VARDAAN LEARNING INSTITUTE

1. Convex Mirror: $u = -10$ cm, $f = +15$ cm.
$1/v = 1/f - 1/u$
$\Rightarrow 1/v = 1/15 - 1/(-10)$ $= 2/30 + 3/30$ $= 5/30$.
$v = +6$ cm. Virtual, Erect, Diminished ($m = -v/u = -6/-10 = 0.6$).
Difference: Plane mirror image is same size, Convex image is diminished.

6. Security Mirror: The image formed by a convex mirror is always formed between the Pole (P) and the Focus (F) behind the mirror. So, the shopkeeper sees the image behind the mirror.

7. Convex Mirror Advantages: (i) It gives a wider field of view (curved outwards). (ii) It always forms an erect image.

8. Half-covered Mirror: Yes, the full image will still be formed. However, the intensity (brightness) of the image will be reduced because fewer rays are involved in image formation.

2. Formula: $n = (360/\theta) - 1$ if $360/\theta$ is even.
$\theta = 60^\circ$. $360/60 = 6$ (Even).
Number of images = $6 - 1 = 5$.

3. $m = -3$ (Real). $m = -v/u \Rightarrow -3 = -v/u \Rightarrow v = 3u$.
$u = -10$ cm, so $v = 3(-10) = -30$ cm.
$f$ was $v u / (u+v) = -300 / -40 = -7.5$ cm.
If moved 5 cm towards it, $u' = -5$ cm. Since $u < f$ (5 < 7.5), image becomes Virtual, Erect, and Magnified.

4. A ray passing through C falls normally on the mirror surface.
Angle of Incidence $i = 0^\circ$. By laws of reflection, Angle of Reflection $r = 0^\circ$. Thus, it retraces its path.

5. For an erect image in a concave mirror, object must be placed between Pole (P) and Focus (F).
Range: 0 cm to 20 cm from the mirror.

9. Concave Lens: $u = -25$ cm, $f = -15$ cm.
Lens formula: $1/v - 1/u = 1/f$.
$1/v = 1/f + 1/u$ $= -1/15 - 1/25$.
$1/v = (-5-3)/75$ $= -8/75$.
$v = -9.375$ cm.
$m = v/u = (-9.375)/-25$ $= +0.375$. Height $= 4 \times 0.375 = 1.5$ cm.

10. Real images are formed by actual intersection of rays (Inverted). Virtual images formed by apparent intersection (Erect).

11. Speed in vacuum = $c$. Speed in diamond = $c/2.42$.
Reduction $= c - (c/2.42)$ $= c(1 - 1/2.42)$ $= c(1.42/2.42)$ $\approx 0.586 c$.
Percentage reduction = $58.6\%$.

12. Light rays from coin (denser) bend away from normal at surface (air), reaching observer's eye appearing to come from a higher point.

13. Convex Lens: $u = -30$ cm, $f = +20$ cm.
$1/v = 1/f + 1/u$ $= 1/20 - 1/30$ $= (3-2)/60$ $= 1/60$.
$v = +60$ cm. Real, Inverted.
$m = v/u = 60/-30 = -2$. Image Height $= 5 \times (-2) = -10$ cm.

14. Concave Lens: $f = -15$ cm, $v = -10$ cm (Virtual).
$1/u = 1/v - 1/f$ $= 1/(-10) - 1/(-15)$ $= -3/30 + 2/30$ $= -1/30$.
$u = -30$ cm.
Magnification $m = v/u = -10/-30$ $= +1/3 = 0.33$.

15. $f_1 = +40$ cm = +0.4 m. $P_1 = 1/0.4 = +2.5$ D.
$f_2 = -25$ cm = -0.25 m. $P_2 = 1/(-0.25) = -4.0$ D.
$P_{net} = +2.5 - 4.0 = -1.5$ D. Since power is negative, nature is Diverging.

16. Diagrams showing image formation at F, 2F, and behind object (virtual).

17. Virtual Image by Convex Lens: Object must be placed between the Optical Center (O) and the Focus (F1).

18. Normal Incidence: When a ray falls normally (perpendicularly), angle of incidence $i=0$. By Snell's Law ($\sin i / \sin r = n$), $\sin r = 0$, so $r=0$. The ray travels undeviated along the normal.

19. Glass Slab: Refraction at first surface (Air to Glass): $n = \sin i / \sin r_1$.
Refraction at second surface (Glass to Air): $1/n = \sin r_2 / \sin e$. (Since surfaces are parallel, $r_1 = r_2$).
Thus, $\sin i = \sin e \Rightarrow \angle i = \angle e$. Emergent ray is parallel to incident ray.

20. Covered Lens: Yes, the full image is still formed because light rays from every point of the object still pass through the uncovered half of the lens. However, since less light passes through, the brightness (intensity) of the image will be reduced to roughly half.

21. Rear-view Mirrors: Convex mirrors curve outwards, reflecting light from a wider angle.

22. Solar Furnace: Large concave mirrors are used to focus parallel sun rays onto a single point (Focus). This concentration of solar energy generates immense heat at the focal point to melt metals.

23. Lens as Prisms: A convex lens can be modeled as a series of prisms increasing in angle towards the edges, with the central slab being rectangular. The upper prisms (base down) bend light down, lower prisms (base up) bend light up, causing convergence at the axis.

24. Real vs Virtual:
(a) Real image is Inverted; Virtual image is Erect.
(b) Real image can be obtained on a screen; Virtual image cannot be obtained on a screen.

25. Reversibility of Light: The principle states that light will follow the exact same path if its direction of travel is reversed. Example: In reflection, if angle $i = 30$, angle $r = 30$. If light is sent back at 30, it reflects at 30. Use in fiber optics and periscopes.

26. Bent Stick: Light rays coming from the immersed part of the stick travel from water (denser) to air (rarer). They bend away from the normal. To the observer, these rays appear to diverge from a point higher than the actual position, making the stick look bent and raised.

27. Lens in Water: Refractive index of glass relative to water (${}_w n_g$) is less than refractive index of glass relative to air (${}_a n_g$) because $n_w > n_a$. Bending power depends on $(n-1)$. Since effective $n$ decreases, Power decreases. Since $P = 1/f$, Focal Length Increases.

28. Refractive Index > 1: Absolute refractive index $n = c/v$. Since the speed of light in vacuum ($c$) is the maximum possible speed in nature, $v$ (speed in medium) is always less than $c$. Therefore, the ratio $c/v$ is always greater than 1.

29. Lateral Displacement Factors: (i) Thickness of the glass slab (directly proportional), (ii) Angle of incidence (directly proportional), (iii) Refractive index of the material (directly proportional).

30. Broken Lens: Even if a lens is broken into pieces, each piece still acts as a lens portion with the same curvature (same focal length). Each piece forms a complete image of the object, but the image will be faint due to very low intensity.

31. Glass Slab Relations: (i) Angle of incidence is equal to Angle of emergence ($\angle i = \angle e$). (ii) Angle of incidence is greater than Angle of refraction ($\angle i > \angle r$).

32. Without Touching: Look at your image in the mirror.
- Plane Mirror: Image is erect and same size.
- Concave Mirror: Image is erect and magnified (if close); inverted (if far).
- Convex Mirror: Image is erect and diminished.

33. Fastest Speed: Light travels fastest in the medium with the lowest optical density (lowest refractive index).
Water ($n=1.33$), Kerosene ($n=1.44$), Turpentine ($n=1.47$), Diamond ($n=2.42$).
Lowest is Water. So, light travels fastest in Water.

34. Moving Lens: For a distant object ($u = \infty$), image forms at Focus. As lens moves towards screen (Screen fixed), $v$ decreases. If $u$ decreases (relatively object comes closer), $v$ should increase. If he moves lens towards screen, he is reducing $v$. This is only possible if $u$ increases. But object is distant. Practically, if he moves the lens towards the screen, the image will go out of focus (become blurred) because the sharp image is formed at $F$.

35. Focal Length of Concave Mirror:
1. Mount the concave mirror on a stand facing a distant object (tree/building).
2. Place a white screen in front of it.
3. Move the screen back and forth until a sharp, inverted image of the distant object is obtained.
4. Measure the distance between the mirror (Pole) and the screen. This distance is the approx focal length ($f$).