Geometrical Optics & Matrix Methods: Master Notes

Chapter Overview

Geometrical optics analyzes light propagation using Fermat's Principle of least time. In IIT-JEE (Advanced) and Olympiads, top questions focus on single spherical refracting surfaces, silvered lens equivalents, prism minimum deviation & grazing emergence, and optical instruments magnification.

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1. Refraction at Spherical Surfaces & Thin Lenses

Multi-Mode DiagramGeometrical Optics: Refraction, Lenses & Prisms

Option 1: Publication-Grade Scientific Vector SVG

Prism minimum deviation ray path and silvered lens equivalent focal length.

Prism Formula: $\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin(A/2)}$

Option 2: Interactive 60 FPS Canvas Physics Engine

Interactive ray tracer through glass prism with minimum deviation tuning.

Refractive Index μ: 1.5

Minimum Deviation ($\delta_m$):37.2°

Option 3: Interactive 3D Spatial Viewport

3D dispersive optical prism.

+X

+Y

+Z

👆 Click and drag anywhere to rotate in 3D space

Azimuth Angle (ψ):35°

Elevation Angle (θ):-22°

Prism Refraction:$\mu = \frac{\sin((A+\delta)/2)}{\sin(A/2)}$

Option 5: Reactive Parametric Physics with Live Calculus

Prism deviation calculator.

Prism Angle A: 60 °

Index μ: 1.5

Delta min ($\delta_m$):$37.2°$

Active Mode:Option 1 (Print & Derivation Vector SVG)Switch modes using the tabs above to explore different visualizations.

$$\boxed{{\displaystyle \frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}}}$$$$\text{Transverse Magnification: }m=\frac{h_i}{h_o}=\frac{{\mu }_{1}v}{{\mu }_{2}u}$$

1.1 Lens Maker's Formula

$$\boxed{{\displaystyle \frac{1}{f}=(\frac{{\mu }_{\text{lens}}}{{\mu }_{\text{surrounding}}}-1)(\frac{1}{R_1}-\frac{1}{R_2})}}$$

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2. Silvered Lens Systems (Equivalent Mirror Method)

When one surface of a lens is silvered to create a reflecting system:

<!-- Lens Front Surface --> <path d="M 180 20 Q 160 70 180 120" fill="none" stroke="var(--vp-c-brand-1)" stroke-width="2.5"/> <!-- Silvered Rear Surface --> <path d="M 210 20 Q 230 70 210 120" fill="none" stroke="#ef4444" stroke-width="3.5"/> <g stroke="#ef4444" stroke-width="1.5"> <line x1="210" y1="30" x2="225" y2="25"/> <line x1="218" y1="50" x2="233" y2="45"/> <line x1="220" y1="70" x2="235" y2="65"/> <line x1="218" y1="90" x2="233" y2="85"/> <line x1="210" y1="110" x2="225" y2="105"/> </g> <text x="245" y="75" fill="#ef4444" font-size="12" font-weight="700">Silvered Mirror</text> 

Figure Silvered Lens System: Light refracts through the lens, reflects at the silvered rear surface, and refracts back through the lens (P_net = 2 P_L + P_M).

$$\boxed{{\displaystyle P_{\text{net}}=2P_{\text{lens}}+P_{\text{mirror}}⟹-\frac{1}{F_{\text{eq}}}=\frac{2}{f_{\text{lens}}}-\frac{1}{f_{\text{mirror}}}}}$$

(The entire system always behaves as an equivalent curved mirror with focal length $F_{\text{eq}}$!)

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3. Prism Optics: Dispersion & Minimum Deviation

$$\text{Angle of Prism: }A=r_1+r_2$$$$\text{Angle of Deviation: }\delta =i+e-A$$

<!-- Incident Ray --> <line x1="40" y1="140" x2="155" y2="117" stroke="#f59e0b" stroke-width="2.5" marker-end="url(#arrow-red)"/> <text x="75" y="125" fill="#f59e0b" font-size="13" font-weight="700">i</text> <!-- Internal Refracted Ray --> <line x1="155" y1="117" x2="295" y2="128" stroke="#f59e0b" stroke-width="2.5" marker-end="url(#arrow-red)"/> <text x="180" y="135" fill="var(--vp-c-text-2)" font-size="11">r_1</text> <text x="260" y="140" fill="var(--vp-c-text-2)" font-size="11">r_2</text> <!-- Emergent Ray --> <line x1="295" y1="128" x2="420" y2="160" stroke="#f59e0b" stroke-width="2.5" marker-end="url(#arrow-red)"/> <text x="365" y="145" fill="#f59e0b" font-size="13" font-weight="700">e</text> 

Figure Prism Optics: Refraction geometry with incident angle i, internal angles r_1, r_2, emergent angle e, and deviation angle delta.

3.1 Condition for Minimum Deviation ($\delta ={\delta }_m$)

At minimum deviation, the ray passes symmetrically through the prism:

$$i=e\text{and}{r}_1=r_2=\frac{A}{2}$$$$\boxed{{\displaystyle \mu =\frac{\sin \left(\frac{A+{\delta }_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}}}$$

• Thin Prism ($A\le {10}^{\circ }$): $\delta \approx (\mu -1)A$.
• Dispersive Power: $\omega =\frac{{\delta }_V-{\delta }_R}{{\delta }_Y}=\frac{{\mu }_V-{\mu }_R}{{\mu }_Y-1}$.

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4. Optical Instruments

Instrument	Magnification (Near Point $D=25\text{ cm}$)	Magnification (Normal / Relaxed Eye)	Tube Length $L$
Simple Microscope	$M=1+\frac{D}{f}$	$M=\frac{D}{f}$	—
Compound Microscope	$M=-\frac{v_o}{u_o}(1+\frac{D}{f_e})$	$M=-\frac{v_o}{u_o}\left(\frac{D}{f_e}\right)$	$L=v_o+u_e$
Astronomical Telescope	$M=-\frac{f_o}{f_e}(1+\frac{f_e}{D})$	$M=-\frac{f_o}{f_e}$	$L=f_o+f_e$

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5. Authentic Previous Years Questions (PYQs)

PYQ 1: JEE Advanced 2022 (Paper 2) — Silvered Plano-Convex Lens

Question:
A plano-convex lens of refractive index $\mu =1.5$ has a curved surface of radius $R=20\text{ cm}$. The curved surface is silvered. Find the position of the image when an object is placed at distance $u=30\text{ cm}$ in front of the flat face.

Step-by-Step Solution:

1. Focal Length of the Lens (before silvering):$$\frac{1}{f_L}=(\mu -1)(\frac{1}{\mathrm{\infty }}-\frac{1}{-R})=(1.5-1)\left(\frac{1}{20}\right)=\frac{0.5}{20}=\frac{1}{40}⟹f_L=40\text{ cm}$$
2. Focal Length of the Silvered Curved Mirror: $f_M=-\frac{R}{2}=-\frac{20}{2}=-10\text{ cm}$.
3. Equivalent Power and Focal Length $F_{\text{eq}}$:$$-\frac{1}{F_{\text{eq}}}=\frac{2}{f_L}-\frac{1}{f_M}=\frac{2}{40}-(-\frac{1}{10})=\frac{1}{20}+\frac{1}{10}=\frac{3}{20}⟹F_{\text{eq}}=-\frac{20}{3}\text{ cm}$$(The system behaves as a concave mirror of focal length $20/3\text{ cm}$!)
4. Mirror Formula:$$\frac{1}{v}+\frac{1}{u}=\frac{1}{F_{\text{eq}}}⟹\frac{1}{v}+\frac{1}{-30}=\frac{1}{-20/3}=-\frac{3}{20}$$$$\frac{1}{v}=-\frac{3}{20}+\frac{1}{30}=\frac{-9+2}{60}=-\frac{7}{60}⟹\boxed{{\displaystyle v=-\frac{60}{7}\text{ cm}\approx -8.57\text{ cm}}}$$

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PYQ 2: JEE Advanced 2020 — Grazing Incidence on Equilateral Prism

Question:
An equilateral glass prism ($A={60}^{\circ }$) has refractive index $\mu =\sqrt{2}$. A ray of light is incident on one face at grazing incidence ($i={90}^{\circ }$). Find the angle of emergence $e$.

Step-by-Step Solution:

1. First Surface Refraction ($i={90}^{\circ }$):$$\sin ({90}^{\circ })=\mu \sin r_1⟹1=\sqrt{2}\sin r_1⟹\sin r_1=\frac{1}{\sqrt{2}}⟹r_1={45}^{\circ }$$
2. Angle at Second Surface:$$A=r_1+r_2⟹{60}^{\circ }={45}^{\circ }+r_2⟹r_2={15}^{\circ }$$
3. Second Surface Refraction:$$\mu \sin r_2=1\cdot \sin e⟹\sqrt{2}\sin ({15}^{\circ })=\sin e$$Since $\sin ({15}^{\circ })=\frac{\sqrt{6}-\sqrt{2}}{4}$:$$\sin e=\sqrt{2}\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)=\frac{\sqrt{12}-2}{4}=\frac{2\sqrt{3}-2}{4}=\frac{\sqrt{3}-1}{2}$$$$\boxed{{\displaystyle e=\arcsin \left(\frac{\sqrt{3}-1}{2}\right)\approx {21.47}^{\circ }}}$$

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6. High-Yield Formula Sheet

Entity	Formula	Notes
Spherical Refraction	$\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}$	Single boundary
Lens Maker	$\frac{1}{f}=({\mu }_{\text{rel}}-1)(\frac{1}{R_1}-\frac{1}{R_2})$	Thin lens
Silvered Lens	$-\frac{1}{F_{\text{eq}}}=\frac{2}{f_L}-\frac{1}{f_M}$	Acts as curved mirror
Prism Formula	$\mu =\frac{\sin \left(\frac{A+{\delta }_m}{2}\right)}{\sin (A/2)}$	Symmetrical ray path
Telescope Normal Length	$L=f_o+f_e$	Focus at infinity