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 )