Ray Optics and Optical Instruments MCQs

For a spherical mirror the focal length f is half its radius of curvature R, i.e. f = R/2.

Given R = 20 cm, substitute into the formula: f = 20 cm / 2.

Thus the focal length equals 10 cm.

n = c / v, where c = 3×10^8 m/s (speed of light in vacuum) and v is the speed in the medium.

Substituting v = 2×10^8 m/s gives n = (3×10^8) / (2×10^8).

Simplify the fraction: 3/2 = 1.5.

Thus the refractive index is 1.5.

Critical angle c satisfies sin c = n₂/n₁ where n₁=1.5 (medium) and n₂≈1 (air).

Compute sin c = 1/1.5 = 0.6667.

Take inverse sine: c = arcsin 0.6667 ≈ 41.8°.

≈ 42 degrees.

Using the lens formula 1/f = 1/v + 1/u, set the object distance u = –2f.

Solving gives the image distance v = +2f, so the image forms at the same distance on the other side.

Since magnification m = –v/u = –1, the image is real, inverted and of the same size; therefore the object is placed at 2F.

Use lens‑maker’s formula : 1/f = (n – 1)(1/R₁ – 1/R₂).

For an equiconvex lens R₁ = +20 cm and R₂ = –20 cm, so 1/f = 0.5(1/20 – (–1/20)) = 0.5·(2/20) = 0.05.

Hence f = 20 cm.

For thin lenses in contact, the powers add: \(P = \frac{1}{f} = \frac{1}{f_1}+\frac{1}{f_2}\).

Substituting \(f_1=20\) cm and \(f_2=30\) cm gives \( \frac{1}{f}= \frac{1}{20}+\frac{1}{30}= \frac{5}{60}= \frac{1}{12}\).

Hence the combined focal length \(f = 12\) cm.

In Cartesian sign convention the direction of incident ray is taken as positive.

Any distance measured opposite to this direction is assigned the opposite sign.

Hence distances measured against the incident light are taken as negative. Negative

When light leaves a denser medium (e.g., water) into air, it bends away from the normal, making the emergent rays diverge.

The observer traces these diverging rays backward, assuming they travel in straight lines, which makes the point of origin appear higher than its true position.

Thus the object seems to be raised toward the surface rather than at its actual depth.

Using the paraxial approximation, the incident and refracted rays make small angles with the normal, so sin i≈i and sin r≈r.

From Snell’s law n₁i = n₂r and the geometry of a spherical surface, i = (h/u) – (h/R) and r = (h/v) – (h/R).

Substituting i and r into n₁i = n₂r and cancelling the common factor h gives n₂/v – n₁/u = (n₂ – n₁)/R.

For each spherical surface, refraction gives 1/v − 1/u = (n−1)/R, where R is taken positive if the centre of curvature lies on the outgoing side.

Treat the lens as two such surfaces separated by a negligible thickness and set the object at infinity (u→∞) to obtain the focal length; the contributions add as (n−1)(1/R1 − 1/R2).

Thus the thin‑lens maker’s formula is 1/f = (n−1)(1/R1 − 1/R2).

Focal length in metres = 25 cm = 0.25 m; for a concave lens the focal length is taken negative, so f = –0.25 m.

Lens power P is defined as P = 1/f (metres).

Substituting f gives P = 1/(–0.25) = –4 dioptres.

Different wavelengths travel at different speeds in a material, so the refractive index n varies with λ.

Snell’s law, n₁ sinθ₁ = n₂ sinθ₂, gives a different θ₂ for each λ because n₂ is λ‑dependent.

Thus the white beam separates into its constituent colours when passing through the prism.

Hence the dispersion of white light by a prism occurs because refractive index depends on Wavelength.

For a concave mirror the focal length f = –R/2, where R is the radius of curvature.

The object is placed at the centre of curvature, so the object distance u = –R.

Using the mirror formula 1/f = 1/v + 1/u gives 1/(–R/2) = 1/v – 1/R ⇒ v = –R, i.e., the image forms at the centre of curvature.

Since v is negative the image is real, inverted and of the same size as the object → Real, inverted, same size at C

In a medium the light speed is v = c/n, so entering water (n>1) reduces its speed.

Since frequency f is set by the source and does not change at an interface, λ = v/f must also decrease.

Thus wavelength and speed decrease while frequency remains constant.

When light passes from a medium of higher refractive index (denser) to one of lower index (rarer), Snell’s law gives sin θ₂ = (n₁/n₂) sin θ₁, where n₁>n₂.

If the incidence angle θ₁ exceeds the critical angle θ_c = sin⁻¹(n₂/n₁), sin θ₂ would be >1, which is impossible, so the ray cannot refract.

Hence the ray is reflected completely inside the denser medium: total internal reflection occurs for denser to rarer medium at angle greater than critical angle.

For a concave lens the focal length is negative, so parallel rays diverge as if coming from the focal point on the same side as the object.

When an object is placed anywhere in front of the lens, the refracted rays appear to diverge from a point between the lens and the focal point, giving an image on the same side as the object.

Since the rays do not actually meet, the image cannot be projected on a screen; it is virtual, upright (erect) and smaller than the object.

Thus a concave lens always forms an image that is Virtual, erect and diminished.

For a thin lens in a medium, 1/f = (n_lens/n_medium – 1)(1/R₁ – 1/R₂).

Immersing the glass lens (n=1.5) in water (n=1.33) reduces the relative refractive index from 1.5 to 1.5/1.33 ≈1.13, so the factor (n_lens/n_medium –1) becomes smaller.

A smaller factor means lower lens power, therefore the focal length becomes larger → Increases.

Net power of combination = +5 D + (–5 D) = 0 D.

Zero power means focal length f = 1/0 → ∞, so the system does not converge or diverge rays.

A lens system with infinite focal length transmits light without changing its convergence, just like a plane glass plate.

For a convex mirror the focal length f is negative, so using 1/u + 1/v = 1/f gives v negative for any positive object distance u, indicating a virtual image.

Magnification m = –v/u is positive because both v and u have opposite signs, so the image is erect.

Since |v| < |u|, |m| < 1, the image size is smaller than the object, i.e., diminished.

sin r = (n₁ sin i)/n₂ = (1 × sin45°)/1.414 = (√2/2)/1.414 ≈ 0.5

r = arcsin 0.5 = 30°

Thus the angle of refraction is 30 degrees.