convex: One surface is plane and the other is convex.
• Concavo-convex (Convex meniscus): One surface is concave and the other convex; thicker at the centre.
• Concavo-concave (Bi-concave): Both surfaces are concave; radii of curvature may be equal or different.
• Plano-concave: One surface is plane, and the other is concave.
• Convexo-concave (Concave meniscus): One surface is convex and the other concave; thinner at the centre.

• The focal length of a lens depends on its refractive index and the radii of curvature of its surfaces (lens maker’s formula).
• Changing the surrounding medium changes a lens's focal length; it increases in a denser medium and may even alter the lens's properties.

• For two coaxial lenses separated by distance d, the equivalent focal length and power depend on f₁, f₂, and d.
• A concave lens always forms a virtual image; therefore, its focal length is determined by combining it with a mirror.
• A convex mirror also always forms a virtual image, so it is combined with a convex lens to find its focal length.
• With a convex lens and a plane mirror, if the object and image coincide without parallax, the object position determines the lens's focal length.
• Focal lengths in lens–mirror combinations are calculated using the lens formula and non-parallax positions.

• Scattering of light by air molecules and fine dust particles explains many atmospheric optical phenomena.
• The sky appears blue because shorter-wavelength blue light scatters more strongly than red light in the atmosphere.
• If there were no atmosphere, the sky would appear black, as no scattering of sunlight would occur.
• Clouds appear white because water droplets and ice crystals are large and scatter all wavelengths nearly equally.
• The Sun appears reddish at sunrise and sunset because blue light scatters more strongly over a longer atmospheric path.
• Red light is used in danger signals because it suffers the least scattering and can be seen from long distances.
• Infra-red rays suffer very little scattering, so infra-red photography is possible in fog and mist.

• Rainbows are formed due to the dispersion of sunlight in raindrops.
• The primary rainbow is formed after one internal reflection in a raindrop and is brighter, with violet inside and red outside.
• The secondary rainbow is formed after two internal reflections and is fainter, with red inside and violet outside.
• The primary rainbow is seen at about 41°–43°, while the secondary rainbow is seen at about 51°–54° from the antisolar direction.
• Primary and secondary rainbows appear as concentric arcs with a common centre on the line joining the sun and the observer.

• The limiting angle of incidence is the angle at which a ray just emerges from the prism; for angles smaller than this, total internal reflection occurs at the second face.
  \[i_1=\sin^{-1}\left[\sqrt{(n^2-1)}\sin A-\cos A\right]\]
• For grazing incidence and grazing emergence, both angles of incidence are 90^∘, and the condition for emergence is
  A ≤ 2Ca​
  where C is the critical angle.
• Maximum deviation by a prism occurs when the angle of incidence at the first face is 90^∘ (grazing incidence).
  δ_max = δ₁ + δ₂ = (90° – C) + (i – r).

• The deviation produced by a prism depends on the angle of incidence, the angle of the prism, and the material of the prism.
• As the angle of incidence increases, the angle of deviation first decreases, becomes minimum, and then increases.
• For minimum deviation, the angle of incidence equals the angle of emergence (i = i′).
• In the condition of minimum deviation, the refracted ray inside the prism travels parallel to the base of the prism.
• For a thin prism, the deviation depends only on the refractive index of the material and the angle of the prism, and not on the angle of incidence.

• An astronomical refracting telescope uses two convex lenses—an objective near the object and an eyepiece near the eye.
• The objective lens has a large focal length and a large aperture, so it can collect more light from distant objects.
• The objective forms a real, inverted, and diminished image of the distant object at its focal plane.
• This image serves as an object for the eyepiece, producing a magnified virtual image for the observer.
• Normal adjustment is done by making the final image at infinity, so the eye observes without strain.
• Refracting telescopes suffer from chromatic and spherical aberrations and have limited magnification and resolution.

• For relaxed eye adjustment, the final image is formed at infinity, and the intermediate image lies at the focus of the eyepiece.
• Magnifying power (relaxed eye) is
  M = \[\frac {L}{f_o}\]\[\frac {D}{f_e}\].
• Maximum magnification is obtained when the object is placed very close to the focal point of the objective.
• A bright, highly magnified image requires lenses with short focal lengths, with the objective having a small aperture.
• A compound microscope is used instead of a simple microscope to achieve higher magnification without sacrificing image quality.

• A compound microscope uses two convex lenses, an objective (short focal length) and an eyepiece (longer focal length).
• The objective forms a real, inverted, magnified image, which acts as a virtual object for the eyepiece.
• The eyepiece produces a final virtual and highly magnified image, usually at the least distance of distinct vision or at infinity.
• Total magnifying power is the product of the magnifications of the objective and the eyepiece.
• Large magnification is achieved when the object is placed close to the objective's focal point and the eyepiece has a short focal length.

• According to Rayleigh’s criterion, two-point objects are just resolved when the principal maximum of one diffraction pattern falls on the first minimum of the other.
• Resolving power increases when the limit of resolution decreases; smaller separation means better resolution.
• For a telescope, the limit of resolution depends on wavelength and aperture, and a larger aperture gives higher resolving power.
• For a microscope, resolving power improves with a smaller wavelength of light and a larger numerical aperture.
• Electron microscopes have very high resolving power because electrons have extremely small wavelengths compared to visible light.

• For a spherical mirror of small aperture, rays close to the principal axis (paraxial rays) obey the law of reflection accurately.
• In both concave and convex mirrors, using geometrical construction and the law of reflection, the focus lies midway between the pole and the centre of curvature.
• Hence, for a small-aperture spherical mirror, the focal length is half the radius of curvature:
  f = \[\frac {R}{2}\]​

• Many rays start from an object point, but only two or three rays are sufficient to locate the image.
• In mirrors, reflected rays remain on the same side of the mirror as the object; no real rays exist on the other side.
• If reflected rays actually meet, a real image is formed, which is inverted.
• If reflected rays diverge and meet only on backward extension, a virtual image is formed, which is erect.
• For lenses, real images are formed on the opposite side of the lens and are inverted, while virtual images are formed on the same side as the object and are erect.

• A concave mirror is used for shaving (erect, magnified image).
• Concave (parabolic) mirrors are used in telescopes to observe distant stars.
• Concave mirrors are used in torches, searchlights and headlights to produce a parallel beam.
• Concave mirrors are used by ENT doctors and eye specialists for examination.
• Convex mirrors are used in street lights to illuminate a large area.
• Convex mirrors are used as rear-view mirrors (erect, diminished image, wide view).
• Image identification:
  Erect & same size → Plane mirror
  Erect & magnified → Concave mirror
  Erect & diminished → Convex mirror

• Parallel ray rule: A ray parallel to the principal axis passes through the focus (concave) or appears to come from the focus (convex) after reflection.
• Focus ray rule: A ray passing through the focus (concave) or directed towards the focus (convex) becomes parallel to the principal axis after reflection.
• Centre of curvature rule: A ray passing through or directed towards the centre of curvature retraces its path after reflection.
• Law of reflection rule: A ray striking the mirror surface reflects according to the laws of reflection.