Light — Reflection and Refraction

Light rays that are parallel to the principal axis of a concave mirror converge at a specific point on its principal axis after reflecting from the mirror. This point is known as the principal focus of the concave mirror.

Radius of curvature, R = 20 cm Radius of curvature of a spherical mirror = 2 × Focal length ( f ) R = 2 f Hence, the focal length of the given spherical mirror is 10 cm.

When an object is placed between the pole and the principal focus of a concave mirror, the image formed is virtual, erect, and enlarged.

Convex mirrors give a virtual, erect, and diminished image of the objects placed in front of them. They are preferred as a rear-view mirror in vehicles because they give a wider field of view, which allows the driver to see most of the traffic behind him.

Radius of curvature, R = 32 cm Radius of curvature = 2 × Focal length ( f ) R = 2 f Hence, the focal length of the given convex mirror is 16 cm.

Magnification produced by a spherical mirror is given by the relation, Let the height of the object, ho = h Then, height of the image, hI = −3h (Image formed is real) Object distance, u = −10 cm v = 3 × (−10) = −30 cm Here, the negative sign indicates that an inverted image is formed at a distance of 30 cm in front of the given concave mirror.

The light ray bends towards the normal. When a ray of light travels from an optically rarer medium to an optically denser medium, it gets bent towards the normal. Since water is optically denser than air, a ray of light travelling from air into the water will bend towards the normal.

The speed of light in vacuum is 3 × 10⁸ m s⁻¹. Refractive index of a medium is given by: n_m = c / v, so v = c / n_m

Speed of light in vacuum, c = 3 × 10⁸ m s⁻¹
Refractive index of glass, n_g = 1.50

Speed of light in the glass, v = c / n_g = (3 × 10⁸) / 1.50 = 2 × 10⁸ m s⁻¹

Highest optical density = Diamond Lowest optical density = Air Optical density of a medium is directly related with the refractive index of that medium. A medium which has the highest refractive index will have the highest optical density and vice- versa. It can be observed from table 10.3 that diamond and air respectively have the highest and lowest refractive index. Therefore, diamond has the highest optical density and air has the lowest optical density.

Speed of light in a medium is given by the relation for refractive index (nm). The relation is given as It can be inferred from the relation that light will travel the slowest in the material which has the highest refractive index and travel the fastest in the material which has the lowest refractive index. It can be observed from table 10.3 that the refractive indices of kerosene, turpentine, and water are 1.44, 1.47, and 1.33 respectively. Therefore, light travels the fastest in water.

Refractive index of a medium nm is related to the speed of light in that medium v by the relation: Where, c is the speed of light in vacuum/air The refractive index of diamond is 2.42. This suggests that the speed of light in diamond will reduce by a factor 2.42 compared to its speed in air.

Power of lens is defined as the reciprocal of its focal length. If P is the power of a lens of focal length Fin metres, then The S.I. unit of power of a lens is Dioptre. It is denoted by D. 1 dioptre is defined as the power of a lens of focal length 1 metre. 1 D = 1 m−1

When a real, inverted image is the same size as the object, the object must be placed at the centre of curvature (2F) of the convex lens, forming the image at 2F on the other side.

Object distance, u = −50 cm
Image distance, v = +50 cm
Using the lens formula: 1/f = 1/v − 1/u
1/f = 1/50 − 1/(−50) = 1/50 + 1/50 = 2/50
∴ f = +25 cm = +0.25 m

Power, P = 1/f = 1/(0.25) = +4 D

The needle is placed 50 cm in front of the lens and the power of the lens is +4 D.

Focal length of concave lens, f = 2 m Here, negative sign arises due to the divergent nature of concave lens. Hence, the power of the given concave lens is −0.5 D.

(d) A lens allows light to pass through it. Since clay does not show such property, it cannot be used to make a lens.

(d) Between the pole of the mirror and its principal focus. A concave mirror produces a virtual, erect, and enlarged image when the object is placed between its pole (P) and the principal focus (F). This is the same principle used in shaving/makeup mirrors.

(b) When an object is placed at the centre of curvature in front of a convex lens, its image is formed at the centre of curvature on the other side of the lens. The image formed is real, inverted, and of the same size as the object.

(a) By convention, the focal length of a concave mirror and a concave lens are taken as negative. Hence, both the spherical mirror and the thin spherical lens are concave in nature.

(d) A convex mirror always gives a virtual and erect image of smaller size of the object placed in front of it. Similarly, a plane mirror will always give a virtual and erect image of same size as that of the object placed in front of it. Therefore, the given mirror could be either plane or convex.

(c) A convex lens gives a magnified image of an object when it is placed between the radius of curvature and focal length. Also, magnification is more for convex lenses having shorter focal length. Therefore, for reading small letters, a convex lens of focal length 5 cm should be used.

Range of object distance = 0 cm to15 cm A concave mirror gives an erect image when an object is placed between its pole (P) and the principal focus (F). Hence, to obtain an erect image of an object from a concave mirror of focal length 15 cm, the object must be placed anywhere between the pole and the focus. The image formed will be virtual, erect, and magnified in nature, as shown in the given figure.

(a) Concave (b) Convex (c) Concave Explanation (a) Concave mirror is used in the headlights of a car. This is because concave mirrors can produce powerful parallel beam of light when the light source is placed at their principal focus. (b) Convex mirror is used in side/rear view mirror of a vehicle. Convex mirrors give a virtual, erect, and diminished image of the objects placed in front of it. Because of this, they have a wide field of view. It enables the driver to see most of the traffic behind him/her. (c) Concave mirrors are convergent mirrors. That is why they are used to construct solar furnaces. Concave mirrors converge the light incident on them at a single point known as principal focus. Hence, they can be used to produce a large amount of heat at that point.

The convex lens will form complete image of an object, even if its one half is covered with black paper. It can be understood by the following two cases. Case I When the upper half of the lens is covered In this case, a ray of light coming from the object will be refracted by the lower half of the lens. These rays meet at the other side of the lens to form the image of the given object, as shown in the following figure. Case II When the lower half of the lens is covered In this case, a ray of light coming from the object is refracted by the upper half of the lens. These rays meet at the other side of the lens to form the image of the given object, as shown in the following figure.

Object distance, u = −25 cm, Object height, h_o = 5 cm, Focal length, f = +10 cm

Using the lens formula: 1/v − 1/u = 1/f

1/v = 1/f + 1/u = 1/10 + 1/(−25) = 1/10 − 1/25 = (5−2)/50 = 3/50

∴ v = 50/3 ≈ +16.67 cm (image formed on the other side of the lens)

Magnification m = v/u = (50/3)/(−25) = −2/3

Image height h_i = m × h_o = (−2/3) × 5 = −3.33 cm

The image is formed at 16.67 cm on the other side of the lens. It is real, inverted, and diminished (smaller than the object).

Focal length of concave lens, f = −15 cm; Image distance, v = −10 cm

Using lens formula: 1/v − 1/u = 1/f

1/u = 1/v − 1/f = 1/(−10) − 1/(−15) = −1/10 + 1/15 = (−3+2)/30 = −1/30

∴ u = −30 cm

The object is placed 30 cm in front of the concave lens.

Focal length of convex mirror, f = +15 cm; Object distance, u = −10 cm

Using mirror formula: 1/v + 1/u = 1/f

1/v = 1/f − 1/u = 1/15 − 1/(−10) = 1/15 + 1/10 = (2+3)/30 = 5/30 = 1/6

∴ v = +6 cm (behind the mirror)

Magnification, m = −v/u = −6/(−10) = +0.6

The image is formed 6 cm behind the convex mirror. It is virtual, erect, and diminished (0.6 times the size of the object).

Object distance, u = −20 cm
Object height, h = 5.0 cm
Radius of curvature, R = +30 cm (convex mirror)
∴ Focal length, f = R/2 = +15 cm

Using the mirror formula: 1/v + 1/u = 1/f
1/v = 1/f − 1/u = 1/15 − 1/(−20) = 1/15 + 1/20
1/v = 4/60 + 3/60 = 7/60
∴ v = +60/7 ≈ +8.57 cm

Magnification, m = −v/u = −(60/7)/(−20) = 60/140 = +3/7 ≈ +0.43

Image height, h' = m × h = (3/7) × 5 = +2.14 cm

The positive value of v indicates the image is formed behind the mirror (virtual). The positive magnification and image height indicate the image is erect and smaller in size. Therefore, the image is virtual, erect, and diminished.

Object distance, u = −27 cm
Object height, h = 7.0 cm
Focal length, f = −18 cm (concave mirror)

Using the mirror formula: 1/v + 1/u = 1/f
1/v = 1/f − 1/u = 1/(−18) − 1/(−27) = −1/18 + 1/27
1/v = −3/54 + 2/54 = −1/54
∴ v = −54 cm

The screen should be placed 54 cm in front of the mirror.

Magnification, m = −v/u = −(−54)/(−27) = −2

Image height, h' = m × h = −2 × 7 = −14 cm

The negative magnification and image height indicate the image is real and inverted. The image is real, inverted, and magnified (twice the object size).

Power of the lens, P = −2.0 D
Focal length, f = 1/P = 1/(−2.0) = −0.5 m

Since the focal length is negative, this is a concave (diverging) lens.

Power of the lens, P = +1.5 D
Focal length, f = 1/P = 1/(+1.5) = +0.67 m

Since the focal length is positive, this is a convex (converging) lens. The prescribed lens is converging.