Question

$\red{\Huge{\underline{\underline{\frak{Question : }}}}}$
For a PlanoConvex lens, prove that focal length is always equal from both the surfaces of the lens. (Derivation)
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Answer 1

Introduction : (Thin Lens)

A thin lens is called convex if it is thicker at the middle and it is called concave if it is thicker at the ends.

One surface of a convex lens is always convex. Depending on the surface a convex lens is categorised as

• Biconvex or convexo convex, if the other surface is also convex.
• Plano convex if the other surface is plane.
• Concavo convex is the other surface is concave.

Similarly concave lens is categorised as concavo-concave or biconcave, plano-concave and convexo-concave.

Formula :

For a spherical, thin lens having the same medium on both sides :

$\bigstar\:\underline{\boxed{\bf{\purple{\dfrac{1}{f}=(n-1)\bigg(\dfrac{1}{R_1}-\dfrac{1}{R_2}\bigg)}}}}$

Where R₁ and R₂ are x coordinate of the centre of curvature of the 1st surface and 2nd surface respectively.

Remember :

• Radius of curvature/focal length of concave mirror is taken negative and that for convex mirror is taken positive.
• Radius of curvature or focal length of plane mirror = ∞

Case - I :

• Surface 1 : convex mirror
• Surface 2 : plane morror

$:\implies\tt\:\dfrac{1}{f}=(n-1)\bigg(\dfrac{1}{R_1}-\dfrac{1}{R_2}\bigg)$

• R₁ = + R and R₂ = ∞

$:\implies\tt\:\dfrac{1}{f}=(n-1)\bigg(\dfrac{1}{R}-\dfrac{1}{\infty}\bigg)$

$:\implies\tt\:\dfrac{1}{f}=(n-1)\bigg(\dfrac{1}{R}-0\bigg)$

$:\implies\tt\:\dfrac{1}{f}=(n-1)\bigg(\dfrac{1}{R}\bigg)$

$:\implies\bf\:f=\dfrac{R}{(n-1)}\:\dots\:(1)$

Case - 2 :

• Surface 1 : plane mirror
• Surface 2 : concave morror

$:\implies\tt\:\dfrac{1}{f}=(n-1)\bigg(\dfrac{1}{R_1}-\dfrac{1}{R_2}\bigg)$

• R₁ = ∞ and R₂ = - R

$:\implies\tt\:\dfrac{1}{f}=(n-1)\bigg(\dfrac{1}{\infty}-\dfrac{1}{-R}\bigg)$

$:\implies\tt\:\dfrac{1}{f}=(n-1)\bigg(0+\dfrac{1}{R}\bigg)$

$:\implies\tt\:\dfrac{1}{f}=(n-1)\bigg(\dfrac{1}{R}\bigg)$

$:\implies\bf\:f=\dfrac{R}{(n-1)}\:\dots\:(2)$

Focal length in both cases are same!

Hence Proved!!