Question

Derive the following expression for the refraction at concave spherical surface: .

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Updated on : 2022-09-05
Solution
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Let MPN be concabe spherical surface of a medium of refractive index . P is the pole o is the centre of curvature PC is the principle axis. Let O be a point object or it's principle axis kept in a rarer medium first incident ray travelling through C is normal to the spherical surface therefor it will not deviate and goes along PX another mident ray OA will refract at Point A and it bends towards the normal. AB and PX are proceeded behind then at I, a virtual image will be form.
Let are the ray angle normal with the principle axis respectively.
then by snell's law
           ........(1)
but here i and r are the small then we put.
      in equation (1)

                ...........(2)
now by using exterior angle theorem in AOC

            .........(3)
now in IAC by using exterior angle theorem

           ..........(4)
putting value of i and r in equation (2)
          ...........(5)
now angle


now
using value of in equation (5)


        .........(6)
now by using sign convention
               

using these value in equation (6)



This is the required expression for the refraction formula for the concave spherical surface.

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