Ray Optics And Optical Instruments Class 12 Chapter Notes

Key Optical Definitions

Refraction

The bending of light as it passes obliquely from one transparent medium to another, due to a change in its speed.

Total Internal Reflection (TIR)

The phenomenon where a ray of light travelling from a denser to a rarer medium is reflected back into the denser medium if the angle of incidence exceeds the critical angle.

Dispersion

The phenomenon of splitting of white light into its constituent colours when it passes through a transparent medium like a prism.

Power of a Lens

The reciprocal of its focal length (in metres). Measured in Dioptres (D). P = 1/f. Converging lenses have positive power, diverging lenses have negative power.

Magnification (m)

The ratio of the height of the image (h') to the height of the object (h). For mirrors, m = -v/u; for lenses, m = v/u.

Critical Angle (θc)

The angle of incidence in the denser medium for which the angle of refraction in the rarer medium is 90 degrees. sin θc = n2/n1 (where n1 > n2).

Must Remember Formulas & Concepts

• Mirror Formula: 1/f = 1/v + 1/u (f is focal length, v is image distance, u is object distance).
• Lens Formula: 1/f = 1/v - 1/u.
• Lens Maker's Formula: 1/f = (n2/n1 - 1) (1/R1 - 1/R2). Used to determine focal length based on refractive indices and radii of curvature.
• Refraction at Spherical Surface: (n2/v) - (n1/u) = (n2 - n1)/R.
• Snell's Law of Refraction: n1 sin i = n2 sin r (where n1, n2 are refractive indices, i is angle of incidence, r is angle of refraction).
• Compound Microscope: Magnification M = (Lo/fo) (1 + D/fe) (when final image is at near point) or M = (Lo/fo) (D/fe) (when final image is at infinity).
• Astronomical Telescope: Magnification M = -fo/fe (normal adjustment). Length L = fo + fe.
• Sign Convention: Always use the New Cartesian Sign Convention consistently for all numericals. Object is typically placed to the left, light travels left to right.

Understanding Refraction at Spherical Surfaces and Lens Maker's Formula

The concept of refraction at a single spherical surface is foundational to understanding lenses. When light travels from a medium with refractive index n1 to another with refractive index n2 through a spherical boundary, its path bends. The formula governing this is given by: (n2/v) - (n1/u) = (n2 - n1)/R, where 'u' is the object distance, 'v' is the image distance, and 'R' is the radius of curvature of the spherical surface. This equation is crucial for analysing light passing through curved boundaries.

Building upon this, the Lens Maker's Formula is derived by applying the refraction formula successively to the two spherical surfaces of a lens. Consider a thin lens with two spherical surfaces having radii of curvature R1 and R2. The refractive index of the lens material is n2, and it is placed in a medium of refractive index n1.

1. For the first surface, light from an object at 'u' forms an image at 'v1'. We apply the formula: (n2/v1) - (n1/u) = (n2 - n1)/R1.
2. This intermediate image acts as a virtual object for the second surface. For the second surface, light travels from medium n2 to n1. The object distance for this surface is 'v1' (with appropriate sign), and it forms the final image at 'v'. The formula becomes: (n1/v) - (n2/v1) = (n1 - n2)/R2 = -(n2 - n1)/R2.
3. Adding these two equations eliminates 'v1' and gives: (n1/v) - (n1/u) = (n2 - n1) (1/R1 - 1/R2).
4. If the object is at infinity (u = ∞), the image is formed at the focal point (v = f). Substituting these into the combined equation, we get n1/f = (n2 - n1) (1/R1 - 1/R2). Rearranging, the Lens Maker's Formula is derived: 1/f = (n2/n1 - 1) (1/R1 - 1/R2).

This formula explicitly shows how the focal length of a lens depends on the refractive index of its material relative to the surrounding medium, and the curvature of its surfaces. Remember to apply the New Cartesian Sign Convention diligently for R1 and R2.

New Cartesian Sign Convention for Spherical Mirrors and Lenses

1. —
2. —
3. —
4. —
5. —
6. —

Real vs. Virtual Images

Worked Examples

• {"title":"Example 1: Concave Mirror Image Formation","example":"Q: An object is placed 15 cm in front of a concave mirror of focal length 10 cm. Find the position and nature of the image.\nA: Given: u = -15 cm, f = -10 cm (concave mirror).\nUsing mirror formula: 1/f = 1/v + 1/u\n1/(-10) = 1/v + 1/(-15)\n1/v = 1/(-10) - 1/(-15) = -1/10 + 1/15 = (-3 + 2)/30 = -1/30\nv = -30 cm.\nThe image is formed 30 cm in front of the mirror. Since v is negative, it's a real image. Magnification m = -v/u = -(-30)/(-15) = -2. Image is real, inverted, and magnified."}
• {"title":"Example 2: Convex Lens Power","example":"Q: A convex lens forms a real image at a distance of 40 cm from it when an object is placed 20 cm away. Calculate the power of the lens.\nA: Given: v = +40 cm (real image behind lens), u = -20 cm.\nUsing lens formula: 1/f = 1/v - 1/u\n1/f = 1/(+40) - 1/(-20) = 1/40 + 1/20 = (1 + 2)/40 = 3/40\nf = 40/3 cm = 40/300 m = 0.133 m.\nPower P = 1/f = 1/(40/300) = 300/40 = 7.5 Dioptres (D)."}

Exam Tip: Mastering Ray Diagrams and Sign Conventions

Ray diagrams are not just illustrative; they are a critical part of problem-solving. Practice drawing accurate ray diagrams for all mirror and lens positions. Use a scale, draw clear principal axes, and mark focal points (F) and centres of curvature (C or 2F) correctly. Always use arrows to indicate the direction of light. A common trap is forgetting to apply the New Cartesian Sign Convention correctly and consistently in numerical problems. Make a habit of writing down the signs for u, v, f, R, and h at the very beginning of solving any numerical. Incorrect signs lead to entirely wrong answers, even with correct formulas. For derivations, clearly state your assumptions and draw neat, labeled diagrams.

Practice Questions with Solutions

• Q: What are the conditions necessary for Total Internal Reflection to occur? A: Two conditions: 1) Light must travel from a denser medium to a rarer medium. 2) The angle of incidence in the denser medium must be greater than the critical angle for that pair of media.
• Q: Distinguish between a real image and a virtual image. A: Real images are formed by the actual convergence of light rays and can be obtained on a screen. Virtual images are formed when rays appear to diverge and cannot be obtained on a screen.
• Q: Why does a convex lens have a positive focal length according to the New Cartesian Sign Convention? A: For a convex lens, parallel rays of light converge at a point (focal point) on its right side. Since distances measured in the direction of incident light (left to right) are positive, the focal length of a convex lens is positive.
• Q: State two applications of optical fibres. A: Optical fibres are used in telecommunications for transmitting data over long distances with minimal loss, and in medical endoscopy for visual examination of internal organs.

Frequently Asked Questions