Ray and Wave Optics — NEET Physics

Light is the strangest thing in the syllabus: it travels in straight rays that reflect and refract through mirrors and lenses, yet it also spreads as waves that interfere and diffract. NEET tests both faces heavily (3–4 marks), and the questions reward two things — clean handling of the sign convention and genuine understanding of why each formula is what it is. So this chapter does not just list formulas; it derives them, works every standard image-formation case, and drills each idea with fully solved examples. Read it as a complete module, not a summary.

PART A — RAY OPTICS

1. Reflection and the plane mirror

Light striking a surface obeys two laws of reflection: (i) the incident ray, reflected ray and normal lie in one plane; (ii) the angle of incidence equals the angle of reflection, both measured from the normal.

A plane mirror forms an image that is virtual, erect, laterally inverted, the same size as the object, and as far behind the mirror as the object is in front. If you walk toward a plane mirror at speed $v$, your image approaches you at $2v$ (each of you closes the gap at $v$). To see your full height you need a mirror only half your height, fixed at the right level — a favourite NEET result that follows from the equal-angle law.

2. Spherical mirrors — the sign convention

For curved mirrors we need bookkeeping. The Cartesian sign convention:

• All distances are measured from the pole $P$.
• Distances measured against the incident light (which travels left→right by convention) are negative; along it, positive.
• Heights above the axis are positive, below negative.

Consequences you must internalise:

• For a concave mirror, $f$ and $R$ are negative; for a convex mirror, positive.
• A real object distance $u$ is negative; a real image distance $v$ is negative (same side as the object); a virtual image distance is positive.

3. The mirror formula — derived

Consider a concave mirror of centre of curvature $C$, focus $F$, pole $P$. An object $AB$ on the axis forms an image $A^{\prime }{B}^{\prime }$. Using the two rays — one parallel to the axis reflecting through $F$, one through $C$ reflecting back on itself — the image forms where they cross. From the similar triangles produced by these rays (triangles $ABP$ and $A^{\prime }{B}^{\prime }P$ for magnification, and $FPN\sim F{B}^{\prime }{A}^{\prime }$ for the focal relation), and applying the sign convention, one obtains the mirror formula:

$\boxed{\frac{1}{v}+\frac{1}{u}=\frac{1}{f}},f=\frac{R}{2}$

and the magnification

$m=\frac{h^{\prime }}{h}=-\frac{v}{u}.$

A negative $m$ means an inverted (real) image; positive means erect (virtual). $|m|>1$ is magnified, $<1$ diminished.

The relation $f=R/2$ follows because a ray parallel and close to the axis striking at height $h$ reflects to cross the axis at the focus; geometry gives the focal length as half the radius.

Worked example 3.1. An object is placed 30 cm in front of a concave mirror of focal length 20 cm. Find the image. Sign convention: $u=-30$ cm, $f=-20$ cm. $\frac{1}{v}=\frac{1}{f}-\frac{1}{u}=\frac{1}{-20}-\frac{1}{-30}=-\frac{1}{20}+\frac{1}{30}=\frac{-3+2}{60}=-\frac{1}{60}.$ So $v=-60$ cm — a real, inverted image 60 cm in front. Magnification $m=-v/u=-(-60)/(-30)=-2$: inverted and twice the size.

Worked example 3.2. The same object at 10 cm ($u=-10$, inside the focus). $\frac{1}{v}=\frac{1}{-20}-\frac{1}{-10}=-\frac{1}{20}+\frac{1}{10}=\frac{1}{20}$, so $v=+20$ cm — a virtual, erect image behind the mirror, $m=-v/u=-20/(-10)=+2$, twice the size. (This is the make-up/shaving mirror at close range.)

4. Image formation by a concave mirror — every case

A convex mirror always gives a virtual, erect, diminished image between $P$ and $F$, whatever the object position — which is why it is the wide-field rear-view mirror ("objects are closer than they appear").

5. Refraction and Snell's law

Crossing into a new medium, light changes speed and bends, obeying Snell's law:

$n_1\sin {\theta }_1=n_2\sin {\theta }_2,n=\frac{c}{v}=\frac{\text{speed in vacuum}}{\text{speed in medium}}.$

• Refractive index $n\ge 1$; higher $n$ means a slower, "denser" (optically) medium. In glass ($n=1.5$) light slows to $v=c/n=2\times 10^8$ m/s.
• Entering a denser medium light bends toward the normal; entering a rarer medium, away from it.
• Relative index: $n_{21}=n_2/n_1=v_1/v_2={\lambda }_1/{\lambda }_2$. Note the frequency is unchanged on refraction — only speed and wavelength change.

Apparent depth. An object under water (real depth $d$) appears raised: the apparent depth is

$d_{\text{app}}=\frac{d}{n}.$

A pool 4 m deep with $n=4/3$ looks $4/(4/3)=3$ m deep. The same effect makes a straw look bent at the water surface.

Worked example 5.1. Light passes from air into a medium at $30°$ incidence, refracting to $19.5°$ (where $\sin 19.5°\approx 1/3$). Find $n$. $n=\frac{\sin 30°}{\sin 19.5°}=\frac{0.5}{1/3}=\mathrm{1.5.}$

6. Total internal reflection

When light travels from a denser to a rarer medium, the refracted ray bends away from the normal; at the critical angle ${\theta }_c$ it grazes along the surface (${\theta }_2=90°$). Beyond ${\theta }_c$, it cannot escape and reflects entirely back — total internal reflection (TIR). Setting ${\theta }_2=90°$ in Snell's law:

$n\sin {\theta }_c=1\cdot \sin 90°\Rightarrow \sin {\theta }_c=\frac{1}{n}.$

• For $n=1.5$, $\sin {\theta }_c=0.67$, ${\theta }_c\approx 42°$; for $n=2$, ${\theta }_c=30°$.
• Both conditions are required: denser → rarer, and angle $>{\theta }_c$.

Applications: optical fibres (light zig-zags down the core by repeated TIR — the basis of endoscopy and high-speed internet); the sparkle of diamond (very high $n=2.42$, tiny ${\theta }_c=24°$, so light bounces many times before exiting); totally-reflecting prisms in binoculars; and the mirage (light curving through hot, less-dense air near a road).

7. Refraction at a spherical surface → the lensmaker's formula

Refraction at a single spherical surface (radius $R$, media $n_1$ and $n_2$) obeys

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

A thin lens is two such surfaces back to back. Applying the relation to each surface and adding gives the lensmaker's formula:

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

where $n$ is the lens material's index relative to its surroundings. This is why a lens loses power in water (the relative $n$ drops toward 1) and why a lens of a given glass can be made stronger by curving it more (smaller $R$).

8. The thin lens formula, image cases and power

For a thin lens the object–image relation is

$\boxed{\frac{1}{v}-\frac{1}{u}=\frac{1}{f}},m=\frac{v}{u},P=\frac{1}{f(\text{metres})}\text{[dioptres]}.$

• Convex (converging): $f>0$, $P>0$. Concave (diverging): $f<0$, $P<0$.
• A lens of $f=20$ cm has $P=1/0.20=+5$ D. Lenses in contact add powers, $P=P_1+P_2$; a $+5$ D and $-2$ D pair gives $+3$ D.

Convex-lens image cases mirror the concave mirror: object beyond $2f$ → real, inverted, diminished; at $2f$ → same size; between $f$ and $2f$ → real, inverted, magnified (projector); inside $f$ → virtual, erect, magnified (magnifying glass). A concave lens always gives a virtual, erect, diminished image.

Worked example 8.1. An object is 15 cm from a convex lens of focal length 10 cm. Image? $u=-15$, $f=+10$: $\frac{1}{v}=\frac{1}{f}+\frac{1}{u}=\frac{1}{10}+\frac{1}{-15}=\frac{3-2}{30}=\frac{1}{30}$, so $v=+30$ cm — real, inverted, on the far side. $m=v/u=30/(-15)=-2$: inverted, magnified 2×.

9. The prism: deviation, minimum deviation and dispersion

A ray through a prism of apex angle $A$ deviates by an angle $\delta$. Geometry gives two relations:

$A=r_1+r_2,\delta =(i+e)-A,$

where $i,e$ are the incidence and emergence angles and $r_1,r_2$ the internal refraction angles. As $i$ varies, $\delta$ passes through a minimum ${\delta }_m$, where the ray passes symmetrically ($i=e$, $r_1=r_2=A/2$). Then the refractive index is

$n=\frac{\sin \left(\frac{A+{\delta }_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}.$

Dispersion. Because $n$ is slightly larger for violet than red, a prism splits white light into a spectrum: violet deviates most, red least. The spread is measured by the dispersive power $\omega =\frac{n_V-n_R}{n_Y-1}$.

10. Scattering: why the sky is blue

Sunlight scatters off air molecules with an intensity $\propto 1/{\lambda }^4$ (Rayleigh scattering), so short-wavelength blue scatters far more than red — filling the daytime sky with blue. At sunset, light travels a long slanted path, the blue is scattered away, and the transmitted light we see turns red-orange. The same $1/{\lambda }^4$ law explains why danger signals are red (least scattered, seen farthest).

11. Optical instruments and the eye

Simple microscope (magnifying glass): a convex lens with the object inside $f$; angular magnification $M=1+\frac{D}{f}$ (near point $D=25$ cm).

Compound microscope: objective + eyepiece; total magnification $M=m_o\times m_e\approx \frac{L}{f_o}\cdot \frac{D}{f_e}$ — short focal lengths for both give high power.

Astronomical telescope (normal adjustment): $M=\frac{f_o}{f_e}$, with a large objective focal length and small eyepiece; tube length $=f_o+f_e$.

The human eye focuses by accommodation (the ciliary muscles change the lens's focal length). Defects and their corrections:

• Myopia (short sight): image forms in front of the retina; far point is finite. Corrected by a concave (diverging) lens.
• Hypermetropia (long sight): image forms behind the retina; near point is distant. Corrected by a convex (converging) lens.
• Presbyopia: ageing loss of accommodation; corrected with bifocals.
• Astigmatism: unequal curvature; corrected with a cylindrical lens.

PART B — WAVE OPTICS

12. Huygens' principle and wavefronts

Every point on a wavefront acts as a source of secondary wavelets; the envelope of these wavelets is the new wavefront. This simple idea derives the laws of reflection and refraction, and — crucially — Snell's law with $n=c/v$, confirming that light slows in a denser medium (the particle theory wrongly predicted it speeds up). Wavefronts are spherical near a point source and effectively plane far away.

13. Coherence, superposition and interference

When two waves overlap, their displacements add (superposition). Sustained interference needs coherent sources — a constant phase relationship and equal frequency — which is why we split one source (two slits) rather than use two lamps. For two coherent waves meeting with path difference $\Delta x$:

$\text{Constructive (bright): }\Delta x=n\lambda ,\text{Destructive (dark): }\Delta x=\left(n+\frac{1}{2}\right)\lambda .$

The resultant intensity is $I=I_1+I_2+2\sqrt{I_1{I}_2}\cos \varphi$, so it ranges from $(\sqrt{I_1}+\sqrt{I_2}{)}^2$ (bright) down to $(\sqrt{I_1}-\sqrt{I_2}{)}^2$ (dark) — energy is redistributed, not lost.

14. Young's double-slit experiment — full derivation

Two slits separated by $d$ are illuminated by coherent light; a screen sits a distance $D$ away ($D\gg d$). For a point $P$ at height $y$ on the screen, the extra path from the far slit is

$\Delta x=\frac{dy}{D}.$

• Bright fringe: $\frac{dy}{D}=n\lambda \Rightarrow y_n=\frac{n\lambda D}{d}$.
• Dark fringe: $\frac{dy}{D}=\left(n+\frac{1}{2}\right)\lambda$.

The spacing between consecutive bright (or dark) fringes — the fringe width — is

$\boxed{\beta =\frac{\lambda D}{d}}.$

Fringes are equally spaced and of equal brightness. Widen the slit gap $d$ and fringes crowd together; increase $\lambda$ or $D$ and they spread.

Worked example 14.1. $\lambda =600$ nm, $d=1$ mm, $D=1$ m: $\beta =\frac{600\times 10^{-9}\times 1}{10^{-3}}=0.6$ mm. Worked example 14.2. If the whole apparatus is dipped in water ($n=4/3$), the wavelength shrinks to $\lambda /n$, so $\beta$ shrinks by $4/3$: new $\beta =0.6\times 3/4=0.45$ mm.

15. Diffraction

Light bending around an obstacle or through a narrow slit is diffraction. A single slit of width $a$ produces a broad central maximum flanked by dimmer ones; the first minima occur at $a\sin \theta =\lambda$, so the central maximum has angular width $\approx 2\lambda /a$ — wider for a narrower slit. Diffraction is significant only when the aperture is comparable to $\lambda$, which is why we rarely notice it for visible light through everyday openings.

Interference vs diffraction: interference comes from a finite number of coherent sources (equal, sharp fringes); diffraction comes from a continuous spread of wavelets across one aperture (a bright central band with rapidly fading side bands).

16. Resolving power

Diffraction sets the ultimate limit on how finely any instrument can resolve two close points (Rayleigh criterion). For a telescope the smallest resolvable angle is $\theta \approx 1.22\lambda /D$ — a larger aperture $D$ resolves finer detail (and gathers more light), which is why research telescopes and electron microscopes (tiny $\lambda$) are built large or short-wavelength.

17. Polarisation

Ordinary light oscillates in all directions perpendicular to travel; polarisation restricts it to one plane. Because only transverse waves can be polarised, polarisation is direct proof that light is transverse (sound, being longitudinal, cannot be polarised).

• Malus's law: light of intensity $I_0$ through a polariser at angle $\theta$ to the polarisation axis transmits $I=I_0{\cos }^2\theta$. (Two crossed polarisers, $\theta =90°$, block all light.)
• Brewster's law: at the polarising angle ${\theta }_p$, reflected light is fully polarised, with $\tan {\theta }_p=n$.
• Uses: Polaroid sunglasses (cut glare, which is partially polarised), LCD screens, and polarimetry to measure optically active molecules such as sugars — a direct link to biochemistry.

18. Common traps NEET sets here

• Sign-convention slips — the single biggest source of mirror/lens errors; fix the convention before substituting.
• $f=R/2$, not $R$ — focal length is half the radius of curvature.
• TIR direction — only denser → rarer, and only beyond the critical angle.
• Frequency changes on refraction — it does not; only speed and wavelength do.
• Power in cm — dioptres need $f$ in metres; lenses in contact add powers.
• Which lens for which defect — concave (myopia), convex (hypermetropia).
• Fringe width in a medium — $\lambda$ (and so $\beta$) shrinks by a factor $n$.
• Violet vs red deviation — violet deviates and scatters most (short $\lambda$).
• Interference ≠ diffraction — two coherent sources vs one continuous aperture.

19. Memory aids

• "f is half of R; against the light is negative."
• "Denser → bend toward the normal; apparent depth = real/n."
• "sin θc = 1/n; denser to rarer only."
• "Dioptre = 1/f in metres; add in contact."
• "Concave for myopia, convex for hyper; cylindrical for astigmatism."
• "β = λD/d; in water it shrinks by n."
• "Blue scatters (1/λ⁴), so blue sky, red sunset."
• "Malus cos², Brewster tan θ = n."

20. Exam protocol

1. Mirrors: $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$, $m=-v/u$, $f=R/2$ — keep the sign convention; know the six concave cases.
2. Refraction: $n_1\sin {\theta }_1=n_2\sin {\theta }_2$, $n=c/v$; frequency unchanged; apparent depth $=d/n$.
3. TIR: $\sin {\theta }_c=1/n$, denser→rarer, angle > critical; fibres, diamond, mirage.
4. Lenses: lensmaker $\frac{1}{f}=(n-1)(\frac{1}{R_1}-\frac{1}{R_2})$; $\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$; power $1/f$ (m); add in contact.
5. Prism: $A=r_1+r_2$, $\delta =(i+e)-A$; minimum deviation gives $n$; violet deviates most.
6. Scattering $\propto 1/{\lambda }^4$: blue sky, red sunset. Eye: concave (myopia), convex (hypermetropia).
7. Instruments: microscope $M=1+D/f$; telescope $M=f_o/f_e$.
8. Wave optics: coherent sources; Young's $\beta =\lambda D/d$ (shrinks by $n$ in a medium); single-slit minima $a\sin \theta =\lambda$.
9. Polarisation proves light transverse; Malus $I=I_0{\cos }^2\theta$; Brewster $\tan {\theta }_p=n$.