2.1 Refraction from a spherical surface

1.1 Refractive index of light
1.2 Types of medium

2.1 Refraction from a spherical surface

3.1 Shift due to a glass slab
3.2 Lateral Magnification

4.1 Sign convention
4.2 Some important terms
4.3 Ray tracing
4.4 Image formed by covex lens
4.5 Image formed by concave lens

5.1 Thin Lens Formula
5.2 Magnification and Power of lens
5.3 Combination of lenses
5.4 Displacement method to find focal length.
5.5 Silvering of lens

6.1 Critical angle
6.2 Application of total internal reflection

7.1 Dispersive power

8.1 Rainbow

9.1 Spectrometer
9.2 Simple microscope
9.3 Compound microscope
9.4 Astronomical telescope (Refracting type)
9.5 Terrestrial telescope
9.6 Galileo's terrestrial telescope
9.7 Reflecting type telescope

The portion of a refracting medium whose curved surface forms the part of a sphere is known as spherical refracting surface.

Spherical refracting surface are of two types,

Sign conventions for spherical refracting surface are same as those for spherical mirrors.

Consider two transparent media having indices of refraction ${\mu _1}$ and ${\mu _2}$, where the boundary between the two media is a spherical surface of radius $R$.

Assume ${\mu _2} > {\mu _1}$,

Let us consider a single ray leaving point $O$ and forms a image at point $I$ as shown.

$CP$: Normal

$OP$: Incident ray

$PI$: Refracted ray

$i$: Angle of incidence

$r$: Angle of refraction

$MO$: $-u$

$MI$: $+v$

$MC$: $+R$

From Snell's law we can write,

$${\mu _1}\sin i = {\mu _2}\sin r$$

As (angle $i$ and $r$ are very small). So,

$${\mu _1}i = {\mu _2}r$$ or $$\frac{i}{r} = \frac{{{\mu _2}}}{{{\mu _1}}}\quad ...(i)$$

$$\alpha + \beta = i\quad ...(ii)$$

$$\gamma + r = \beta $$ or $$r = \beta - \gamma \quad ...(iii)$$

Dividing equation $(ii)$ and $(iii)$ we get,

$$\frac{i}{r} = \left( {\frac{{\alpha + \beta }}{{\beta - \gamma }}} \right)$$

Using equation $(i)$,

$$\frac{{{\mu _2}}}{{{\mu _1}}} = \left( {\frac{{\alpha + \beta }}{{\beta - \gamma }}} \right)$$

$$\begin{equation} \begin{aligned} {\mu _2}\beta - {\mu _2}\gamma = {\mu _1}\alpha + {\mu _1}\beta \\ \beta \left( {{\mu _2} - {\mu _1}} \right) = {\mu _1}\alpha + {\mu _2}\gamma \quad ...(iv) \\\end{aligned} \end{equation} $$

Now we can write,

$$\begin{equation} \begin{aligned} \tan \alpha = \frac{{PN}}{{NO}}\quad {\text{or}}\quad \tan \alpha = \frac{{PN}}{{MO}} \\ \tan \beta = \frac{{PN}}{{NC}}\quad {\text{or}}\quad \tan \beta = \frac{{PN}}{{MC}} \\ \tan \gamma = \frac{{PN}}{{NI}}\quad {\text{or}}\quad \tan \gamma = \frac{{PN}}{{MI}} \\\end{aligned} \end{equation} $$

As angles $\alpha $, $\beta $ & $\gamma $ are very small due to paraxial approximation.

\[\left. \begin{gathered} \tan \alpha \approx \alpha = \frac{{PN}}{{MO}} \hspace{1em} \\ \tan \beta \approx \beta = \frac{{PN}}{{MC}} \hspace{1em} \\ \tan \gamma \approx \gamma = \frac{{PN}}{{MI}} \hspace{1em} \\\end{gathered} \right\}\quad ...(v)\]

From equation $(iv)$ and $(v)$ we get,

$$\frac{{PN}}{{MC}}\left( {{\mu _2} - {\mu _1}} \right) = {\mu _1}\left( {\frac{{PN}}{{MO}}} \right) + {\mu _2}\left( {\frac{{PN}}{{MI}}} \right)$$

or

$$\left( {\frac{{{\mu _2} - {\mu _1}}}{{MC}}} \right) = \frac{{{\mu _1}}}{{MO}} + \frac{{{\mu _2}}}{{MI}}$$

As $\left( {MC = + R,\;MO = - u,\;MI = + v} \right)$. So,

$$\frac{{{\mu _2} - {\mu _1}}}{{ + R}} = \frac{{{\mu _1}}}{{ - u}} + \frac{{{\mu _2}}}{v}$$

So,

This formula is valid for both convex and concave spherical surfaces.