Question

Define refraction of light waves.
Draw a ray diagram for refraction at a spherical separating two media. For refraction at a spherical surface, derive the relation $\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$ in object distance $\left(u\right)$, image distance $\left(v\right)$, refractive index of media $(n_1,n_2)$ and radius of curvature $\left(R\right)$.

Answer 1

When light ray enters a medium of different refractive index from another medium, it undergoes change of velocity. This is known as refraction of light. This change in velocity happens both in direction and magnitude of the light wave.

Assumption:

$i$ and $r$ are small

$⟹x,y,z$ are small

$⟹MN\approx$ length of arc ($l$)

$OM\approx u:$ Object distance

$IM\approx v:$ Image distance

$Cm\approx R:$ Radius

Using geometry,

$i=x+y..............\left(i\right)$

$r=y-z..............\left(ii\right)$

Using snell's law,

$n_1\sin i=n_2\sin r$

$n_1\sin (x+y)=n_2\sin (y-z)...............\left(iii\right)$ using (i) and (ii)

Using approximation, $\sin (x+y)\approx x+y\approx \sin x+\sin y$

$\sin (y-z)\approx y-z\approx \sin y-\sin z$

Using definition of sinusoid,

$\sin x\approx \frac{MN}{u}$

$\sin z\approx \frac{MN}{v}$

$\sin y\approx \frac{MN}{R}$

Using above approximations, equation (iii) can be rewritten as:

$n_1(\frac{MN}{u}+\frac{MN}{R})=n_2(\frac{MN}{R}-\frac{MN}{v})$

Rearranging the terms, we get:

$\frac{n_1}{u}+\frac{n_2}{v}=\frac{n_2-n_1}{R}$

Applying sign convention, $u\to -u$

$v\to +v$

$R\to +R$

$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$