Ray Optics and Optical Instruments

1. Check this before you revise anything

The "Additional Exercises" section has been removed from this chapter, as from all 14 chapters of the current Class 12 Physics book. The questions run contiguously from 9.1 to 9.31 with no gaps.

Even after that removal this is the largest exercise set in the book — 31 questions, a third of them on optical instruments alone.

Dispersion has been removed, but the Introduction still promises it. Older editions had a full section on dispersion by a prism, with dispersive power and the spectrum. The section is gone and "dispersive" returns zero hits.

Yet the Introduction still says: "In this chapter, we consider the phenomena of reflection, refraction and dispersion of light." The only survivals are a one-line Summary entry, "Dispersion is the splitting of light into its constituent colour," and a passing remark in Points to Ponder.

Natural phenomena due to sunlight are gone entirely. Searching the chapter returns zero hits for "rainbow", "scattering", "Rayleigh" and "blue sky". The section explaining why the sky is blue and sunsets are red no longer exists.

The human eye is promised and then handed back to Class X. The Introduction says the chapter will "describe the construction and working of some important optical instruments, including the human eye". But section 9.7 opens by saying: "We have already studied about the human eye in Class X. We now go on to describe the principles of working of the microscope and the telescope."

Consistently, "myopia", "hypermetropia", "presbyopia", "astigmatism", "accommodation" and "cataract" all return zero hits. Eye defects and their correction are not in this chapter.

An internal inconsistency worth knowing. Fig. 9.27(b), used in Exercise 9.4, shows light entering water at $60^{\circ }$ and refracting at $47^{\circ }$, which implies $n_{\text{water}}=1.18$. Exercises 9.3 and 9.5 in the same chapter use the real value, 1.33. The figure's angles are illustrative, so answer 9.4 from the data it gives you, not from the physical value.

2. Spherical Mirrors and the Mirror Equation (Textbook 9.2)

The Cartesian sign convention (9.2.1) is where most marks are lost, and it is worth stating in full before any calculation:

• All distances are measured from the pole of the mirror.
• Distances measured along the incident light are positive; those against it are negative.
• Heights above the principal axis are positive, below it negative.

The practical consequences: a real object always has $u<0$; a concave mirror has $f<0$ and a convex mirror $f>0$; and a negative $v$ means a real image for mirrors.

Focal length (9.2.2). For a spherical mirror the focus lies halfway to the centre of curvature:

$f=\frac{R}{2}$

The mirror equation (9.2.3) and the magnification:

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f},m=-\frac{v}{u}=\frac{h^{\prime }}{h}$

A negative $m$ means an inverted image, which for a mirror also means a real one.

Reading a convex mirror off the algebra. Writing $v=\frac{f|u|}{f+|u|}$ shows immediately that $v$ is always positive and always less than $f$. So a convex mirror gives a virtual image, always between pole and focus, always diminished — whatever the object distance. That is Exercise 9.15 done algebraically rather than by ray diagram.

3. Refraction and Total Internal Reflection (Textbook 9.3 to 9.4)

Snell's law relates the angles at a boundary:

$n_1\sin i=n_2\sin r,n_{21}=\frac{n_2}{n_1}=\frac{\sin i}{\sin r}$

Apparent depth. An object under a denser medium looks shallower than it is:

$n=\frac{\text{real depth}}{\text{apparent depth}}$

which is the whole of Exercise 9.3. A parallel-sided slab produces a normal shift:

$\text{shift}=t\left(1-\frac{1}{n}\right)$

This depends only on the slab's thickness and index, not on where the slab sits — the point Exercise 9.16 is testing.

Total internal reflection (9.4) occurs when light travels from a denser to a rarer medium and strikes the boundary beyond the critical angle:

$\sin C=\frac{n_2}{n_1}$

Both conditions are needed: denser to rarer, and angle greater than $C$. Light going the other way never totally reflects.

Two consequences the chapter builds on. A point source underwater illuminates only a circle of radius $h\tan C$ at the surface, with everything outside acting as a mirror — Exercise 9.5. And an optical fibre guides light by repeated total internal reflection, with the cladding's lower index setting the acceptance cone, which Exercise 9.17 works out both with and without the cladding.

Mirages are mentioned as the natural example, though the wider treatment of atmospheric optics has been removed.

4. Lenses, Power and Combinations (Textbook 9.5)

Refraction at a single spherical surface (9.5.1):

$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$

Applying this twice, once at each face of a thin lens, gives the lens maker's formula (9.5.2):

$\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

For a double convex lens $R_1=+R$ and $R_2=-R$, so the two surfaces reinforce and $f=\frac{R}{2(n-1)}$. Dropping the opposite signs is the standard error in Exercise 9.7.

The thin lens equation looks similar to the mirror equation but with a crucial sign difference:

$\frac{1}{v}-\frac{1}{u}=\frac{1}{f},m=\frac{v}{u}$

Note the minus where the mirror has a plus, and $m=v/u$ with no leading minus. For lenses a positive $v$ means a real image, the opposite of the mirror convention.

Virtual objects. If light is already converging when it meets a lens, the object distance is positive. This is what makes Exercise 9.8 work, where a converging beam meets a lens 12 cm short of its focus.

Power (9.5.3) measures converging ability, in dioptres:

$P=\frac{1}{f\text{(in metres)}}$

Thin lenses in contact (9.5.4) add reciprocally, or equivalently their powers add:

$\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2}+\cdots ,P=P_1+P_2+\cdots$

Separated lenses are a different matter. Exercise 9.20 places the same two lenses 8 cm apart and asks for an "effective focal length". Tracing a parallel beam from one side gives $-220$ cm and from the other $-420$ cm, while the combination formula gives $-300$ cm. Since no single number predicts both measurements, the notion is not useful here — a separated pair needs its principal planes specified too.

5. The Prism (Textbook 9.6)

For a ray passing through a prism of refracting angle $A$, the internal angles satisfy:

$r_1+r_2=A,\delta =i_1+i_2-A$

At minimum deviation the path becomes symmetric, with $i_1=i_2$ and $r_1=r_2=A/2$, giving the relation used for measuring refractive index:

$n=\frac{\sin \left(\frac{A+D_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}$

In a medium the relative index is what counts. A prism immersed in water bends light far less, because $n_{gw}=n_g/n_w$ replaces $n_g$. Exercise 9.6 shows the minimum deviation dropping from $40^{\circ }$ to about $10^{\circ }$ on moving from air to water.

Grazing emergence. Setting $r_2$ equal to the critical angle gives the smallest angle of incidence for which light still escapes the second face. Below it the ray is totally internally reflected inside the prism, which is Exercise 9.21.

6. The Microscope and the Telescope (Textbook 9.7)

Both instruments work by increasing the angle subtended at the eye, not the physical size of anything.

The simple magnifier. Two settings matter, and they give different answers:

Angular magnification is not linear magnification. Exercises 9.22 to 9.24 exist to force this distinction. In 9.22 the card sits exactly at the focal point, so the image goes to infinity and the linear magnification is unbounded — yet the magnifying power is a perfectly ordinary $25/9=2.8$. In 9.23, with the image at the near point, the two happen to coincide at 3.8, but only because the image then sits at exactly the distance the unaided eye would use.

The compound microscope (9.7.1). The objective forms a real, magnified, inverted image, which the eyepiece then views as a magnifier:

$M=m_o\times m_e=\frac{v_o}{|u_o|}\left(1+\frac{D}{f_e}\right)$

Both focal lengths must be short, since the magnification varies inversely with each.

The telescope (9.7.2). The objective has a long focal length and the eyepiece a short one — the reverse of the microscope:

$M=\frac{f_o}{f_e}\text{(normal adjustment)},M=\frac{f_o}{f_e}\left(1+\frac{f_e}{D}\right)\text{(image at }D)$

with tube length $f_o+f_e$ in normal adjustment.

The image size at the objective depends only on the angle subtended by the object, $h=f_o\theta$. That single relation answers Exercises 9.14(b) and 9.28(b).

The eye ring. All rays emerging from the eyepiece pass through a small region a short way beyond it. Placing the eye there, rather than pressed against the lens, collects all the light and gives the widest field of view — the answer to Exercise 9.25(e).

Reflecting telescopes avoid chromatic aberration and can be made much larger. The Cassegrain design folds a long focal length into a short tube using a convex secondary, which Exercise 9.29 works through numerically.

Summary

• Cartesian convention: distances from the pole, positive along the incident light; $u<0$ for a real object.
• Mirrors: $f=R/2$, with $f<0$ concave and $f>0$ convex; $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$ and $m=-v/u$.
• For mirrors a negative $v$ means a real image; for lenses a positive $v$ does.
• A convex mirror always gives a virtual, erect, diminished image between pole and focus.
• Snell's law $n_1\sin i=n_2\sin r$; apparent depth $=$ real depth$/n$.
• Normal shift through a slab is $t(1-1/n)$, independent of where the slab is placed.
• Total internal reflection needs denser to rarer and $i>C$, where $\sin C=n_2/n_1$.
• A submerged source lights a circle of radius $h\tan C$; optical fibres guide light by repeated total internal reflection.
• Refraction at a spherical surface: $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$.
• Lens maker's formula $\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$; for a double convex lens $R_1$ and $R_2$ have opposite signs.
• Thin lens: $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$, $m=v/u$; a converging incident beam makes $u$ positive.
• Power $P=1/f$ in dioptres; lenses in contact add their powers.
• Separated lenses have no single useful "effective focal length" — the traced answer depends on which side the light enters.
• Prism: $r_1+r_2=A$, $\delta =i_1+i_2-A$, and $n=\sin \frac{A+D_m}{2}/\sin \frac{A}{2}$ at minimum deviation.
• In water a prism deviates far less, since the relative index $n_g/n_w$ replaces $n_g$.
• Magnifier: $D/f$ with the image at infinity, $1+D/f$ at the near point.
• Angular magnification and linear magnification are different quantities and are generally unequal.
• Microscope $M=\frac{v_o}{|u_o|}\left(1+\frac{D}{f_e}\right)$, both focal lengths short.
• Telescope $M=f_o/f_e$ with a long objective focal length and tube length $f_o+f_e$.
• Image height at a telescope objective is $h=f_o\theta$, set by the angle subtended.
• Dispersion, the rainbow, scattering and the human eye have all been removed, although the Introduction still promises dispersion and the eye.