Overview: Reflection of Light: Spherical Mirrors

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 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.

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

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

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

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

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

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

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

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

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.

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

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°.

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.

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

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

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.

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.

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

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

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

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'.

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

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.

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

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

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

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

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

OR

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

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.

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

OR

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

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

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

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

θ = (n_V - n_R) A

• 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.

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

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.

ω = \[\frac{n_{V}-n_{R}}{n_{Y}-1}\]

Internationally Accepted: 

ω = \[\frac{n_{F}-n_{C}}{n_{D}-1}\]

• 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 = \[\frac {R}{2}\]​

• 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.

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.

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.

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

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

• 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.

• 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.

• 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.

• 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).

• 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.

• 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.

• 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

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

• 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 − \[\frac {1}{n}\])

• 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.

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

• 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.

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

• 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.

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.

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.

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

Image at least distance of distinct vision:

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

Eye relaxed, image at infinity:

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

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:

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

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

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

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

• 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.

• 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.

• 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.

• 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.