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Class 10, Science, Chapter-9, Lecture-6, Lens Formula (Notes) 

LENS FORMULA:

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

mathematically ${1 \over v}\,\, - \,{1 \over u}\,\, = \,\,{1 \over f}\,$
where, 
v=Image distance 
u=object distance
f=focal length

LINEAR MAGNIFICATION OF LENS:

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

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

POWER OF LENS:

The measure of the degree of convergence or divergence of light rays falling on a lens is termed as the power of the lens.
Mathematically, the power of a lens is the reciprocal of its focal length expressed in metres.
$P\, = \,{1 \over {\,f\,({\rm{in m}})}}$
where, 
P = power of a lens
f = focal length of the lens

  • Unit of power is dioptre ( D )
  • Power of a concave lens is – ve.
  • Power of a convex lens is + ve.
  • Instrument used to measure power of a lens is DIOPTREMETER.

ONE DIOPTRE:

The power of a lens of focal length 1 metre is one dioptre (1D).

${\therefore}~~1{\rm{ D  =  1 }}{{\rm{m}}^{ - 1}}$

POWER OF COMBINATION OF LENSES:

Power (P) of a combination of lenses is the algebraic sum of the individual powers P1 , P2 , P3 . . . of the lenses.

Mathematically P = P1 + P2 + P3 + . . . 

DERIVATION OF LENS 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 $\Delta ABC$ 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 Þ PB = – u 

image distance , v = – PB/ Þ PB/ = – v

focal length , f = – PF Þ PF = – f 

radius of curvature, 2f = – PC Þ 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}$

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

According to the sign convention:

object distance, u = – PB Þ PB = – u 

image distance, v = – PB/ Þ PB/ = – v

object height, h1 = ­ AB Þ AB = h1

image height, h2 = – A/B/ Þ A/B/ = – h2

Substituting these values in equation ( A ):

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

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

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