Ray Optics and Optical Instruments: Class 12 Physics · Chapter 9

9.1 Introduction

Sunlight slanting through a gap in a tin roof, or a torch beam in a dusty room, draws a sharp straight line through the air. Shadows have crisp edges, and a tall building hides the Sun exactly when it blocks the straight line from your eye. Everyday experience says two things about light: it goes very fast, and it goes in straight lines. This chapter builds a whole toolkit for mirrors, lenses, prisms and instruments on the second idea.

In Chapter 8 of Part I you met light as an electromagnetic wave. Our eyes respond to a narrow slice of that spectrum, wavelengths from about $400\mathrm{n}\mathrm{m}$ (violet) to about $750\mathrm{n}\mathrm{m}$ (red), and we call that slice light (प्रकाश). In vacuum it travels at $c=2.99792458\times {10}^8\mathrm{m}\mathrm{/}\mathrm{s}$, a value now fixed by definition; $3.00\times {10}^8\mathrm{m}\mathrm{/}\mathrm{s}$ is good enough for almost every calculation. Nothing carries energy or information faster. Even so, sunlight needs about $500\mathrm{s}$ to cover the $1.5\times {10}^{11}\mathrm{m}$ to the Earth.

How can a wave travel in straight lines? The answer is size. A wave squeezing through an opening spreads out noticeably only when the opening is comparable to its wavelength. Visible light has $\lambda \approx 5\times {10}^{-7}\mathrm{m}$, while a window, a lens or a coin is a centimetre to a metre across, a hundred thousand to a million times larger. Past such objects the spreading is far too small to notice, and the energy follows straight lines (Fig. 9.1). Chapter 10 deals with what happens when openings do become tiny.

With the ray picture we study three things in this chapter: reflection, refraction and the images they form at plane and curved surfaces. First come spherical mirrors, then refraction at flat surfaces, total internal reflection, curved surfaces and thin lenses, and the prism. The chapter ends with the simple microscope, the compound microscope and the telescope, instruments built from these parts.

9.2 Reflection of light by spherical mirrors

You already know the two laws of reflection. The angle of reflection equals the angle of incidence, both measured from the normal; and the incident ray, the normal and the reflected ray lie in one plane. Nothing in these laws says the surface must be flat. At any point of a curved mirror, a tiny patch around that point is effectively flat, so the laws apply point by point, with the normal to that patch.

For a spherical mirror, a piece cut from a hollow sphere, the normal at every point is easy to find: it lies along the radius through that point, so every normal passes through the centre of curvature C. The middle point of the mirror's surface is its pole P, the distance PC is the radius of curvature R, and the line through P and C is the principal axis. A mirror silvered on its hollow side is concave; one silvered on its bulging side is convex. For a lens the corresponding middle point is the optical centre, and its principal axis is the line through the optical centre and the principal foci.

One more requirement hides in the word image. A polished mirror reflects regularly: neighbouring rays meet neighbouring patches that face the same way, so rays from one object point stay together. A whitewashed wall reflects most of the light that falls on it, yet you cannot see your face in it: its rough grains face every direction, so rays from one point scatter everywhere and never come together at one image point.

9.2.1 Sign convention

A mirror formula has to cover objects in front of a mirror and images behind it, concave and convex mirrors, real and virtual images. Instead of a separate formula for every case, we give every distance a sign and use one formula throughout. This book uses the Cartesian sign convention (Fig. 9.2):

1. Origin. Measure from the middle of the mirror or lens, its pole P (for a lens, its optical centre O), with the principal axis as the $x$-axis.
2. Distances. A point reached by moving with the incident light from the origin has a positive distance; one reached by moving against the light has a negative distance.
3. Heights. Points above the axis have positive heights, points below it negative heights.

We always draw the incident light travelling from left to right. Then a real object is on the left and has a negative $u$; the focus of a concave mirror is in front of it, so its $f$ is negative; the focus of a convex mirror is behind it, so its $f$ is positive. Do not decide signs by looking at whether an image is real. Put the numbers with their signs into the formula, and let the sign of the answer tell you where the image is.

9.2.2 Focal length of spherical mirrors

Send a narrow beam parallel to the principal axis onto a concave mirror. If the rays are paraxial (they hit the mirror near its pole and are only slightly inclined to the axis), they all reflect through a single point F on the axis: the principal focus. A convex mirror spreads such a beam out, and the reflected rays appear to come from a point F behind it (Fig. 9.3). The distance PF is the focal length $f$.

A parallel beam that is tilted to the axis also comes to a point, but off the axis. All such points lie in the plane through F perpendicular to the axis, the focal plane. This is why a distant scene, whose points send parallel beams in slightly different directions, forms a flat image near the focus.

9.2.3 The mirror equation

Light from a point on an object spreads in every direction. If, after reflection, all those rays actually pass through one point, that point is a real image (वास्तविक प्रतिबिंब) of the object point; if they only seem to diverge from a point when extended backwards, it is a virtual image (आभासी प्रतिबिंब). An image is a point-to-point copy of the object made by reflection or refraction.

To find an image point we need only two rays, because two lines fix their crossing point. Four rays from a concave mirror are easy to draw (Fig. 9.5):

• a ray parallel to the axis reflects through F (for a convex mirror, it reflects as if coming from F);
• a ray through F (or heading for F of a convex mirror) reflects parallel to the axis;
• a ray through C (or heading for C of a convex mirror) meets the mirror along a normal and comes straight back;
• a ray striking the pole reflects at an equal angle on the other side of the axis.

The diagram shows only a few rays, but the object point sends out countless rays. Every one that reaches the mirror passes through the same image point, which is what makes the image sharp. Now we turn the drawing into an equation.

How tall the image is, compared with the object, is measured by the linear magnification:

For a real image $u$ and $v$ are both negative, so $m$ is negative: real images formed by a single mirror are inverted. For a virtual image $v$ is positive, $m$ is positive and the image is erect. The size of $\mathrm{\mid }m\mathrm{\mid }$ says whether the image is enlarged ($\mathrm{\mid }m\mathrm{\mid }>1$) or diminished. Fig. 9.6 shows the two everyday virtual images: a concave mirror with the object inside F (a shaving or make-up mirror) and a convex mirror (the mirror on an auto-rickshaw).

Try the mirror bench: start with a concave mirror and slide the object from far away towards the mirror. The real image grows and moves outwards, races off to infinity as the object reaches F, and reappears behind the mirror as a large virtual image. A convex mirror never does anything dramatic: its image is always virtual, upright and smaller.

9.3 Refraction

Put a straw in a glass of nimbu-pani and it looks snapped at the surface. When light meets the boundary between two transparent media, part of it reflects and the rest crosses into the second medium. If it arrives obliquely ($0^{\circ }<i<{90}^{\circ }$), the transmitted light changes direction at the boundary. This change of direction is refraction (अपवर्तन), and its rules were found by measurement long before anyone could explain them (Fig. 9.9).

The constant $n_{21}$ is the refractive index (अपवर्तनांक) of medium 2 relative to medium 1. It belongs to the pair of media and changes slightly with colour (wavelength), but it does not depend on the angle. If $n_{21}>1$, then $r<i$: the ray bends towards the normal, and medium 2 is called optically denser. If $n_{21}<1$, the ray bends away from the normal on entering the optically rarer medium.

Refraction happens because light travels at different speeds in different media. The absolute refractive index of a medium is $n=c\mathrm{/}v$, where $v$ is the speed of light in it; for vacuum $n=1$ and for air $n=1.0003$, which we take as 1. In terms of absolute indices, $n_{21}=n_2\mathrm{/}{n}_1=v_1\mathrm{/}{v}_2$, and Snell's law takes the symmetric form $n_1\sin i=n_2\sin r$. Water ($n=1.33$) slows light to $2.26\times {10}^8\mathrm{m}\mathrm{/}\mathrm{s}$; diamond ($n=2.42$) slows it to $1.24\times {10}^8\mathrm{m}\mathrm{/}\mathrm{s}$.

Because a light path can always be run backwards (Fig. 9.9, step 3), the index of medium 1 relative to medium 2 is just the reciprocal:

Relative indices also chain together. Since $n_{32}=n_3\mathrm{/}{n}_2=(n_3\mathrm{/}{n}_1)(n_1\mathrm{/}{n}_2)$, we have $n_{32}=n_{31}\times n_{12}$. For example, glass relative to water is $n_{\text{glass, water}}=1.50\mathrm{/}1.33=1.13$: a ray going from water into glass bends only a little.

A parallel-sided slab. A ray crossing a glass block refracts twice (Fig. 9.10). The two faces are parallel, so the normals are parallel and the ray meets the second face at the same angle $r$ at which it left the first. Snell's law at the exit, $n\sin r=\sin e$, then gives $e=i$. The ray leaves in the same direction it arrived: there is no deviation, only a sideways lateral shift $d$. From the geometry of the figure, $d=t\sin (i-r)\mathrm{/}\cos r$ for a slab of thickness $t$. A $6.0\mathrm{c}\mathrm{m}$ slab with $n=1.50$ and $i={50}^{\circ }$ gives $r={30.7}^{\circ }$ and $d=2.3\mathrm{c}\mathrm{m}$.

Apparent depth. A coin at the bottom of a bucket of water looks nearer the surface than it is. Rays from the coin bend away from the normal as they leave the water, so when your eye traces them back in straight lines they seem to come from a point higher up (Fig. 9.11).

9.4 Total internal reflection

Now send light the other way, from a denser medium into a rarer one, say from water into air. Some light is reflected back into the water at the surface, which is called internal reflection, and the rest refracts out, bending away from the normal so that $r>i$. As $i$ grows, $r$ grows faster, and at one particular angle of incidence the refracted ray skims along the surface with $r={90}^{\circ }$. That angle is the critical angle $i_c$ (क्रांतिक कोण) (Fig. 9.12).

Beyond $i_c$, Snell's law would need $\sin r>1$, which is impossible: there is no refracted ray at all, and all the light is reflected back into the denser medium. This is total internal reflection (पूर्ण आंतरिक परावर्तन). Even the best silvered mirror absorbs a few per cent of the light that falls on it; total internal reflection loses essentially none, which is why it is so useful.

Setting $r={90}^{\circ }$ in Snell's law for light going from medium 1 (denser) into medium 2 (rarer), $n_1\sin i_c=n_2$:

For a material of index $n$ in air this is $\sin i_c=1\mathrm{/}n$, so measuring the critical angle gives the refractive index: $n=1\mathrm{/}\sin i_c$. The larger the index, the smaller the critical angle (Table 9.2).

9.4.1 Total internal reflection in nature and its technological applications

(i) Prisms that act as perfect mirrors. A glass prism whose angles are ${45}^{\circ }$, ${45}^{\circ }$ and ${90}^{\circ }$ can turn light without any silvering. Light entering one face normally meets the long face at ${45}^{\circ }$, more than the critical angle of glass ($41$–${42}^{\circ }$, Table 9.2), and is totally reflected. Depending on how the prism is turned, it bends the beam through ${90}^{\circ }$, sends it back through ${180}^{\circ }$, or flips an image upside down without changing its size (Fig. 9.14). Periscopes and binoculars use such prisms; their reflections stay bright for decades because there is no coating to tarnish.

(ii) Optical fibres. An optical fibre (प्रकाशिक तंतु) is a glass or plastic thread thinner than a hair. Its core has a slightly higher refractive index than the surrounding cladding. Light entering one end at a suitable angle strikes the core–cladding boundary at more than the critical angle, is totally reflected, crosses to the other side, and is reflected again, thousands of times per metre (Fig. 9.15). A fibre can even be bent gently: the light follows it round the bend, which is why it is often called a light pipe.

Fibres carry telephone calls, television and internet data as flashes of light, both under the oceans and into homes. Bundles of fibres also carry light into the body and an image back out: with such an endoscope doctors can look inside the stomach, the airways or a damaged knee without large cuts. The tufted lamps sold at fairs, whose plastic strands glow at their tips, are decorative optical fibres lit from a bulb at the base.

The real challenge is purity. Ordinary window glass looks clear, but a block a few metres thick would be dark green. Silica for fibres is purified so carefully that more than $95\mathrm{\%}$ of the light survives a full kilometre.

9.5 Refraction at spherical surfaces and by lenses

A drop of water on a leaf, a glass marble and the lens of your eye all refract light at curved surfaces. As with curved mirrors, we treat a tiny patch of a spherical surface as flat, apply Snell's law there, and remember that the normal at any point runs along the radius, through the centre of curvature. We first find what a single spherical surface does. A thin lens is then just two such surfaces, close together, at least one of them curved: applying the single-surface result twice gives the lens maker's formula and the lens formula.

9.5.1 Refraction at a spherical surface

Medium 1 (index $n_1$, on the left) meets medium 2 (index $n_2$, on the right) at a convex spherical boundary whose centre is C and whose radius is $R$. A point object O sits on the axis in the first medium, and P is where the axis meets the surface. We want the point I where the refracted rays cross the axis. Keep the aperture small compared with all other distances, so every angle in Fig. 9.16 is small.

9.5.2 Refraction by a lens

A thin lens of index $n_2$ in a medium of index $n_1$ has two surfaces with radii $R_1$ (the first surface the light meets) and $R_2$. The trick is to let the first surface act alone, as if glass filled all the space beyond it, and find the image $I_1$ it would form. That image then serves as the object for the second surface (Fig. 9.17).

The lens maker's formula (9.21) is what a lens designer uses: pick a glass, then choose radii that give the focal length wanted. A lens has two principal foci, F on the side of the incoming light (the first focal point) and F′ on the other side (the second focal point). When the same medium is on both sides of a thin lens, the two are at the same distance $\mathrm{\mid }f\mathrm{\mid }$ from the optical centre O.

For drawing, three rays through a thin lens are easy (Fig. 9.18):

• a ray parallel to the axis passes through the second focus F′ of a convex lens, or diverges as if from the first focus F of a concave lens;
• a ray through the optical centre O goes straight on, undeviated;
• a ray through F of a convex lens (or heading for F′ of a concave lens) emerges parallel to the axis.

The magnification of a lens is defined exactly as for a mirror, and similar triangles through O give

For a real image $u<0$ and $v>0$, so $m<0$: the image is inverted. A virtual image from a lens lies on the object side, $v<0$, so $m>0$ and the image is erect. Try both lenses on the lens bench below and check $1\mathrm{/}v-1\mathrm{/}u=1\mathrm{/}f$ with the signed readouts.

9.5.3 Power of a lens

How strongly a lens bends light is measured by its power. A ray parallel to the axis at height $h$ is bent by an angle $\delta$ so that it crosses the axis at the focus: $\tan \delta =h\mathrm{/}f$. For a ray at unit height and small angles, $\delta \approx 1\mathrm{/}f$: the shorter the focal length, the more sharply the lens converges (or, for negative $f$, diverges) a beam. So we define the power of a lens (लेंस की क्षमता) as

Opticians write prescriptions in dioptres. Reading glasses of $+2.0\mathrm{D}$ are converging lenses with $f=+50\mathrm{c}\mathrm{m}$; glasses for a short-sighted person of $-2.5\mathrm{D}$ are diverging lenses with $f=-40\mathrm{c}\mathrm{m}$.

9.5.4 Combination of thin lenses in contact

Camera and microscope objectives are made of several lenses cemented together. Two thin lenses of focal lengths $f_1$ and $f_2$ in contact share practically the same optical centre. What single lens would do the same job?

Combining lenses lets a designer reach a power that no single lens has, and, by pairing glasses that spread colours differently, cancel defects such as coloured fringes. Lens systems of this kind sit inside every camera, microscope and telescope.