Derive mirror formula for concave mirror class 12

    In this blog post, we will derive the mirror formula for concave mirror class 12 students. You will also get the definition of a spherical mirror, diagram, MCQ's and numerical questions.

We will provide step-by-step instructions for its derivation.

What is a spherical mirror?

        The cut part of a hollow glass sphere is called a spherical surface, if one side of the spherical surface is polished with silver or nickel, then it is called a spherical mirror.

 Definition related to Mirror Formula

Object Distance

    The distance between the pole (O) and the object  (PQ) is called the focal object distance, it is represented by `OQ = u.`

Image Distance

    The distance between the pole (O) and the image  `(P^'Q^')` is called the focal object distance, it is represented by `OQ^' = v`.

Focal length

    The distance between the pole (O) and the focus point of a spherical mirror (F) is called the focal length, it is represented by f. `OF = f`

Mirror formula derivation class 12

 

   Let there be a concave mirror whose pole is O, the center of curvature is C and the focal point is F.

    Its radius of curvature is R and its focal length is u. An object AB is placed at a distance from its pole. The distance between the pole (O) and the object  (PQ) is called the focal object distance, it is represented by `OQ = u`.  It is placed beyond the center of curvature its image is formed between the center of curvature  (C) and the focus point (F).

    First, we get the image P'Q' of an object PQ with the help of a spherical mirror.

We are to find out the relation among u, v, and f. We are to find out the relation among u, v, and f. For this, we will use this ray diagram.

Mirror Formula Derivation

Consider `\triangle OPQ` and `\triangle OP'Q'`

`\frac{P'Q'}{PQ} = \frac {OQ'}{OQ}`

`\frac{P'Q'}{PQ} = \frac {- v}{- u}`

`\frac{P'Q'}{PQ} = \frac { v}{ u}`  ...eq. (1)

In triangle MNF and P'Q'F

`\frac{P'Q'}{MN} = \frac {Q'F}{NF}`

`\frac{P'Q'}{PQ} = \frac {OQ' - OF}{OF}`

`\frac{P'Q'}{PQ} = \frac {(-v) - (-f)}{- f}`

`\frac{P'Q'}{PQ} = \frac {f - v}{- f}`

`\frac{P'Q'}{PQ} = \frac {-(v-f)}{- f}`

`\frac{P'Q'}{PQ} = \frac {v-f}{f}`    ...eq. (2)

 

By eq. (1) and (2)

`\frac { v}{ u} = \frac {v-f}{f}`

`vf = u(v-f)`

`vf = uv-uf`

Dividing through by uvf, we will get :

`\frac{vf}{uvf} = \frac{uv}{uvf}-\frac{uf}{uvf}`

`\frac{1}{u} = \frac{1}{f}-\frac{1}{v}`

`\frac{1}{u} + \frac{1}{v} = \frac{1}{f}`

`\frac{1}{f} = \frac{1}{u} + \frac{1}{v}`

This is the Mirror Formula

Where

`u =` object distance

`v =` image distance

`f =` focal length

Spherical mirrors Question  1   

Q.   The radius of curvature of a concave mirror is 50 cm. Find out its focal length.

Sol. Given, R = 50 cm

`f = \frac {R}{2}`

`f = \frac {50}{2}`

`f = 25 cm` or `f = 0.25 m`

Spherical mirrors Question  2

Q.   An object is placed at a 10 cm distance in front of a concave mirror with a radius of curvature of 15 cm. Find the nature of the image, size, and magnification.

Sol. Given, R = - 15 cm `

` f = - \frac{15}{2}`

` and u = - 10 cm

From mirror formula

`\frac{1}{u} + \frac{1}{v} = \frac{1}{f}`

`\frac{1}{v} = \frac{1}{f} - \frac{1}{u}`

`\frac{1}{v} = \frac{1}{(-\frac{15}{2})} - \frac{1}{(- 10)}`

`\frac{1}{v} = - \frac{2}{15} + \frac{1}{10}`

`\frac{1}{v} = - \frac{ - 4 + 3}{30}`

`\frac{1}{v} = - \frac{ - 1}{30}`

`v = - 30 cm`

Magnification

`m = - \frac{v}{u}`

`m = - \frac{ - 30}{ -10 }`

`m =  - 3 `    (`\because` Unitless)

    So Image will be formed real and invented, three times in size and at a 30 cm distance from the mirror.

Related Physics articles:

Prism Formula Derivation Class 12 

Questions on concave and convex mirrors class 10

Spherical mirrors Question  3   

Q. Mirrors having a curved reflecting surface are called:

(1)  plane mirror

(2)  spherical mirrors

(3) Simple mirror

(4)  none of the above

Ans: 

(2)  spherical mirrors

Spherical mirrors Question  4

How many types of spherical mirrors?

(1)  2

(2)  4

(3)  5

(4)  3

Ans: 

(1) 2

Hint-    There are two types of spherical mirrors: concave mirrors and convex mirrors.

Spherical mirrors Question  5

A spherical mirror with a reflecting surface curved inwards is called ……………

(1)  Convex mirror

(2)  Concave mirror

(3)  Curved mirror

(4)  None of the above

Ans: 

(2)  Concave mirror

Spherical mirrors Question  6

Type of spherical mirrors are:

(1)  Concave mirror

(2)  Convex mirror

(3)  Both 1 and 2

(4)  None of above

Ans: 

(3)   Both 1 and 2

Spherical mirrors Question  7

Q.  Pole lies on the surface of ………………..

(1)  spherical mirrors 

(2)  simple mirror

(3)  plane mirror

(4)  none of the above

Ans: 

(1)  spherical mirrors 

Spherical mirrors Question  8

The Pole is generally represented by ……………

(1)  R

(2)  P

(3)  C

(4)  O

Ans: 

(2)   P

Spherical mirrors Question  9

The center of a sphere of which the reflecting surface of a spherical mirror is a part is called ……………

 

(1)  Pole

(2)  Centre of curvature

(3)  The radius of Curvature

(4)   Aperture

Ans: 

(2)   Centre of Curvature

Spherical mirrors Question  10

The Centre of curvature is not a part of the spherical mirror rather it lies ………….. the mirror

(1)  Boundary

(2)  Inside

(3)  outside

(4)  none of the above

Ans: 

(3)  outside

Spherical mirrors Question  11

In the case of a concave mirror center of curvature lies in ………… of the reflecting surface

(1)  Boundary

(2)  Inside

(3)  Outside 

(4)  Front

Ans: 

(4)  Front

Spherical mirrors Question  12

The light reflected by a plane mirror may form a real image

(1)  If the rays incident of the mirror are converging.

(2)  Under no circumstances

(3)  If the object is placed very close to the mirror.

(4)  If the rays incident on the mirror are diverging.

Ans: 

(1)  If the rays incident of the mirror are converging.

Spherical mirrors Question  13

In image formation from spherical mirrors, only paraxial rays are considered because they

(1)  From nearly a point image of a point source.

(2)  Show minimum dispersion effect.

(3)  Contain most of the intensity of the incident light.

(4)   Are easy to handle geometrically.

Ans: 

(1)  From nearly a point image of a point source.

Spherical mirrors Question  14

For reflection through spherical surfaces, the normal at the point of incidence is

(1)  Perpendicular to the tangent plane at the point of incidence and passes through the center of curvature

(2)  Perpendicular to the tangent plane at the pole and passes through through the focus.

(3)  Perpendicular to the focal plane and passes through the pole.

(4)   Perpendicular to the principal axis and passes through the center of curvature

Ans: 

(1)  Perpendicular to the tangent plane at the point of incidence and passes through the center of curvature

Spherical mirrors Question  15

A convex mirror is used to form the image of an object. Then which of the following statements is/are true?

  i)  The Image lies between the pole and the focus

 ii) The image is diminished in size

iii) The image is real

(1)  i and ii

(2)  ii only

(3)  i only

(4)  i and iii

Ans: 

(1)  i and ii

Spherical mirrors Question  16

A person is six feet tall. How tall must a plane mirror be if he is able to see his entire length?

(1)  3 ft

(2)  4.5 ft

(3)  7.5 ft

(4)  6 ft

Ans: 

(1)  3 ft

Spherical mirrors Question  17

An object is placed from a concave mirror of a focal length of 20 cm. The image formed is

(1)  Real, inverted, and same in size

(2)  real, inverted, and smaller

(3)  virtual, erect, and larger

(4)  virtual, erect, and smaller

Ans: 

(1)  Real, inverted, and same in size

Spherical mirrors Question  18

Which of the following is an incorrect statement?

(1)  a real, inverted, same-sized image can be formed using a convex mirror

(2)  a virtual, magnified image can be formed using a concave mirror

(3)  a virtual, erect, same-sized image can be obtained using a plane mirror

(4)  The magnification produced by a convex mirror is always less than one

Ans: 

(1)  a real, inverted, same-sized image can be formed using a convex mirror

Spherical mirrors Question  19

A rod of length 10 cm lies along the principal axis of a concave mirror of focal length 10 cm in such a way that its end closer to the pole is 20 cm away from the mirror. The length. The length of the image is

(1)    5 cm

(2)  2.5 cm

(3)  15 cm

(4)  10 cm

Ans: 

(1)    5 cm

Spherical mirrors Question  20

Two plane mirrors are inclined to each other at a certain angle. A ray of light first incident on one of them at an inclination of `10^\circ` with the mirror retraces its path after five reflections. The angle between the mirrors is

(1)  `20^\circ`

(2)  `30^\circ`

(3)  `22^\circ`

(4)  `12^\circ`

Ans: 

(1)  `20^\circ`

Spherical mirrors Question  21

A concave mirror is placed on a horizontal table with its axis directed vertically upwards. Let O be the pole of the mirror and C be it's center of curvature. A mirror is now filled with water, the image will be

(1)  Real, and located at a point between C and O

(2)  Real, and will remain at C

(3)  Virtual, and located at a point between C and O

(4)  None of these

Ans: 

(1)  Real, and located at a point between C and O