Ray Optics & Optical Instruments | Class 12 Physics (NCERT Chapter 9)

CBSE Class 12 › Unit VI: Optics

Ray Optics & Optical Instruments

NCERT Chapter 9 • Reflection, Refraction, Lenses & Microscopes

NCERT 2025–26 Unit VI • ~14 Marks JEE Main • 2-3 Questions

1. Reflection by Spherical Mirrors

In ray optics, spherical mirrors (concave and convex) form images by reflecting light. Real or virtual, inverted or erect — the nature of the image depends on the object’s position and the mirror’s curvature. The focal point and radius of curvature are geometrically linked, leading to the mirror equation — the foundation of all mirror problems.

Diagram showing arc MP for f = R/2 derivation in concave mirror. — Ray tracing for concave mirror.

Derivation: Relation $f = \dfrac{R}{2}$ 2 Marks

Assumptions: Paraxial rays (small aperture), point object on principal axis.
Consider a concave mirror with center of curvature $C$ and pole $P$ . A ray parallel to the principal axis strikes the mirror at $M$ .
After reflection, it passes through focus $F$ .

Arc Length Method (Precise):
$\theta = \dfrac{\text{arc } MP}{CP} = \dfrac{\text{arc } MP}{R}$
$2\theta = \dfrac{\text{arc } MP}{FP} = \dfrac{\text{arc } MP}{f}$
$R \theta = f \cdot 2\theta \implies f = \dfrac{R}{2}$

Ray diagrams showing image formation by concave and convex mirrors. — Images formed by spherical mirrors follow standard ray tracing rules.

Derivation: Mirror Equation & Magnification 3 Marks

Mirror Equation Setup:
Consider object $AB$ at distance $u$ from pole $P$ of a concave mirror. Image $A'B'$ is formed at distance $v$ .
Draw ray $BM$ parallel to axis → reflects through $F$ .
Draw ray $BP$ through center $C$ → reflects back on itself.
Let the two reflected rays meet at $B'$ .

In $\triangle ABP$ and $\triangle A'B'P$ :
$\angle APB = \angle A'PB'$ (vertically opposite)
$\angle PAB = \angle PA'B' = 90^\circ$
So $\triangle ABP \sim \triangle A'B'P$ → $\dfrac{AB}{A'B'} = \dfrac{PB}{PB'} = \dfrac{-u}{-v} = \dfrac{u}{v}$ …(1)

In $\triangle PDF$ and $\triangle A'B'F$ :
$\angle PDF = \angle A'B'F = 90^\circ$
$\angle PFD = \angle A'FB'$ (vertically opposite)
So $\triangle PDF \sim \triangle A'B'F$ → $\dfrac{PD}{A'B'} = \dfrac{PF}{A'F}$
But $PD = AB$ , and for small aperture $A'F \approx PB' = v$ , so:
$\dfrac{AB}{A'B'} = \dfrac{f}{v - f}$ …(2)

From (1) and (2): $\dfrac{u}{v} = \dfrac{f}{v - f}$
Cross-multiply: $u(v - f) = vf$ → $uv - uf = vf$
Divide both sides by $uvf$ : $\dfrac{1}{f} - \dfrac{1}{v} = \dfrac{1}{u}$
Rearranged: $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$
(Use sign convention: $u$ negative, $v$ negative for real image, $f$ negative for concave mirror.)

Derivation of Magnification (m):
Magnification is defined as the ratio of image height to object height:

$m = \frac{\text{Height of Image }(A'B')}{\text{Height of Object }(AB)} = \frac{h'}{h}$

From the similarity of $\triangle ABP$ and $\triangle A'B'P$ (Step 2):

$\frac{AB}{A'B'} = \frac{PB}{PB'} \Rightarrow \frac{h}{h'} = \frac{-u}{-v} = \frac{u}{v}$

(Note: Object height $h$ is taken as positive; image height $h'$ is negative for inverted image in sign convention.)

Therefore:

$\frac{h'}{h} = -\frac{v}{u}$

Magnification formula for spherical mirrors:

$m = \dfrac{h'}{h} = -\dfrac{v}{u}$

Interpretation:
– If $m$ is negative → image is inverted (real).
– If $m$ is positive → image is erect (virtual).
– If $|m| > 1$ → image is magnified.
– If $|m| < 1$ → image is diminished.

Sign Convention Summary: Mirrors and Lenses

In the Cartesian sign convention (used in CBSE/NCERT):

All distances are measured from the pole (P) of mirror or optical center (O) of lens.
Direction of incident light is taken as the positive X-axis.
Distances measured against incident light → negative.
Distances measured along incident light → positive.

Optical Element	Focal Length (f)	Type	Focus is…	Nature
Concave Mirror	–ve	Converging	Real (in front)	Converges parallel rays to a real focus
Convex Mirror	+ve	Diverging	Virtual (behind)	Diverges parallel rays as if from virtual focus
Convex (Converging) Lens	+ve	Converging	Real (on opposite side)	Converges parallel rays to a real focus
Concave (Diverging) Lens	–ve	Diverging	Virtual (on same side)	Diverges parallel rays as if from virtual focus

2. Refraction and Total Internal Reflection

When light crosses the boundary between two transparent media, it bends — a phenomenon called refraction. This bending explains everyday effects like a straw appearing bent in water or a swimming pool looking shallower than it is. The extent of bending is governed by Snell’s Law and the media’s refractive indices.

A. Snell’s Law & Refractive Index

$n_1 \sin i = n_2 \sin r$
$\dfrac{\sin i}{\sin r} = n_{21} = \dfrac{n_2}{n_1}$
Also, $n_{12} = \dfrac{1}{n_{21}}$

B. Apparent Depth

When an object is placed in a denser medium (like water) and viewed from a rarer medium (like air), it appears at a smaller depth than its real depth.

Ray diagram showing how refraction causes an object to appear raised at an apparent depth.

Derivation: Apparent Depth 3 Marks

Setup:
Object $O$ is at real depth $h$ in denser medium (index $n_1$ ).
Observer is in rarer medium (index $n_2$ ).
Consider two rays from $O$ :
1. Normal ray $OA$ passes straight.
2. Ray $OB$ incident at angle $i$ , refracts at angle $r$ away from normal.
The backward extension of the refracted ray meets normal $OA$ at $I$ (Apparent position). Apparent depth $AI = h'$ .

Small Angle Approximation:
For near-normal viewing, angles $i$ and $r$ are small.
$\sin i \approx \tan i \approx \dfrac{AB}{AO} = \dfrac{x}{h}$
$\sin r \approx \tan r \approx \dfrac{AB}{AI} = \dfrac{x}{h'}$

Applying Snell’s Law:
$n_1 \sin i = n_2 \sin r$
Substituting approximations: $n_1 \left(\dfrac{x}{h}\right) = n_2 \left(\dfrac{x}{h'}\right)$

Result:
$\dfrac{n_1}{h} = \dfrac{n_2}{h'}$ implies $h' = h \dfrac{n_2}{n_1}$ .
If observer is in air ( $n_2 \approx 1$ ) and object in medium $n$ :
Apparent Depth $h' = \dfrac{h}{n}$

Shift $= \text{Real Depth} - \text{Apparent Depth} = h \left(1 - \dfrac{1}{n}\right)$

C. Total Internal Reflection (TIR)

When light travels from a denser to a rarer medium, it bends away from the normal. If the angle of incidence exceeds a certain value — the critical angle — no refraction occurs. Instead, 100% of the light reflects back into the denser medium. This is Total Internal Reflection (TIR), the principle behind optical fibers and brilliant diamond sparkle.

Diagram explaining the critical angle and total internal reflection.

$\sin i_c = \dfrac{n_2}{n_1}$ where $n_1 > n_2$

D. Optical Fibers: Application of TIR

An optical fiber is a flexible, transparent strand of glass (silica) or plastic, slightly thicker than a human hair, that transmits light signals over long distances with minimal loss. It works on the principle of Total Internal Reflection.

Structure of an optical fiber showing core, cladding, and light propagation via total internal reflection. — An optical fiber consists of a core (high n) surrounded by cladding (lower n). Light undergoes TIR at the core-cladding interface.

Working Principle 2 Marks

Structure:
– Core: Central part made of high-refractive-index material ( $n_1$ ).
– Cladding: Outer layer with slightly lower refractive index ( $n_2 < n_1$ ).
– Buffer Coating: Protective plastic layer (not involved in light guidance).

Light Propagation:
Light enters the core at the acceptance angle ( $\theta_a$ ).
At the core–cladding boundary, the angle of incidence is greater than the critical angle ( $i > i_c$ ), so TIR occurs repeatedly, guiding light along the fiber—even around gentle curves.

Acceptance Angle & Numerical Aperture (NA):
The maximum angle of incidence at the fiber’s entrance for which TIR is possible is called the acceptance angle.
It is given by:

$\sin \theta_a = \sqrt{n_1^2 - n_2^2}$

The term $\sqrt{n_1^2 - n_2^2}$ is called the Numerical Aperture (NA) — a measure of light-gathering ability.

Why optical fibers? Key Advantages:

Low Loss: Signals can travel 100+ km without amplification.
High Bandwidth: Carry thousands of phone calls or HD video streams simultaneously.
No EMI: Immune to electromagnetic interference (unlike copper wires).
Security: Light doesn’t leak; hard to tap without detection.

Applications: Telecommunications, endoscopy, fiber lasers, sensors, and internet backbone.

3. Refraction at Spherical Surfaces & Lenses

Lenses are transparent optical devices that refract light to converge or diverge rays, enabling magnification, focusing, and vision correction. A lens’s power is determined by its curvature and material. The Lens Maker’s Formula connects these physical properties to its focal length — essential for designing camera lenses, eyeglasses, and microscopes.

Geometry for deriving the refraction formula at a spherical surface.

Derivation: Refraction at Single Spherical Surface 3 Marks

Consider convex spherical surface separating medium $n_1$ (object side) and $n_2$ (image side), radius $R$ .
Object at $O$ (dist. $u$ ), image at $I$ (dist. $v$ ).
Ray from $O$ strikes surface at $M$ , refracts to $I$ .
Normal is $MC$ (C = center of curvature).

In $\triangle OMC$ : exterior angle $\angle i = \angle OMC = \angle MOC + \angle OCM = \alpha + \gamma$
In $\triangle IMC$ : exterior angle $\angle r = \angle IMC = \angle MIC + \angle ICM = \beta + \gamma$
For paraxial rays: $\alpha \approx \tan \alpha = \dfrac{h}{-u}$ , $\beta \approx \dfrac{h}{v}$ , $\gamma \approx \dfrac{h}{R}$
(Use sign convention: $u$ negative, $R$ positive for convex surface).

Snell’s law: $n_1 \sin i = n_2 \sin r \approx n_1 i = n_2 r$
So: $n_1(\alpha + \gamma) = n_2(\beta + \gamma)$
Substitute angles: $n_1\left(\dfrac{h}{-u} + \dfrac{h}{R}\right) = n_2\left(\dfrac{h}{v} + \dfrac{h}{R}\right)$
Divide by $h$ : $n_1\left(-\dfrac{1}{u} + \dfrac{1}{R}\right) = n_2\left(\dfrac{1}{v} + \dfrac{1}{R}\right)$

Rearranged: $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$

Derivation: Lens Maker’s Formula 5 Marks

A thin lens has two spherical surfaces: first radius $R_1$ , second $R_2$ .
Refractive index of lens material = $n$ , surrounding = $n_1 = 1$ (air).
Object at infinity → image at first focus $F_1$ .
Apply refraction formula to each surface.

Surface 1 (convex):
$u = \infty$ , $v = v_1$ , $R = R_1$
$\dfrac{n}{v_1} - \dfrac{1}{\infty} = \dfrac{n - 1}{R_1}$ → $\dfrac{n}{v_1} = \dfrac{n - 1}{R_1}$ …(1)

Surface 2 (concave):
The image from Surface 1 acts as virtual object for Surface 2.
So $u_2 = v_1$ , $v = f$ (final image at focus), $R = -R_2$ (since center is on left)
$\dfrac{1}{f} - \dfrac{n}{v_1} = \dfrac{1 - n}{-R_2} = \dfrac{n - 1}{R_2}$ → $\dfrac{1}{f} = \dfrac{n - 1}{R_2} + \dfrac{n}{v_1}$ …(2)

Substitute (1) into (2):
$\dfrac{1}{f} = \dfrac{n - 1}{R_2} + \dfrac{n - 1}{R_1} = (n - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$
$\dfrac{1}{f} = (n - 1) \left( \dfrac{1}{R_1} - \dfrac{1}{R_2} \right)$

Formulas for Lenses

Thin Lens Formula: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$
Magnification: $m = \dfrac{h'}{h} = \dfrac{v}{u}$
Power of a Lens: $P = \dfrac{1}{f(\text{in meters})}$ (Unit: Dioptre, D)

Combination of Thin Lenses in Contact

Ray diagram showing two convex lenses in contact converging parallel rays to a combined focal point, illustrating the formula 1/f = 1/f1 + 1/f2

Derivation: Combination of Thin Lenses in Contact 3 Marks

Setup:
Consider two thin lenses L₁ and L₂ placed in close contact (separation ≈ 0).
Let their focal lengths be $f_1$ and $f_2$ respectively.
An object is placed at a distance $u$ from the combination.

Step 1: Effect of First Lens (L₁)
The first lens L₁ forms an intermediate image $I_1$ at a distance $v_1$ , given by the lens formula:

$\frac{1}{v_1} - \frac{1}{u} = \frac{1}{f_1} \quad \text{...(1)}$

Step 2: Effect of Second Lens (L₂)
Since the lenses are in contact, the intermediate image $I_1$ acts as the virtual object for the second lens L₂.
Object distance for L₂ = $v_1$
Let the final image be formed at distance $v$ .
Applying lens formula to L₂:

$\frac{1}{v} - \frac{1}{v_1} = \frac{1}{f_2} \quad \text{...(2)}$

Step 3: Combine the Two Equations
Add equations (1) and (2):

$\left( \frac{1}{v_1} - \frac{1}{u} \right) + \left( \frac{1}{v} - \frac{1}{v_1} \right) = \frac{1}{f_1} + \frac{1}{f_2}$

The $\frac{1}{v_1}$ terms cancel out:

$\frac{1}{v} - \frac{1}{u} = \frac{1}{f_1} + \frac{1}{f_2}$

Step 4: Define Equivalent Focal Length
The combination behaves like a single lens with focal length $f$ , so:

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

Comparing with the result above:

$\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}$

Generalization:
For $n$ thin lenses in contact:

$\frac{1}{f_{\text{eq}}} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + \cdots + \frac{1}{f_n}$

The power of the combination is also additive:

$P_{\text{eq}} = P_1 + P_2 + P_3 + \cdots + P_n$

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

Combination of Thin Lenses in Contact:
Effective Focal Length: $\dfrac{1}{f_{eq}} = \dfrac{1}{f_1} + \dfrac{1}{f_2} + \dots$
Effective Power: $P = P_1 + P_2 + \dots$
Total Magnification: $m = m_1 \times m_2 \times \dots$

4. Refraction through a Prism

A prism is a transparent optical element with flat, polished surfaces that refract light. Unlike a parallel slab, a prism’s non-parallel faces cause light to bend twice in the same direction, resulting in a net deviation. This property is exploited in spectrometers to disperse white light into its constituent colors (dispersion).

Path of a light ray through a glass prism showing angle of deviation.

Derivations: Prism Relations 3 Marks

1. Angle Relations (Geometry):
In quadrilateral $AQNR$ : $\angle A + \angle QNR = 180^\circ$ → $\angle QNR = 180^\circ - A$
In triangle $QNR$ : $r_1 + r_2 + \angle QNR = 180^\circ$ → $r_1 + r_2 + (180^\circ - A) = 180^\circ$
So: $r_1 + r_2 = A$

2. Angle of Deviation ( $\delta$ ):
Deviation at first surface: $\delta_1 = i - r_1$
Deviation at second surface: $\delta_2 = e - r_2$
Total deviation: $\delta = \delta_1 + \delta_2 = (i + e) - (r_1 + r_2)$
Using $r_1 + r_2 = A$ : $\delta = i + e - A$

3. Minimum Deviation Condition:

Angle of deviation graph, at minimum deviation i = e.

For a given prism and wavelength, $\delta$ is minimum when $i = e$ → then $r_1 = r_2 = r$ .
From $r_1 + r_2 = A$ : $2r = A$ → $r = \dfrac{A}{2}$
From $\delta = i + e - A = 2i - A$ : $i = \dfrac{A + \delta_m}{2}$

4. Refractive Index:
Snell’s law at first surface: $n = \dfrac{\sin i}{\sin r} = \dfrac{\sin\left(\dfrac{A + \delta_m}{2}\right)}{\sin\left(\dfrac{A}{2}\right)}$
$n = \dfrac{\sin\left(\dfrac{A + \delta_m}{2}\right)}{\sin\left(\dfrac{A}{2}\right)}$

5. Optical Instruments

Human eyes have limited resolution and magnification. Optical instruments like microscopes and telescopes overcome these limits by using combinations of lenses (or mirrors) to produce highly magnified or resolved images of tiny or distant objects. Understanding their working principle is key to applications from biology to astronomy.

A. Simple Microscope (Magnifying Glass)

Image at Near Point (D = 25 cm): $m = 1 + \dfrac{D}{f}$
Image at Infinity: $m = \dfrac{D}{f}$

B. Compound Microscope

Consists of Objective (short $f_o$ ) and Eyepiece (longer $f_e$ ). Tube length $L = v_o - f_o \approx 16\,\text{cm}$ (standard).

Derivation: Magnification (Image at Infinity) 3 Marks

Objective: Forms real, inverted image $I_1$ near focus of eyepiece.
Magnification $m_o = \dfrac{\text{size of } I_1}{\text{object size}} = \dfrac{L}{f_o}$ (since $v_o \approx L + f_o \approx L$ , and $u_o \approx -f_o$ )

Eyepiece: Acts as simple microscope with image at infinity.
Magnification $m_e = \dfrac{D}{f_e}$

Total magnification: $m = m_o \times m_e = \left(\dfrac{L}{f_o}\right) \times \left(\dfrac{D}{f_e}\right)$
$m = \dfrac{L D}{f_o f_e}$

C. Astronomical Telescope (Refracting)

Objective: large $f_o$ , Eyepiece: small $f_e$ . Used for distant objects ( $u = \infty$ ).

Derivation: Magnifying Power (Normal Adjustment) 3 Marks

For distant object, rays from top are parallel, inclined at angle $\alpha$ .
Objective forms image $I$ in focal plane. Angle subtended by image at eye (via eyepiece) = $\beta$ .
Magnifying power $m = \dfrac{\beta}{\alpha}$

From objective: size of image $h = f_o \tan \alpha \approx f_o \alpha$
From eyepiece (used as magnifier): $\beta = \dfrac{h}{f_e} = \dfrac{f_o \alpha}{f_e}$
So: $m = \dfrac{\beta}{\alpha} = \dfrac{f_o}{f_e}$

Magnifying Power: $m = \dfrac{f_o}{f_e}$
Tube Length (Normal Adjustment): $L = f_o + f_e$

6. Reflecting Telescope (Cassegrain)

Refracting telescopes (using lenses) suffer from chromatic and spherical aberrations, especially at large sizes. Reflecting telescopes replace the objective lens with a curved mirror, eliminating color fringing and allowing for massive apertures. The Cassegrain design folds the light path using a secondary mirror, creating a compact yet powerful instrument — the backbone of modern astronomy (e.g., Hubble Space Telescope).

Ray diagram of a Cassegrain reflecting telescope showing the path of light. — A Cassegrain telescope uses a concave primary mirror and a convex secondary mirror.

The refracting telescope suffers from chromatic aberration (colors focus at different points) and spherical aberration. To overcome this, Reflecting Telescopes use mirrors instead of lenses for the objective.

Construction:
Primary Mirror: A large concave parabolic mirror. Parabolic shape removes spherical aberration.
Secondary Mirror: A small convex mirror placed before the focus of the primary mirror.
Eyepiece: Light reflects off the secondary mirror and passes through a hole in the primary to the eyepiece.

Magnifying Power: $m = \dfrac{f_o}{f_e} = \dfrac{R/2}{f_e}$

Advantages over Refracting Telescope:

No Chromatic Aberration: Mirrors reflect all wavelengths equally; lenses disperse light.
No Spherical Aberration: Parabolic mirrors focus all rays to a single point.
Large Aperture: Mirrors are lighter and can be supported from the back, allowing for huge diameters (high resolving power). Lenses are heavy and can only be supported at the rim.
Cost: Large mirrors are cheaper to manufacture than large defect-free lenses.

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