Ch 09 - RAY OPTICS AND OPTICAL INSTRUMENTS12th Physics

Visible Light: Electromagnetic radiation with a wavelength range of roughly 400 nm to 750 nm.

Speed in Vacuum: The highest attainable speed in nature, exactly $c = 2.99792458 \times 10^8 \text{ m/s}$, typically approximated as $3 \times 10^8 \text{ m/s}$.

Because the wavelength of light is extremely small compared to everyday objects, it can be treated as a ray traveling in a straight line. A bundle of such rays forms a beam.

Law 1: The angle of reflection ($r$) is equal to the angle of incidence ($i$).

Law 2: The incident ray, the reflected ray, and the normal to the reflecting surface at the point of incidence all lie in the same plane.

Boundary Condition: These laws are strictly valid at each point on any reflecting surface, whether plane or curved. For a spherical mirror, the normal at any point is the line joining that point to the centre of curvature.

Real Image: Formed when light rays actually converge and meet at a physical point in space after reflection or refraction. Real images are always inverted and can be captured on a screen.

Virtual Image: Formed when light rays do not actually meet but appear to diverge from a point when produced backwards. Virtual images are always erect and cannot be captured on a screen.

Variables & Units:

• f = Focal length (m)
• R = Radius of curvature (m)

Conditions/Notes:

• This relation is an approximation strictly valid only for paraxial rays (rays incident close to the pole, making small angles with the principal axis).
• Sign Convention Application: $f$ is negative for concave mirrors and positive for convex mirrors.

Law 1 (Coplanarity): The incident ray, the refracted ray, and the normal to the interface at the point of incidence all lie in the same plane.

Law 2 (Snell's Law): The ratio of the sine of the angle of incidence ($i$) to the sine of the angle of refraction ($r$) is constant for a given pair of media and a specific wavelength of light.

Optical Density: Determined strictly by the ratio of the speed of light in two media. A medium is optically denser if light travels slower in it.

Mass Density: The physical mass per unit volume of a substance.

Distinction: They are entirely independent physical properties. An optically denser medium can possess a lower mass density than an optically rarer medium.

Example: Turpentine and Water. Turpentine floats on water because its mass density is lower, but its refractive index is higher than water's, making it optically denser.

Variables & Units:

• v = Image distance from pole (m)
• u = Object distance from pole (m)
• f = Focal length (m)

Conditions/Notes:

• Valid for both concave and convex mirrors, and for both real and virtual images.
• The New Cartesian Sign Convention MUST be applied when substituting numerical values into this equation.

First Principal Focus ($F_1$): The point on the principal axis where an object must be placed so that the emergent rays become parallel to the principal axis (image at infinity).

Second Principal Focus ($F_2$): The point on the principal axis where a beam of light initially parallel to the axis converges (or appears to diverge from) after passing through the lens.

In a standard thin lens bounded by the same medium on both sides, $F_1$ and $F_2$ are equidistant from the optical centre.

Variables & Units:

• m = Linear magnification (dimensionless ratio)
• h' = Height of image (m)
• h = Height of object (m)
• v = Image distance (m)
• u = Object distance (m)

Conditions/Notes:

• $m$ is negative: The image is real and inverted.
• $m$ is positive: The image is virtual and erect.

Definition: A plane passing through the principal focus ($F$) and strictly perpendicular to the principal axis.

Significance: If a parallel beam of light is incident making a small angle with the principal axis (not parallel to it), the reflected or refracted rays will converge (or appear to diverge) from a specific point that lies precisely on this focal plane.

Linear Magnification ($m$): The ratio of the physical size of the image to the physical size of the object ($h'/h$).

Angular Magnification: The ratio of the angle subtended by the image at the eye ($\beta$) to the angle subtended by the object at the unaided eye when placed at the near point ($\alpha$).

Importance: A simple microscope doesn't actually make the physical image drastically larger on the retina. Instead, it provides angular magnification by allowing the object to be brought much closer to the eye than the normal 25 cm near point, greatly increasing the visual angle.

Variables & Units:

• i = Angle of incidence (rad or degrees)
• r = Angle of refraction (rad or degrees)
• $n_{21}$ = Relative refractive index of medium 2 with respect to medium 1 (dimensionless)

Conditions/Notes:

• $n_{21}$ is dependent on the pair of media and the wavelength of light, but independent of the angle of incidence.
• If $n_{21} > 1$, $r < i$, and the refracted ray bends towards the normal (Medium 2 is optically denser).

Definition: Dispersion is the splitting of a beam of white light into its constituent colors (the visible spectrum) upon passing through a transparent medium (like a prism).

Underlying Cause: The refractive index ($n$) of a medium is not constant; it depends on the wavelength ($\lambda$) of the incident light. Because $n$ is different for different colors, each color bends by a different angle (as governed by $D_m = (n-1)A$), causing them to physically separate.

Origin: All distances are measured from the pole (P) of the mirror or the optical centre of the lens.

X-axis (Principal Axis):

• Distances measured in the same direction as the incident light are Positive (+).
• Distances measured opposite to the incident light are Negative (-).

Y-axis (Transverse Heights):

• Heights measured upwards (above the principal axis) are Positive (+).
• Heights measured downwards (below the principal axis) are Negative (-).

Variables & Units:

• $n_{ab}$ = Refractive index of medium $a$ with respect to medium $b$ (dimensionless)

Conditions/Notes:

• These relations originate from the definition of the relative refractive index as the ratio of velocities: $n_{21} = \frac{v_1}{v_2}$.
• Very useful for converting air-to-water and air-to-glass indices into a water-to-glass index.

Condition 1: The light ray must be traveling from an optically denser medium to an optically rarer medium (e.g., water into air, or glass into air).

Condition 2: The angle of incidence ($i$) must be strictly greater than the critical angle ($i_c$) for that specific pair of media.

Unlike regular reflection, which involves some fractional transmission, TIR reflects 100% of the light energy internally with zero transmission.

Net Angular Deviation: The angle of emergence exactly equals the angle of incidence ($e = i$). Therefore, there is zero net angular deviation; the emergent ray is strictly parallel to the incident ray.

Physical Effect: Despite remaining parallel, the ray suffers a lateral shift (or lateral displacement). The position of the emergent ray is shifted sideways from the path of the original incident ray.

Variables & Units:

• $h_1$ = Apparent depth (m)
• $h_2$ = Real depth (m)
• n = Refractive index of the denser medium with respect to the observer's (rarer) medium (dimensionless)

Conditions/Notes:

• This formula is strictly valid only for near-normal viewing (looking almost straight down).
• It conceptually explains why the bottom of a water tank appears raised to an observer looking from the air.

Setup: A converging (convex) lens of short focal length is held such that the object is placed within its focal length ($u < f$).

Image Nature: It forms an erect, virtual, and magnified image.

Mechanism: It achieves angular magnification by allowing the object to be brought much closer to the eye than the normal near point ($D = 25 \text{ cm}$), thereby subtending a much larger angle at the eye for comfortable viewing.

Objective Lens: A convex lens with a very short focal length and small aperture. Placed near the specimen, it forms a real, inverted, and magnified intermediate image.

Eyepiece Lens: A convex lens with a slightly larger focal length and aperture. It functions essentially like a simple microscope, using the intermediate image as its object.

Final Image: The eyepiece forms a highly magnified, virtual, and inverted final image (inverted relative to the original object).

Variables & Units:

• $i_c$ = Critical angle (rad or degrees)
• $n_{21}$ = Refractive index of the rarer medium (2) with respect to the denser medium (1)
• $n_{12}$ = Refractive index of the denser medium (1) with respect to the rarer medium (2)

Conditions/Notes:

• The critical angle is defined as the specific angle of incidence in the denser medium for which the angle of refraction in the rarer medium is exactly $90^\circ$.

Structure: Uses an objective lens and an eyepiece lens.

Objective Characteristics: Unlike a microscope, the telescope objective must have a large focal length and a large aperture. The large aperture gathers maximum light from faint distant objects and provides high resolving power.

Image Formation: The objective forms a real, inverted image of the distant object precisely at its focal plane. The eyepiece (with a short focal length) then magnifies this intermediate image, producing a final virtual and inverted image.

1. No Chromatic Aberration: Mirrors reflect all wavelengths equally, inherently preventing the color-fringing defect (chromatic aberration) caused by dispersion in thick glass lenses.

2. Mechanical Stability: A massive mirror is much lighter than an equivalent lens and can be physically supported across its entire back surface. Lenses can only be supported by their thin edges, causing sagging due to gravity.

3. Ease of Manufacturing: Grinding and polishing just one surface of a mirror is cheaper and easier than perfectly shaping both sides of a massive, defect-free glass block.

Arrangement: It utilizes a large concave primary mirror to gather light, and a smaller convex secondary mirror placed before the primary's focus. The secondary mirror reflects light back down through a small central hole in the primary mirror to the eyepiece.

Problem Solved: This folded optical path eliminates the need for the observer (or heavy equipment) to be suspended inside the top of the telescope tube. It provides an effectively huge focal length within a mechanically short and compact tube.

Variables & Units:

• $n_1$ = Refractive index of the medium containing the object
• $n_2$ = Refractive index of the refracting medium forming the surface
• v = Image distance from the pole (m)
• u = Object distance from the pole (m)
• R = Radius of curvature of the spherical interface (m)

Conditions/Notes:

• Strictly relies on small-angle approximations (paraxial rays).
• Must rigorously apply the Cartesian sign convention to $v$, $u$, and $R$.

Variables & Units:

• f = Focal length of the lens (m)
• $n_{21}$ = Relative refractive index of the lens material w.r.t the surrounding medium ($n_2 / n_1$)
• $R_1$ = Radius of curvature of the first refracting surface (m)
• $R_2$ = Radius of curvature of the second refracting surface (m)

Conditions/Notes:

• Derived assuming a "thin lens" (thickness is negligible compared to other distances).
• Apply standard sign conventions. For a double convex lens, $R_1$ is positive and $R_2$ is negative.

Variables & Units:

• v = Image distance from optical centre (m)
• u = Object distance from optical centre (m)
• f = Focal length of the lens (m)

Conditions/Notes:

• Though derived for a real image formed by a convex lens, this equation is universally valid for both convex and concave lenses, and for both real and virtual images, provided Cartesian sign conventions are strictly followed.

Variables & Units:

• m = Linear magnification (dimensionless)
• h' = Height of image (m)
• h = Height of object (m)
• v = Image distance (m)
• u = Object distance (m)

Conditions/Notes:

• Notice the positive relation ($+v/u$) compared to the negative relation in mirrors.
• A positive $m$ implies an erect and virtual image.
• A negative $m$ implies an inverted and real image.

Variables & Units:

• P = Power in Dioptres (D), where $1\text{ D} = 1\text{ m}^{-1}$
• f = Focal length in Metres (m)

Conditions/Notes:

• Conceptually, power is a measure of the convergence or divergence a lens introduces to incident light.
• Convex lenses converge light ($+f$), giving Positive (+) power.
• Concave lenses diverge light ($-f$), giving Negative (-) power.

Principle of Least Time (Fermat's Principle): Light always takes the path between two points that requires the minimum time to traverse.

The Lifeguard Analogy: A lifeguard runs faster on the beach than they swim in the water. To reach a drowning child in the shortest time, they don't take a straight line; they run further on land to minimize time spent in the slower medium (water).

Conclusion: This mathematical optimization of time naturally yields the condition $\frac{\sin i}{\sin r} = \frac{v_1}{v_2}$, which is Snell's Law.

Variables & Units:

• f = Effective focal length (m)
• f_1, f_2 = Focal lengths of individual lenses (m)
• P = Effective power (Dioptres, D)
• P_1, P_2 = Powers of individual lenses (D)

Conditions/Notes:

• The lenses must be thin and placed in direct contact (optical centres assumed coincident).
• The algebraic sum strictly requires applying the correct sign convention (Positive for convex, Negative for concave).

Variables & Units:

• m = Total effective magnification (dimensionless)
• m_1, m_2, m_3 = Linear magnifications of the individual lenses

Conditions/Notes:

• The total magnification is the product (not sum) of individual magnifications.
• Commonly used in designing multi-lens instruments like microscopes and telescopes to enhance image sharpness and achieve high magnification.

Variables & Units:

• r_1, r_2 = Angles of refraction at the first and second faces (degrees or rad)
• A = Apex/Prism Angle (degrees or rad)
• $\delta$ = Angle of deviation (degrees or rad)
• i = Angle of incidence (degrees or rad)
• e = Angle of emergence (degrees or rad)

Conditions/Notes:

• The deviation $\delta$ depends on the angle of incidence. A plot of $\delta$ vs. $i$ yields a parabolic curve with a distinct minimum.

Variables & Units:

• n_{21} = Refractive index of the prism material w.r.t the surrounding medium (dimensionless)
• A = Angle of the prism (degrees or rad)
• D_m = Angle of minimum deviation (degrees or rad)

Conditions/Notes:

• This formula is the standard experimental method (using a spectrometer) for determining the refractive index of transparent solids.

Variables & Units:

• D_m = Angle of minimum deviation (rad or degrees)
• n_{21} = Relative refractive index of the prism (dimensionless)
• A = Prism angle (rad or degrees)

Conditions/Notes:

• Strictly valid only for small-angle prisms (thin prisms) where small-angle approximations ($\sin \theta \approx \theta$) hold true.
• Implies that thin prisms do not deviate light much.

Variables & Units:

• m = Magnifying power / Angular magnification (dimensionless)
• D = Least distance of distinct vision (m or cm, typically $25\text{ cm}$)
• f = Focal length of the convex lens (m or cm, must match unit of D)

Conditions/Notes:

• The maximum magnification is achieved when the image is at $D$.
• However, viewing the image at infinity is preferred as it avoids eye strain (relaxed eye), even though magnification is one less.

Variables & Units:

• m = Total magnifying power (dimensionless)
• L = Tube length (distance between second focal point of objective and first focal point of eyepiece)
• f_o = Focal length of objective lens
• f_e = Focal length of eyepiece lens
• D = Least distance of distinct vision (usually $25\text{ cm}$)

Conditions/Notes:

• This combines the linear magnification of the objective ($m_o = L/f_o$) and the angular magnification of the eyepiece ($m_e = D/f_e$).
• Clearly shows that to achieve large magnification, both $f_o$ and $f_e$ must be very small.

Variables & Units:

• m = Magnifying power (dimensionless)
• f_o = Focal length of the objective lens
• f_e = Focal length of the eyepiece lens
• L = Length of the telescope tube

Conditions/Notes:

• Valid for normal adjustment, meaning the final image is formed at infinity (relaxed eye).
• Magnifying power is the ratio of the angle subtended at the eye by the image to the angle subtended by the actual object.

Variables & Units:

• $v_i$ = Velocity of the image (m/s)
• $v_o$ = Velocity of the object (m/s)
• $m$ = Linear magnification (dimensionless)
• $v$ = Image distance (m)
• $u$ = Object distance (m)

Conditions/Notes:

• The negative sign indicates that the image always moves in the opposite direction to the object with respect to the mirror (for real objects).

Thermal Gradient: On hot days, the air immediately near the ground is hotter and less dense, therefore possessing a smaller refractive index (optically rarer) than the cooler air above it.

Refraction & TIR: Light descending from a tall object passes through layers of decreasing refractive index, bending away from the normal. Once the angle of incidence at a lower layer exceeds the critical angle, Total Internal Reflection (TIR) occurs, bending the ray upwards.

Illusion: The observer's brain naturally interprets the upward-traveling ray as a straight line, creating an inverted image mimicking reflection off a pool of water.

High Refractive Index: Diamond has an exceptionally high refractive index ($n \approx 2.42$).

Small Critical Angle: Because $\sin i_c = 1/n$, this extreme density gives diamond an extraordinarily small critical angle ($i_c \approx 24.4^\circ$) for the diamond-air interface.

Internal Trapping: Skilled diamond cutters facet the stone so that light entering it almost inevitably strikes internal faces at angles $> 24.4^\circ$. This guarantees multiple Total Internal Reflections before the light finds a geometry allowing it to exit, generating massive sparkle.

Variables & Units:

• n = Absolute refractive index of the medium (dimensionless)
• c = Speed of light in a vacuum ($3 \times 10^8 \text{ m/s}$)
• v = Speed of light in the given medium (m/s)

Conditions/Notes:

• Since $v$ is always less than $c$ in any material medium, the absolute refractive index $n$ is always strictly greater than 1.

Material Properties: They use isosceles right-angled prisms made of crown or flint glass, which have critical angles around $41^\circ \text{ to } 42^\circ$.

Geometry: Light is directed perpendicularly into a face so that it strikes the hypotenuse interface from the inside at exactly $45^\circ$.

The TIR Effect: Because $45^\circ > i_c$, the beam undergoes Total Internal Reflection with 100% efficiency. This allows sharp $90^\circ$ or $180^\circ$ turns without the intensity loss or tarnishing issues inherent to conventional silvered mirrors.

Variables & Units:

• $h_{\text{apparent}}$ = Total apparent depth (m)
• $h_k$ = Real depth (thickness) of the $k$-th liquid layer (m)
• $n_k$ = Refractive index of the $k$-th liquid layer

Conditions/Notes:

• Valid for near-normal viewing.
• The total apparent shift of the object is the sum of the shifts produced by each individual layer.

Principle: Light is guided over vast distances via continuous Total Internal Reflection (TIR) with negligible absorption.

Architecture: Made of high-quality quartz or glass, it consists of a central core completely surrounded by a protective cladding layer.

Critical Requirement: The refractive index of the core must be strictly higher than that of the cladding ($n_{\text{core}} > n_{\text{cladding}}$). Light launched at an appropriate acceptance angle strikes the core-cladding boundary at an angle greater than $i_c$ and propagates endlessly.

At minimum deviation, light passes completely symmetrically through the prism, satisfying:

Condition 1: The angle of incidence exactly equals the angle of emergence ($i = e$).

Condition 2: The two internal angles of refraction are identical ($r_1 = r_2 = A/2$).

Condition 3: The refracted ray traveling inside the prism is strictly parallel to the base of the prism (assuming an isosceles prism profile).

Condition: The refractive index of the liquid must exactly match the refractive index of the lens material ($n_1 = n_2$).

Physics behind it: According to the Lens Maker's Formula, if $n_1 = n_2$, the relative refractive index $n_{21} = 1$. The term $(n_{21} - 1)$ thus becomes zero, making $\frac{1}{f} = 0$.

Result: The focal length becomes infinity ($f = \infty$). The lens behaves exactly like a flat pane of glass, neither converging nor diverging light, effectively rendering its optical boundaries invisible.

Refractive Index: Red light has a longer wavelength than blue light. Therefore, the refractive index is lower for red light than for blue light ($n_{\text{red}} < n_{\text{blue}}$).

Focal Length: Because a lens bends blue light more sharply (higher $n$), the focal length for red light is greater than for blue light ($f_{\text{red}} > f_{\text{blue}}$).

Critical Angle: Since $\sin i_c = 1/n$, a lower refractive index for red light implies that the critical angle for red light is greater than for blue light (${i_c}_{\text{red}} > {i_c}_{\text{blue}}$).

The Parameter: The diameter (or aperture) of the objective lens or primary mirror.

Light Gathering Power: A larger area ($A = \pi r^2$) captures significantly more photons, allowing astronomers to see faint, distant galaxies.

Resolving Power: A larger diameter reduces optical diffraction limits, directly increasing the telescope's angular resolution (its ability to observe two extremely close objects, like binary stars, as distinct, separate entities).

Observation: It is a visibly dark region of the sky located strictly between the primary rainbow and the secondary rainbow.

Cause: It occurs because light entering water droplets is scattered such that the angle made at the observer's eye lies roughly between $42^\circ$ and $50^\circ$.

Essentially, no light is scattered by the raindrops into this specific angular viewing region, leaving it starkly unilluminated compared to the inherently brightened sky above and below the respective bows.