`text(Condition of no emergence :)`

A ray of light incident on a prism of angle A and refractive index `mu` will not emerge out of a prism (whatever may be the angle of incidence) if `A > 2theta_c`, where `theta_c` is the critical angle.

i.e. `mu > 1/(sin(A//2)`

`text(Condition of grazing emergence :)`

By the condition of grazing emergence we mean the angle of incidence i at which the angle of emergence becomes e = 90-.

`i=sin^(-1)[sqrt(mu^2 - 1)sinA - cosA]`

`text(Note :)`
The light will emerge out of a given prism only if the angle of incidence is greater than the condition of grazing emergence.

`text(Deviation)` `delta :`
It is the angle between the emergent and the incident ray. It other words, it is the angle through which incident ray turns in passing through a prism.

`delta=i+e-A`

`text(Condition of maximum deviation :)`

Maximum deviation occurs when the angle of incidence is 90-.

`delta_(max) = 90^o + e -A`

where `e = sin^(-1)[musin(A-theta_c)]`

`text(Condition of minimum deviation :)`

The minimum deviation occurs when the angle of incidence is equal to the angle of emergence, i.e.,

`i=e`
`delta_(m i n)=2i-A`

Using Snell's law

`mu=(sin((delta_(mi n) +A)/2))/sin(A/2)`

`text(Note :)`
In the condition of minimum deviation the light ray passes through the prism symmetrically, i.e. the light ray in the prism becomes parallel to its base.

`text(Thin Prisms :)`

In thin prisms the distance between the refracting surfaces is negligible and the angle of prism (A) is very small.

Since `A=r_1+r_2` therefore, if A is small then both `r_1` and `r_2` are also small, and the same is true for `i_1` and `i_2`.

According to Snell's law

`sin i_1 = musinr_1` `=>` `mur_1`
`sin i_2 = mu sinr_2` `=>` `mur_2`

`:.` Deviation `delta=(i_1-r_1)+(i_2-r_2)`
`delta=(r_1+r_2)-(i_1 +i_2)`
`delta=A(mu-1)`

`text(Note :)` The deviation for a small angled prism is independent of the angle of incidence.

(a) Variation of `theta` versus `i` (shown in diagram).
For one `theta` (except `delta_(mi n)`) there are two values of angle of incidence.
If i and e are interchanged then we get the same value of `theta` because of reversibility principle of light.

(b) There is one and only one angle of incidence for which the angle of deviation is minimum.

(c) When `delta = delta_(mi n)`, the angle of minimum deviation, then i = e and `r_1 = r_2`, the ray passes symmetrically w.r.t. the refracting surfaces. We can show by simple calculation that `delta_(mi n) = 2i_(mi n) - A` where `i_(mi n) =` angle of incidence for minimum deviation, and `r= A//2`

`therefore` `mu_(rel) =(sin(A+delta_(mi n))/2)/sin(A/2)`

where `mu_(rel)=(mu_(prism))/(mu_(surroundi ngs)`

Also `delta_(mi n) = (mu - 1)A` (for small values of `angleA`)

When a beam of light (containing several wavelengths) falls on one face of a prism, it splits into its constituent colours. This phenomenon of splitting of light into its constituent colours is called dispersion of light and the band of colours obtained on a screen is called spectrum. The cause of dispersion is variation of refractive index with wavelength of light. An approximate empirical relation as proposed by Cauchy is given by

`mu(lamda)=A+B/lamda^2`

where A and Bare known as Cauchy's constant. The value of A and B depends on material of prism.

We know that
`lamda_(red) > lamda_(viol et)`
`:.` `mu_(red) < mu_(viol et)`
Hence, `delta_(red) < delta_(viol et)`

The difference in the deviations suffered by two colours in passing through a prism gives the angular dispersion for these colours. The angle between the violet and red colours is known as angular dispersion.
We know that for small angle of prism, deviation is given by

`delta=A(mu-1)`
`:. delta_V=` Deviation in violet colour `= (mu_V- R )A`
`delta_R=` Deviation in violet colour `= (mu_R- R )A`

Hence, Angular Dispersion (AD) `=delta_V -delta_R=(mu_V -mu_R)A`

It is clear from above relation that angular dispersion depends upon
(i) the nature of material of the prism
(ii) the angle of the prism. This is also defined as the rate of change of angle of deviation with wavelength
i.e., AD`=(ddelta)/(dlamda)`.

Dispersive power of a prism is defined as the ratio between angular dispersion to mean deviation produced by the prism.

`omega=` Dispersive Power
`=(delta_V - delta_R)/delta_R=(mu_V - mu_R)/(delta_Y -1)=(dmu)/(mu_Y - 1)`

Where `dmu` denotes the difference between the refractive indices of material of prism for violet and red light. It is also defined as dispersion per unit deviation. Yellow colour is taken as mean colour.

Also, `mu_Y=(mu_V + mu_R)/2` or `(mu_B + mu_R)/2`