Spherical Refracting Surface

A refracting surface, which forms a part of a sphere of transparent refracting material, is called spherical refracting surface. The two types are convex spherical refracting surfaces and concave spherical refracting surfaces.

Convex Spherical Refracting Surface

Concave Spherical Refracting Surface

XY is the refracting surface.

P is the pole of spherical refracting surface. C is centre of curvature of spherical refracting surface.

m₁, m₂ are the absolute refractive indices of the two media.

Assumption: In dealing with refraction at spherical refracting surface, we assume.

• The object to be a point lying on the principal axis of the spherical refracting surface.

• The aperture of the spherical refracting surface is small.

• The incident and refracted rays make small angles with the principal axis of the surface so that sini » i and sinr » i

The sign convention used in mirror is applicable for spherical refracting surfaces.

Refraction from Rarer to Denser Medium at a Convex Spherical Refracting Surface

Real Image

Consider a spherical surface XY convex to the incident ray OA. The point O is a point object and I is the image of the point object where the refracted rays actually meet.

From A draw a perpendicular on the axis so as to meet M

In triangle OAC, i = a + g

According to Snell's law

As the aperture is close, M is close to P.

Using the sign convention, we put

PO = -u , PI = +v, PC = R

OR

Note:

For the virtual image, the point lies close to the pole of refracting surface. In this case the refracted rays PC and AB do not meet actually at any point but appear to come from a point I as shown below.

Refraction from denser to rarer medium at a concave spherical refracting surface

Let the point object lie on the principal axis. A ray of light meets the spherical surface concave to the incident ray at A. The refracted ray bends away from the normal C A N and moves along AI.

Since the two refracted rays AI and BI actually meet, I represent a real image.

Now, from Snell's law

(Since refraction occurs from denser to rarer)

or m₂ sin i = m₁ sin ror m₂ i = m₁ r (as i and r are small angles)

In D OAC

i = g - a

In D AIC

r = g - b

From A, draw AM perpendicular to principal axis

For small aperture, M is close to P

Applying the sign convention

Following the procedure as in previous case we have

PO = -u, PI = +v, PC = -R

We have

or