Chapter 9: Ray Optics and Optical Instruments Long Questions | physics Class 12 CBSE

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Chapter 9: Ray Optics and Optical Instruments Long Questions | physics Class 12 CBSE | ByteNotes.in

Long Answer

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Medium

Question 1

Long Form

Derive the mirror equation 1/v + 1/u = 1/f for a concave mirror forming a real image. State clearly the sign convention used.

Using the Cartesian sign convention (all distances from pole P, incident light direction positive), for a concave mirror with object AB at distance u and image A'B' at distance v: since both object and image are in front of the mirror, u and v are negative, and focal length f is negative.
From the ray diagram, triangles A'B'F and MPF are similar (paraxial approximation makes MP perpendicular to CP), giving B'A'/BA = B'F/FP.
Triangles A'B'P and ABP are also similar, giving B'A'/BA = B'P/BP. Comparing: B'F/FP = B'P/BP, i.e., (B'P - FP)/FP = B'P/BP.
Substituting sign-convention values B'P = -v, FP = -f, BP = -u and simplifying: (-v+f)/(-f) = (-v)/(-u), which gives v/f = 1 + v/u.
Dividing throughout by v: 1/f = 1/v + 1/u, i.e., 1/v + 1/u = 1/f — the mirror equation, valid for all cases of reflection by spherical mirrors with correct sign convention.

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Long Answer

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Question 2

Long Form

Derive the lens maker's formula starting from refraction at a single spherical surface. State the assumptions made.

Assumptions: the lens is thin (thickness negligible compared to object/image distances), rays are paraxial, and the lens is in air (n1 = 1) with lens material of refractive index n2 = n21.
For refraction at the first surface (radius R1): n2/v1 - n1/u = (n2 - n1)/R1; for an object at infinity (u → ∞), the first surface forms image I1 at distance v1.
The image I1 acts as a virtual object for the second surface (radius R2, with medium n2 on the left and n1 = 1 on right): -n2/v1 + n1/v = (n2 - n1)/(-R2); here the sign of R2 is negative per convention for a convex second surface.
Adding the two equations for a thin lens (BI1 = DI1): n1/v - n1/u = (n2 - n1)(1/R1 - 1/R2). For an object at infinity, DI = f, giving 1/f = (n21 - 1)(1/R1 - 1/R2).
This is the lens maker's formula; it applies to both convex and concave lenses and enables design of lenses with desired focal length by selecting appropriate surface radii.

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Long Answer

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Question 3

Long Form

Explain total internal reflection. Derive an expression for the critical angle and describe two technological applications.

When light travels from a denser medium (n1) to a rarer medium (n2), it bends away from the normal. As the angle of incidence i increases, the refracted angle r increases until at i = ic, r = 90° and the refracted ray grazes the interface.
Applying Snell's law at the critical angle: n1 sin ic = n2 sin 90° = n2, giving sin ic = n2/n1 = n21. For i > ic, Snell's law cannot be satisfied and all incident light is totally reflected — total internal reflection.
Unlike partial reflection, total internal reflection involves no energy loss to a refracted ray; the reflected ray has the same intensity as the incident ray.
Application 1 — Optical fibres: A glass core of high refractive index is coated with lower-index cladding. Light entering at a suitable angle undergoes repeated total internal reflections, transmitting signals over long distances with minimal loss (over 95% over 1 km in silica fibres).
Application 2 — Totally reflecting prisms: Prisms designed to bend light by 90° or 180° use total internal reflection. The critical angle of crown glass (~41°) and dense flint glass (~37°) are both less than 45°, ensuring total internal reflection at 45° incidence; such prisms are used in binoculars and periscopes.

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Question 4

Long Form

Compare the construction and working of a compound microscope with a simple microscope, and derive the expression for its total magnification.

A simple microscope uses a single converging lens of short focal length; its maximum magnification is limited to about 9 for practical focal lengths. A compound microscope uses two lens systems — an objective and an eyepiece — compounding the magnification.
In a compound microscope, the objective (short focal length fo, small aperture) is placed close to the object, forming a real, inverted, magnified intermediate image I' near the focal point of the eyepiece. The eyepiece then acts as a simple magnifier for this intermediate image, producing a final enlarged virtual image.
Magnification by the objective: mo = h'/h = L/fo, where L is the tube length (distance between the second focal point of objective and first focal point of eyepiece) and h', h are sizes of intermediate image and object.
Magnification by the eyepiece (image at infinity): me = D/fe, where D = 25 cm and fe is eyepiece focal length.
Total magnification m = mo × me = (L/fo)(D/fe). For large m, both fo and fe must be small; in practice focal lengths shorter than 1 cm are difficult to achieve, and multi-component lenses are used to minimise aberrations.

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Long Answer

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Hard

Question 5

Long Form

Describe the construction and working of a refracting telescope. Derive its magnifying power and compare it with a reflecting telescope.

A refracting telescope consists of an objective lens of large focal length fo and large aperture, and an eyepiece of short focal length fe. Light from a distant object enters the objective and forms a real image at its second focal point. This image lies near the focal plane of the eyepiece.
The eyepiece magnifies the real image, producing a final inverted virtual image at infinity (normal adjustment). The length of the telescope tube is fo + fe.
Magnifying power: m = β/α ≈ (h/fe)/(h/fo) = fo/fe, where h is the size of the intermediate image, β is the angle subtended at the eye by the final image, and α is the angle subtended by the object. A larger fo and smaller fe gives higher magnification.
The main criteria for an astronomical telescope are light-gathering power (depends on objective area) and resolving power (ability to distinguish close objects — also depends on objective diameter). Hence large-aperture objectives are desirable.
A reflecting telescope uses a concave mirror instead of a lens: it has no chromatic aberration, the mirror is lighter and can be supported over its full back surface allowing much larger apertures, and large mirrors are easier to fabricate. The Cassegrain design uses a secondary convex mirror to pass light through a hole in the primary, giving a compact tube. The Keck telescopes (Hawaii) with 10 m diameter mirrors are the world's largest.

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Long Answer

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Hard

Question 6

Long Form

Analyse refraction through a prism. Show that r1 + r2 = A and derive the relation δ = i + e - A. Explain how minimum deviation helps determine refractive index.

For a triangular prism ABC with apex angle A, a ray PQ enters face AB with angle of incidence i, refracts with angle r1, travels to face AC, refracts again with angle r2, and emerges as RS with angle of emergence e. In quadrilateral AQNR (Q and R are right angles), angle A + angle QNR = 180°.
In triangle QNR: r1 + r2 + angle QNR = 180°. Comparing these two: r1 + r2 = A.
Total deviation δ = deviation at face AB + deviation at face AC = (i - r1) + (e - r2) = i + e - A.
At minimum deviation Dm: i = e (by symmetry), hence r1 = r2 = A/2, and Dm = 2i - A, giving i = (A + Dm)/2.
Applying Snell's law at minimum deviation: n21 = sin[(A + Dm)/2] / sin(A/2). Since A and Dm are measurable with a spectrometer, this provides a precise experimental method for finding the refractive index of the prism material.

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Question 7

Long Form

Discuss image formation by a convex lens for different positions of the object, and explain how a convex lens is used as a simple microscope.

For a convex lens: object beyond 2f gives a real, inverted, diminished image between f and 2f on the other side; at 2f gives real, inverted, same-size image at 2f; between f and 2f gives real, inverted, magnified image beyond 2f; at f gives image at infinity; between f and optical centre gives virtual, erect, magnified image on the same side as object.
When used as a simple microscope, the object is placed slightly inside the focal length (u < f) so the lens produces a virtual, erect, magnified image.
For image at near point D: m = 1 + D/f. To obtain a magnification of 6, a lens of focal length f = 5 cm is needed (since D ≈ 25 cm).
For image at infinity (relaxed eye): m = D/f, which is one less but more comfortable for extended viewing.
The simple microscope allows the object to be brought much closer to the eye than the 25 cm near-point distance, making it subtend a larger angle; its maximum practical magnification is about 9 due to limitations on achievable short focal lengths.

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Question 8

Long Form

Analyse how the combination of two thin lenses in contact gives an equivalent single lens. Extend this to n lenses and discuss its significance in instrument design.

For two thin lenses A (focal length f1) and B (focal length f2) in contact, with the optical centres coincident at point P, the image I1 formed by lens A acts as a virtual object for lens B.
For lens A: 1/v1 - 1/u = 1/f1. For lens B (object at v1): 1/v - 1/v1 = 1/f2. Adding: 1/v - 1/u = 1/f1 + 1/f2.
Comparing with the single-lens formula 1/v - 1/u = 1/f, the effective focal length satisfies 1/f = 1/f1 + 1/f2, and in terms of power: P = P1 + P2.
For n lenses: 1/f = 1/f1 + 1/f2 + ... + 1/fn and P = P1 + P2 + ... + Pn (algebraic sum, accounting for signs).
This combination principle is used in cameras, microscopes, and telescopes to achieve required magnification, correct chromatic and spherical aberrations, and produce sharper images; it also enables the use of multiple weaker lenses to replace a single very short focal length lens that would be difficult to fabricate.

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Long Answer

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Hard

Question 9

Long Form

Explain the phenomenon of refraction at a spherical surface separating two transparent media. Derive the relation n2/v - n1/u = (n2 - n1)/R using Snell's law and paraxial ray approximation.

Consider a spherical surface of radius R separating media n1 and n2, with an object O on the principal axis. A paraxial ray from O hits the surface at N; NM is the foot of the perpendicular from N to the principal axis.
Using small angle approximation: angle of incidence i = MN/OM + MN/MC (exterior angle of triangle NOC); angle of refraction r = MN/MC - MN/MI.
Applying Snell's law for small angles (n1 i = n2 r): n1(MN/OM + MN/MC) = n2(MN/MC - MN/MI).
Dividing by MN and applying Cartesian sign convention (OM = -u, MI = +v, MC = +R): n1/(-u) + n1/R = n2/R - n2/(-v), which simplifies to n2/v - n1/u = (n2 - n1)/R.
This equation is universal for any spherical refracting surface and forms the basis for deriving the lens maker's formula by application at both surfaces of a thin lens.

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Question 10

Long Form

Describe the structure of an optical fibre. Explain quantitatively how total internal reflection enables efficient signal transmission and state its practical applications.

An optical fibre consists of a very thin glass or quartz core of high refractive index surrounded by a cladding of slightly lower refractive index. The refractive index difference ensures that the critical angle at the core-cladding interface is less than the angle at which light strikes the wall.
When light enters one end of the fibre at a suitable angle, it strikes the core-cladding interface at an angle greater than the critical angle (sin ic = n_cladding/n_core) and undergoes total internal reflection. This reflection repeats along the full length of the fibre.
Since no light escapes at each reflection, intensity is preserved; in silica glass fibres, over 95% of light is transmitted over 1 km, compared to near-zero transmission through 1 km of ordinary glass.
Even when the fibre is bent, light continues to undergo total internal reflection at each point, so an optical fibre acts as a flexible optical pipe with minimal signal loss.
Applications include: (i) long-distance transmission of audio, video, and digital data in telecommunications; (ii) medical endoscopy — bundles of fibres transmit images from inside the body (esophagus, stomach, intestines); (iii) decorative lamps where light exits at fibre tips; and (iv) sensors and illumination in hard-to-reach locations.

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Chapter 9: Ray Optics and Optical Instruments | Long Questions

Derive the mirror equation 1/v + 1/u = 1/f for a concave mirror forming a real image. State clearly the sign convention used.

Derive the lens maker's formula starting from refraction at a single spherical surface. State the assumptions made.

Explain total internal reflection. Derive an expression for the critical angle and describe two technological applications.

Compare the construction and working of a compound microscope with a simple microscope, and derive the expression for its total magnification.

Describe the construction and working of a refracting telescope. Derive its magnifying power and compare it with a reflecting telescope.

Analyse refraction through a prism. Show that r1 + r2 = A and derive the relation δ = i + e - A. Explain how minimum deviation helps determine refractive index.

Discuss image formation by a convex lens for different positions of the object, and explain how a convex lens is used as a simple microscope.

Analyse how the combination of two thin lenses in contact gives an equivalent single lens. Extend this to n lenses and discuss its significance in instrument design.

Explain the phenomenon of refraction at a spherical surface separating two transparent media. Derive the relation n2/v - n1/u = (n2 - n1)/R using Snell's law and paraxial ray approximation.

Describe the structure of an optical fibre. Explain quantitatively how total internal reflection enables efficient signal transmission and state its practical applications.

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