REFLECTION OF LIGHT BY SPHERICAL MIRROR -

Reflection of light (introduction)

Rays of Light reflects in the same way just like a ball bounces back from the ground. Reflection of light enables us to see the objects around us or we can say that everything we see in this world is because of the reflection of light. Objects do not possess their own light, the object is only visible due to reflection of light. Source of light can be a Sun or any light-emitting blub etc which produces light energy. So here we are going to discuss the topic Reflection of Light By Spherical Mirror. I here discuss reflection by both plane mirror and a spherical mirror.

REFLECTION OF LIGHT

When light strikes the surface of an object, some of the light is returned into the same medium. Then this phenomenon is called a reflection of light

LAWS OF REFLECTION

reflection of light

1. The angle of incidence is equal to the angle of reflection.

$reflection of light$

2. The incident ray, the reflected ray and the normal at the point of incidence all lie in one plane.

NOTE:-

a. If a ray of light falling normally on a mirror retraces its path on reflection.

b. The laws of reflection are the same whether the reflecting surface is plane or curved.

OBJECT AND IMAGE

The concept of an object is before the phenomena of reflection and refraction of light and the concept of an image is after the phenomena of reflection and refraction of light.

The object can be anything that emits light rays and the image is the point where the rays meet (or appear to meet) after reflection or refraction. An object may be real or virtual, the same as an image may be real or virtual.

laws of reflection

Q> How we can see a virtual image?

Ans> The virtual image can not be taken on-screen. But, they can be seen by our eyes. Because for our eye lense, this virtual image acts as an object and finally our eye lens forms their real image on our retina. That’s why we can see a virtual image.

PROPERTIES OF IMAGE FORMED BY PLANE MIRROR

1. In a plane mirror, the line joining the object and the image is always perpendicular to the mirror.

laws of reflection

2. If the mirror $M$ is large compare to $M{}'$

laws of reflection

the intensity of the image $I$ is more than $I{}'$ because $I$ is formed from more number of reflected rays. (The same in case of the lens)

3. Fields of view

laws of reflection

4. suppose a mirror is rotated by an angle $\theta$ , keeping the incident ray fixed, then the reflected ray rotates by $2\theta$ along the same sense.

laws of reflection

5. The minimum length of a plane mirror to see ones full height in it is H/2, where H is the height of a man. But the mirror should be placed in a fixed position.

laws of reflection

6. The number of images:-If angle between two plane mirror is $\theta$ , then the number of images formed is determined as follows:

reflection of light

$\fn_cm \large n=\frac{360^{0}}{\theta }-1$ if $\fn_cm \large 360^{0}$ is completely divided by θ

$\fn_cm \large n=\frac{360^{0}}{\theta }$ if $\fn_cm \large \frac{360^{0}}{\theta }$ is a fraction. (takes only integer parts)

7. The image formed by a plane mirror is virtual, erect and of the same size as the object.

8. In front of a plane mirror if object speed is v towards the mirror then its image speed is 2v towards the mirror.

9. Image formation needs regular reflection/ refraction. In principle, all rays from a given point should reach the same image point. This is why you do not see your image by an irregular reflecting object, say the page of a book.

SPHERICAL MIRROR

A spherical mirror is a part of a hollow sphere whose one side is painted.

reflection of light

APERTURE:- The diameter of the mirror is known as its aperture. From fig line joining M to M’ is the aperture of the mirror.

FOCUS:- The principal focus of a spherical mirror is a point on the principal axis of the spherical mirror at which, the light rays which are parallel to the principal axis actually converge (meet) or appear to diverge after reflection.

reflection of light

FOCAL PLANE:- The plane that is perpendicular to the axis of a mirror/lens and passes through the focus is called the focal plane. All parallel lines are passes through the focus.
f – focal length (P to F)
R- Radius of curvature (P to C)

The spherical mirrors are of two types.

1. Concave mirror (Converging mirror):- Here reflecting surface is on the concave side.

2. Convex mirror (Diverging mirror):- Here reflecting surface is on the convex side.

IMPORTANT RAY PATH

1. The incident ray parallel to the principal axis is reflected through the focus.

reflection of light

2. The incident ray passing through the centre of curvature C is reflected back along its original path.

reflection of light

3. If the incident ray strike on the pole, then it is reflected at an equal angle (opposite side) with the axis

reflection of light

SIGN CONVENTION

reflection of light

a. All distance are measured from the pole of the mirror.

u- Object distance from the pole

v- Image distance from the pole

f- Focal length

R- Radius of curvature

b. The distance measured in the direction of the incident light is taken +ve & in opposite direction taken -ve.

c. Height measured upward and perpendicular to the principal axis are taken +ve while those measured downward are considered -ve.

PARAXIAL RAYS

Rays which pass very close to the principal axis is known as a paraxial ray. For small aperture mirror/ lens, rays are generally taken as paraxial rays.

RELATION BETWEEN f AND R

reflection of light

Consider a concave mirror of the small aperture with a pole at P, F is the focus and C is the centre of curvature of the mirror.
Let a ray of light travelling parallel to the principal axis and after reflection by a mirror, it passes through the focus.

From fig.

$\fn_cm \angle ABC=\angle CBF$ (Laws of reflection)

$\fn_cm \angle ABC=\angle BCF$ (Alternate angle)

$\fn_cm \therefore \Delta CBF$ are isosceles triangle

$\therefore$ CF = FB ——————(1)

for small aperture (or paraxial ray), point B very close to point P.

i.e we can write

FB= PF —————–(2)

$\therefore$ from (1) and (2)

CF= PF

This means that F is the midpoint of CP

$\fn_cm \therefore PF=\frac{PC}{2}$

$\fn_cm \LARGE \left [f =\frac{R}{2} \right ]$

MIRROR FORMULA

Mirror formula gives the relation between object position, image position w.r.t given mirror. i.e it is a simple relation between the u (object distance), v (image distance) and f (focal length).

reflection of light

Consider AB is an object which is placed perpendicular on the principal axis in front of a concave mirror.
P is the pole of the mirror
F is the focus of the mirror
C is the centre of curvature.
After reflection, a real, inverted and diminished image A’B’ are formed as shown in the figure.
from fig

$\Delta APB$ and $\Delta A'PB'$ are similar triangle

i.e $\fn_cm \frac{AB}{A'B'}= \frac {PB}{PB'}$ ————————–(1)

similarly $\Delta MFN$ and $\Delta A'FB'$ are similar triangle

i.e $\fn_cm \frac{MN}{A'B'}= \frac{NF}{FB'}$

$\fn_cm \frac{AB}{A'B'}=\frac{NF}{FB'}$

for paraxial ray N and P are coincide

i.e we can write

$\fn_cm \frac{AB}{A'B'}=\frac{PF}{FB'}$ ——————————(2)

from (1) and (2)

$\fn_cm \Rightarrow \frac{PB}{PB'}=\frac{PF}{FB'}$

$\fn_cm \Rightarrow \frac{PB}{PB'}=\frac{PF}{PB'-PF}$

using sign convention

PB=-u, PB’= -v, PF= -f

Putting these value in the above equation, we get

$\fn_cm \large \Rightarrow \frac{-u}{-v}=\frac{-f}{-v+f}$

$\fn_cm \large \Rightarrow uv - uf =vf$

divide uvf in both side

$\fn_cm \large \Rightarrow \frac{uv}{uvf}-\frac{uf}{uvf}=\frac{vf}{uvf}$

$\fn_cm \large \Rightarrow \frac{1}{f}-\frac{1}{v}=\frac{1}{u}$

$\fn_cm \LARGE \left [ \frac{1}{v}+\frac{1}{u}=\frac{1}{f} \right ]$

proof mirror formula

LINEAR MAGNIFICATION

It means how times image larger/smaller than image. Linear magnification produced by a spherical mirror is defined as the ratio of the size of the image formed by the mirror to the size of the object. It is denoted by m.

i.e $\fn_cm \LARGE \left [ m=\frac{h_{2}}{h_{1}} \right ]$

where $h_{2}$ – the height of the image
and $h_{1}$ – the height of the object

from above fig. (mirror formula)

$\fn_cm \frac{A'B'}{AB}=\frac{PB'}{PB}$

$\fn_cm \frac{-h_{2}}{h_{1}}=\frac{-v}{-u}$

$\fn_cm \frac{h_{2}}{h_{1}}=\frac{-v}{u}$

$\fn_cm \LARGE \left [ m=-\frac{v}{u} \right ]$

RELATION BETWEEN OBJECT SPEED AND IMAGE SPEED w.r.t SPHERICAL MIRROR

We know that $\fn_cm \large \frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

differentiate it w.r.t time, we get

$\fn_cm \large \frac{\mathrm{d} \frac{1}{v}}{\mathrm{d} t}+\frac{\mathrm{d} \frac{1}{u}}{\mathrm{d} t}=\frac{\mathrm{d} \frac{1}{f}}{\mathrm{d} t}$

$\fn_cm \large -\frac{1}{v^{2}}\frac{\mathrm{d} v}{\mathrm{d} t}-\frac{1}{u^{2}}\frac{\mathrm{d} u}{\mathrm{d} t}=0$

$\fn_cm \large -\frac{1}{v^{2}}\frac{\mathrm{d} v}{\mathrm{d} t}=\frac{1}{u^{2}}\frac{\mathrm{d} u}{\mathrm{d} t}$

$\fn_cm \large \frac{\frac{\mathrm{d} v}{\mathrm{d} t}}{\frac{\mathrm{d} u}{\mathrm{d} t}}=-\frac{v^{2}}{u^{2}}$

$\fn_cm \large \left [ \frac{speed of image}{speed of object} = -\frac{v^{2}}{u^{2}} \right ]$

POSITION AND NATURE OF IMAGE w.r.t AN OBJECT BY A SPHERICAL MIRROR

FOR CONCAVE MIRROR

reflection of light

FOR CONVEX MIRROR

reflection of light

USE OF MIRROR

a. a concave mirror is used for shaving or make-up mirror (because for magnified and erect image produced)

b. A concave mirror is used as a reflector in heading of vehicles.

d. A convex mirror is used as a driver’s side mirror. because convex mirror produces virtual, erect and diminished images, so the driver has a large field of views as compared to plane or concave mirror.

e. A convex mirror is used as a reflector in the street light.

MISCONCEPTION

a. If a mirror or lens is blackened on one half, half the image will be formed.

(Ans- No, the full image is formed, only Intensity is reduced)

b. If a hole is formed in the mirror, a hole will appear in the image too.

(Ans- same as previous)

c. Mirror formula is used for any aperture.

(No, mirror formula is used only for paraxial rays (small aperture))

RELATED LINKS OF RAY OPTICS AND OPTICAL INSTRUMENTS
Introduction	Refraction
Total Internal Reflection	Refraction at Spherical Surfaces and by Lenses
Refraction through a Prism	Dispersion by a Prism
Some Natural Phenomena due to Sunlight	Optical Instruments

IMPORTANT QUESTIONS FOR PRACTICE

(1) Suppose that the lower half of the concave mirror’s reflecting surface is covered with an opaque (non-reflective) material. What effect will this have on the image of an object placed in front of the mirror?

(2) An object is placed at (i) 10cm, (ii) 5 cm in front of a concave mirror of radius of curvature 15cm. Find the position, nature and magnification of the image in each case.

(3) A small candle, 2.5 cm in size is placed at 27 cm in front of a concave mirror of radius 36 cm. At what distance from the mirror should a screen be placed in order to obtain a sharp image. If the candle is moved closer to the mirror. How would the screen have to be moved?

(4) A 4.5 cm needle is placed 12 cm away from a convex mirror of focal length 15 cm. Give the location of the image and the magnification. Describe what happens as the needle is moved farther from the mirror.

(5) Light incident normally on a plane mirror attached to a galvanometer coil retraces backwards as shown in the figure. A current in the coil produces a deflection of $3.5^{0}$ of the mirror. What is the displacement of the reflected spot of light on a screen placed 1.5 m away?

reflection of light

(6) Answer the following questions:

(a) You have learnt that plane and convex mirrors produce virtual images of objects. Can they produce real images under some circumstances? Explain.

(b) A virtual image, we always say, cannot be caught on a screen. Yet when we ‘see’ a virtual image, we are obviously bringing it on to the ‘screen’ (i.e the retina) of our eye. Is there a contradiction?

(7) Use the mirror equation to deduce that:

(a) An object placed between f and 2f of a concave mirror produces a real image beyond 2f.

(b) A convex mirror always produces a virtual image independent of the location of the object.

(c) The virtual image produced by a convex mirror is always diminished in size and is located between the focus and the pole.

(d) An object placed between the pole and focus of a concave mirror produces a virtual and enlarge image.

(8) A mobile phone lies along the principal axis of a concave mirror, as shown in fig. Show by the suitable diagram, the formation of its image. Explain why the magnification is not uniform. Will the distortion of the image depending on the location of the phone with respect to the mirror?

reflection of light

(9) Suppose while sitting in a parked car, you notice a jogger approaching towards you in the side view mirror of R= 2 m. If the jogger is running at a speed of 5 m/s, how fast the image of the jogger appear to move when the jogger is (a) 39 m, (b) 29 m, (c) 19 m and (d) 9 m away.

(10) What is the difference between virtual images formed by a plane mirror, a concave mirror and a convex mirror?