Ray Optics and Optical Instruments

In this chapter: reflection by spherical mirrors, the Cartesian sign convention, mirror equation and magnification, refraction and Snell's law, apparent depth, total internal reflection and its uses, refraction at a spherical surface, thin lens formula, lens maker's formula, power and combinations of lenses, refraction through a prism and minimum deviation, simple and compound microscopes, refracting and reflecting telescopes.

Sign convention

NCERT uses the Cartesian sign convention. All distances are measured from the pole of the mirror or the optical centre of the lens. Distances measured in the direction of the incident light are positive, and those against it are negative. Heights above the principal axis are positive and those below are negative. With the object on the left and light going left to right, u is always negative for a real object.

Reflection by spherical mirrors

For a spherical mirror, the focal length is half the radius of curvature, f = R/2, for paraxial rays. By the sign convention a concave mirror has negative f and a convex mirror has positive f.

A negative m means a real, inverted image; a positive m means a virtual, erect image. A convex mirror always gives a virtual, erect, diminished image, which is why it is used as a rear-view mirror.

Worked example: An object is placed 10 cm in front of a concave mirror of focal length 15 cm. Locate the image.
Solution: u = −10 cm, f = −15 cm. 1/v = 1/f − 1/u = −1/15 + 1/10 = 1/30, so v = +30 cm. The image is 30 cm behind the mirror. m = −v/u = −(30)/(−10) = +3: virtual, erect and three times the size. This is how a shaving mirror works.

Refraction

When light passes from one medium to another, the incident ray, refracted ray and normal lie in one plane, and

Light bends towards the normal on entering an optically denser medium. The frequency does not change on refraction; the speed and wavelength both decrease by the factor n. An object at real depth h in water seen from directly above appears raised: apparent depth = h/n. A ray passing through a glass slab with parallel faces comes out parallel to its original direction but laterally shifted.

Total internal reflection

When light goes from a denser to a rarer medium, the refracted ray bends away from the normal. At the critical angle i_c the refracted ray grazes the surface (r = 90°), and for any larger angle of incidence all the light is reflected back into the denser medium.

For crown glass (n ≈ 1.5) the critical angle is about 41.8°; for water about 48.8°; for diamond (n = 2.42) only about 24.4°. The small critical angle of diamond, together with cutting the facets so that light is totally reflected many times inside, is what makes diamonds sparkle.

• Prisms: right-angled isosceles prisms bend rays by 90° or 180°, or invert an image, with no loss of light (used in binoculars and periscopes).
• Optical fibres: a glass or quartz core of higher refractive index is surrounded by cladding of lower index. Light entering at a suitable angle is totally reflected again and again along the fibre, even when it is bent. Used to carry telephone and internet signals and in endoscopes.

Refraction at spherical surfaces and lenses

For refraction at a single spherical surface separating media of indices n₁ (object side) and n₂:

Applying this twice to the two surfaces of a thin lens gives the lens maker's formula and the thin lens formula:

The focal length depends on the surrounding medium through n₂₁. A glass lens (n = 1.5) placed in a liquid of the same index has no effect on light at all, and in a liquid of higher index a convex lens diverges light.

Ray diagrams

Draw any two of these three rays from the top of the object: a ray parallel to the axis, which passes through the second focus after refraction (or appears to come from the first focus for a concave lens); a ray through the optical centre, which goes straight on; and a ray through the first focus, which emerges parallel to the axis.

Worked example: An object is placed 30 cm from a convex lens of focal length 20 cm. Find the image position and magnification.
Solution: u = −30 cm, f = +20 cm. 1/v = 1/f + 1/u = 1/20 − 1/30 = 1/60, so v = +60 cm (on the other side). m = v/u = 60/(−30) = −2: real, inverted, twice the size.

Power and combination of lenses

The power of a lens is P = 1/f, with f in metres; the unit is the dioptre (D). A converging lens has positive power. For thin lenses in contact:

Refraction through a prism

For a prism of angle A, a ray incident at angle i on the first face emerges at angle e from the second face. The angles inside satisfy r₁ + r₂ = A, and the deviation is δ = i + e − A. As i increases, δ first decreases, reaches a minimum D_m, and then increases. At minimum deviation, i = e, r₁ = r₂ = A/2 and the ray inside is parallel to the base.

Worked example: A prism of angle 60° gives a minimum deviation of 30°. Find its refractive index.
Solution: n = sin[(60° + 30°)/2]/sin 30° = sin 45°/sin 30° = 0.707/0.5 ≈ 1.41, that is √2.

Optical instruments

Simple microscope

A convex lens of short focal length held close to the eye, with the object inside its focus, gives an enlarged virtual image. Taking the least distance of distinct vision D = 25 cm:

Compound microscope

An objective of short focal length forms a real, inverted, magnified image of the object; the eyepiece acts as a simple microscope to magnify this image further. The final image is virtual and inverted with respect to the object. For the final image at infinity,

Telescopes

A refracting telescope has an objective of large focal length and large aperture, and an eyepiece of short focal length. In normal adjustment (final image at infinity):

A large objective gathers more light and resolves finer detail, but big lenses are heavy, hard to make and suffer from chromatic aberration. Most large research telescopes are therefore reflecting telescopes, which use a concave (parabolic) mirror as the objective. A mirror has no chromatic aberration, a parabolic shape removes spherical aberration, and it can be supported over its whole back. In the Cassegrain design, a small convex secondary mirror reflects the light back through a hole in the primary mirror to the eyepiece.

Worked example: A telescope has an objective of focal length 100 cm and an eyepiece of focal length 5 cm. Find its magnifying power and length in normal adjustment.
Solution: m = 100/5 = 20; length = 100 + 5 = 105 cm.

Common mistakes: (1) Putting u as positive for a real object; with light going left to right it is negative. (2) Using the mirror formula (1/v + 1/u) for lenses; the lens formula has a minus sign (1/v − 1/u). (3) Writing m = −v/u for a lens; for a lens m = v/u. (4) Forgetting that critical angle and total internal reflection need light going from a denser to a rarer medium. (5) In a compound microscope, taking L as the distance between the lenses instead of between the inner focal points.

JEE and NEET focus

• Mirror and lens formulas with the sign convention, including image nature from the sign of m.
• Lens maker's formula, change of focal length in a liquid, power of combinations.
• Critical angle, apparent depth, optical fibres.
• Prism formula, minimum deviation, thin prism deviation.
• Magnifying power of simple and compound microscopes and of telescopes.

Practice questions