Question

Derive expression for the refraction through a spherical surface

Answer 1

Refraction at Spherical Surface

Let us now see the refraction of light at the spherical surface. Now, the change in direction or bending of a light wave passing from one transparent medium to another caused by the change in wave’s speed is the Refraction. Suppose the above figure is a spherical surface. There is one medium with refractive index n1 and second medium with refractive index  n2.

There is an object O and a ray of light from the object O is incident on the spherical mirror. Since it is moving from a rarer medium to a denser medium, the ray bends towards the normal.  An image is formed and radius of curvature of a spherical surface is R with the center C of the spherical surface.

”u” is the object distance from a pole of a spherical surface

”v” is the image distance from a pole of the spherical surface

Now as we know that,

n1 is the refractive index of a medium from which rays are incident.

n2 is the refractive index of another medium

We get,

tanα = MNOM

tanγ = MNMC

tanβ = MNMI

Now, for Δ NOC, i is the exterior angle.

i = ∠ NOM + ∠ NCM

i= MNOM+MNMC …….1

Similarly,

r =  MNMC–MNMI …….2

Now by using Snell’s law we get

n1 sin i = n2sin r

Substituting i and r from Eq. (1) and (2), we get

n1OM+n2MI=n2−n1MC

As, OM = -u, MI = +v, MC = +R

Hence, the equation becomes n2v–n1u=n2−n1R