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Class 10, Science, Chapter-9, Lecture-2, Mirror Formula (Notes)

MIRROR FORMULA:

A formula that gives the relationship between the image distance, the object distance, and the focal length of a mirror is known as mirror formula.

Mathematically: ${1 \over v}\,\, + \,{1 \over u}\,\, = \,\,{1 \over f}\,$
where ;
v=image distance 
u=object distance
f=focal length

DERIVATION OF MIRROR FORMULA:

From the ray diagram,

$\Delta A'B'P$ and $\Delta ABP$ are similar

${\therefore}~~{{A'B'} \over {AB}}\,\, = \,\,\,{{PB'} \over {PB}}$  --- (1)

$\Delta A'B'P$ and $\Delta ABP$ and $\Delta ABC$ are also similar

${\therefore}~~{{A'B'} \over {AB}}\,\, = \,\,{{B'C} \over {BC}}\,$ --- (2)

From equations (1) and (2) 
${{PB'} \over {PB}}\,\,\, = \,\,\,{{B'C} \over {BC}}\,\,\,\,\,$
$ \Rightarrow $ ${{PB'} \over {PB}}\,\, = \,\,{{PC\, - \,PB'} \over {PB\, - \,PC}}$ --- (3)

According to the sign convention

object distance, $u =  - PB$   $ \Rightarrow $  $PB =  - u$

image distance, $v =  - PB'$  $ \Rightarrow $  $PB' =  - v$

focal length,   $f =  - PF$  $ \Rightarrow $   $PF =  - f$

radius of curvature,  $2f =  - PC$  $ \Rightarrow $  $PC =  - 2f$

Substituting these values in equation (3)

${{ - v} \over { - u}}\,\,\, = \,\,\,{{ - 2f - ( - v)} \over { - u - ( - 2f)}}$$ \Rightarrow \,\,\,{v \over u}\,\, = \,\,{{ - 2f + v} \over { - u + 2f}}\;$$ \Rightarrow \,\,\,fu + vf = vu$$ \Rightarrow \,\,\,{1 \over v}\,\, + \,\,{1 \over u}\,\, = \,\,{1 \over f}$

LINEAR MAGNIFICATION OF MIRROR: 

The ratio of the height of the image produced by a mirror to the height of the object is known as linear magnification produced by the mirror.

Magnification , $m\, = \,{{{\rm{Height~ of~ image}}} \over {{\rm{Height ~of~ object}}}} = {{h'} \over h}$

DERIVATION OF THE FORMULA ${{\rm{M}} = \,\,\, - {{\rm{v}} \over {\rm{u}}}}$

From the ray diagram:

$\Delta A'B'P$ and $\Delta ABP$ are similar  $ \Rightarrow $   ${{A'B'} \over {AB}} = {\rm{ }}{{PB'} \over {PB}}$  --- ( A )

According to the sign convention:

object distance, $u =  - PB$   $ \Rightarrow $ $PB =  - u$

image distance, $v =  - PB'$  $ \Rightarrow $  $PB' =  - v$

object height,   ${h_1} = AB$  $ \Rightarrow $  $AB = {h_1}$

image height,   ${h_2} =  - A'B'$  $ \Rightarrow $  $A'B' =  - {h_2}$

Substituting these values in equation (A),

${{ - h'} \over h}\,\,\, = \,\,\,{{ - v} \over { - u}}$ $ \Rightarrow $ $ - {v \over u}\,\,\, = \,\,\,{{h'} \over h}$  --- ( B )

By definition, magnification, $M\,\,\,\, = \,\,\,\,\,{{h'} \over h}$ --- ( C )

From equation ( B ) and ( C ) $M = \,\, - {v \over u}$

Some Important Points About Linear Magnification:

For Real images , sign of magnification is negative ( – ve )

For Virtual images , sign of magnification is positive ( + ve )

For Magnified images, magnitude of magnification is greater than one ( |M| > 1 )

For Diminished images, magnitude of magnification is less than one ( |M| < 1 )

For Equal images, magnitude of magnification is equal to one ( |M| = 1 )

For Plane mirror, magnitude of magnification is equal to one ( |M| = 1 )