Chapter 10 Light - Reflection and Refraction

The ray of light  that is parallel to the principal axis of a concave mirror converges 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.

Given,

Radius of curvature, R = 20 cm

Radius of curvature of a spherical mirror = 2 × Focal length (f)

R = 2f  

 by putting the values we get

The focal length of the given spherical mirror is 10 cm.

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

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

Given:

Radius of curvature, R = 32 cm

Radius of curvature = 2 × Focal length (f)

R = 2f 

On putting the values we get

Therefore, the focal length of the given convex mirror is 16 cm.

Magnification produced by a spherical mirror is expressed as.

Suppose, 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.

A ray of light when it travels from air to water it 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.As water is  more optically denser than air, a ray of light travelling from air into the water will bend towards the normal.

Refractive index of a medium nm is expressed as

Speed of light in vacuum, c = 3 × 10⁸ m s−1

Refractive index of glass, n_g = 1.50

So , Speed of light in the glass,

From the above provided information the object having, Highest optical density is Diamond and Lowest optical density  is 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.

Speed of light in a medium is given by the relation for refractive index (n_m). The relation 

It can conclude 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. Therefore, light travels the fastest in water.

The refractive index of diamond is 2.42. This means  that the speed of light in diamond will reduce by a factor 2.42 compared to its speed in air.

Refractive index of a medium n_m is related to the speed of light in that medium v by the relation:

Where, c is the speed of light in vacuum/air.

1 dioptre is defined as the power of a lens of focal length 1 metre.

∴1 D = 1 m^-1

Power of lens is defined as the reciprocal of its focal length. Where P is the power of a lens of focal length f  in metres, then

The S.I. unit of power of a lens is Dioptre. It is denoted by D.

The image formed is inverted and of the same size as the object. When an object is placed at the centre of curvature, 2F₁, of a convex lens, its image is formed at the centre of curvature, 2F₂, on the other side of the lens.

Given,

The image of the needle is formed at a distance of 50 cm from the convex lens. Hence, the needle is placed in front of the lens at a distance of 50 cm.

Object distance, u = −50 cm

Image distance, v = 50 cm

Focal length = f

Putting values to the lens formula,

Hence, the power of the given lens is +4 D.

Given,

Focal length of concave lens, f = - 2 m

Power of a lens,

a negative sign is due to the divergent nature of the concave lens.

Hence, the power of the given concave lens is −0.5 D.

d) Clay cannot be used for making lenses because it is an opaque object through which light cannot pass.

(d) The image formed is virtual, erect, and larger than the object, When an object is placed between the pole and principal focus of a concave mirror.

(b) The image formed is real, inverted, and of the same size as the object. 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 i.e at twice of focal length.

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

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

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

A concave mirror gives an erect image when an object is placed between its pole (P) and the principal focus (F).

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:

(a) In the headlights of a car  Concave mirror is used. This is because concave mirrors converging mirrors that can produce a powerful parallel beam of light when the light source is placed at their principal focus.

(b) The side/rear-view mirror of a vehicle Convex mirror is used.Convex mirrors give a virtual, erect, and diminished image of the objects placed in front of it and they have a wide field of view therefore  the driver can see most of the traffic behind. 

(c) In Solar furnaces Concave mirrors are used. Concave mirrors are converging mirrors. That is why they are used to construct solar furnaces. It converges the incident light 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.

Yes, The convex lens will form a complete image of an object, even if its one half is covered with black paper. It can be  experimentally verified:.

In first Case 

If the upper half of the lens is covered then a ray of light coming from the object will be refracted by the lower half of the lens and these rays meet at the other side of the lens to form the image of an object,

In Second Case

When the lower half of the lens is covered then a ray of light coming from the object is refracted by the upper half of the lens and these rays meet at the other side of the lens to form the image of the given object.

Given

Converging lens is convex lens

Object distance, u = −25 cm

Object height, h_o = 5 cm

Focal length of convex lens, f = +10 cm

According to the lens formula,

The value of v  positive this shows that the image formed at the other side of the lens.

The negative sign shows that the image is real and formed behind the lens.

The negative value of image height shows that the image formed is inverted. Ray diagram showing The position, size, and nature of image are:

Given:

Focal length of concave lens, f = −15 cm

Image distance, v = −10 cm

According to the lens formula,

The value of u is negative which shows that the object is placed 30 cm in front of the lens. The ray diagram is 

Given,

Focal length of convex mirror, f = +15 cm

Object distance, u = −10 cm

The mirror formula is

The value of v is  positive which shows  that the image is formed behind the mirror.

The positive value of magnification indicates that the image formed is virtual and erect.

The magnification produced by a plane mirror is +1. It shows that the image formed by the plane mirror is of the same size as that of the object.  Whereas the positive sign shows that the image formed is virtual and erect.   

Given,

Convex mirror

Radius of curvature, R = 30 cm

Radius of curvature = 2 × Focal length

R = 2f

f = 15 cm

Object distance, u = −20 cm

Object height, h = 5 cm

the mirror formula,

The value of v is positive which  indicates that the image is formed behind the mirror.

The value of image height is  positive which explains that the image formed is erect.

Hence, the image formed is virtual, erect, and smaller in size.

Given 

Object distance, u = −27 cm

Object height, h = 7 cm

Focal length of concave mirror , f = −18 cm

the mirror formula,

The screen should be placed at a distance of 54 cm in front of the given mirror.

The negative value of magnification indicates that the image formed is real.

The Value of image height is negative  that shows that the image formed is inverted.

Power of lens = -2.0 D

A concave lens has a negative focal length.therefore , it is a concave lens.

Given,

Power of lens = + 1.5 D

A convex lens has a positive focal length.Therefore , it is a convex lens or a converging lens.