1. Image forms at Focus. So $v = f$ (Focal length).
2. A Rectangular Hyperbola. As $u$ increases, $v$
decreases non-linearly.
3. At Infinity. (Source at Focus $\to$ Parallel rays).
4. Moves away from Focus to Infinity.
5. Angle of incidence is equal to angle of emergence
($\angle i = \angle e$).
6. Lateral Displacement (or Lateral Shift).
7. $v = c/n = (3 \times 10^8) / 1.5 = 2 \times 10^8$ m/s.
8. Divergent (Concave): $f = -30$, $v = -15$ (Always
virtual).
$1/u = 1/v - 1/f$ $= 1/(-15) - 1/(-30)$
$= -2/30 + 1/30 = -1/30$.
$u = -30$ cm. Object placed at Focus.
$m = v/u = (-15)/(-30) = +0.5$. Size = $5 \times 0.5 = 2.5$ cm.
[Generate Image: Ray diagram for a Concave Lens. Object at 30cm (2F). Rays diverge. Virtual image formed
at 15cm (F) on same side. Erect and Diminished.]
9. Refractive index of glass w.r.t water ${}_w n_g = n_g /
n_w$ $= (3/2) / (4/3)$ $= 9/8 = 1.125$.
Speed of light in water $v_w = c/n_w$ $= 3 \times 10^8 / (4/3)$ $= 2.25 \times 10^8$ m/s.
10. Power $P = +3.5 + (-2.5) = +1.0$ D.
Focal length $f = 1/P = 1$ m $= +100$ cm.
Object at $u = -50$ cm. $1/v = 1/f + 1/u$ $= 1/100 - 1/50$ $= -1/100$.
$v = -100$ cm. Image is Virtual, Erect, Magnified.
11. Yes.
Magnified Real: Object placed between
F and 2F.
Magnified Virtual: Object placed between F and Optical Center (O).
12. (i) Focus the reflected light onto the paper for some
time.
(ii) Concave Mirror (Converging).
(iii) Much smaller (Point-sized), as object (Sun) is at
infinity.
13. Real, Inverted, Same Size $\to$ Object is at $2F_1$.
If moved 5 cm towards lens (into F-2F region), image moves beyond $2F_2$, becomes Magnified.
14. Air to Water (Rarer to Denser): Bends towards normal.
Angle of refraction $r < 45^\circ$.
Proof: $\sin i / \sin r = 1.33$. $\sin r = \sin 45 / 1.33$.
Since $1.33 > 1$, $\sin r < \sin 45$, so $r < 45$.
15. Diamond has a very high Refractive Index
(2.42), resulting in a very small Critical Angle ($24.4^\circ$).
Most light entering gets
trapped inside due to multiple Total Internal Reflections, making it sparkle. Glass ($n=1.5$)
has larger critical angle ($42^\circ$), so less trapping.
16. If mirror rotates by $\theta$, the
Reflected Ray rotates by $2\theta$. (Geometric proof involves angle of incidence changing by
$\theta$, so angle of reflection changes by $\theta$, total deviation change $2\theta$).
17. No. A convex mirror is a Diverging mirror.
It spreads sunlight out instead of focusing it to a point. Thus, it cannot generate enough heat
to burn paper.
18. Liquid Lens: $n_{liquid} (1.6) >
n_{lens} (1.5)$.
Since the surrounding medium is optically denser, the nature of the lens
is reversed. A convex lens will behave as a Diverging Lens.
19. Convex mirrors are used as side mirrors.
They form Diminished images. Since we judge distance by image size, smaller images are
perceived as being further away than they actually are. Hence the warning.
20. Formula: $\sin C = 1/n$.
$\sin C =
1/1.5$ $= 2/3 = 0.666$.
$C = \sin^{-1}(0.666) \approx 41.8^\circ$.
