Refraction of Light
7.0 Refraction through prism
7.0 Refraction through prism
It is a homogenous transparent medium enclosed by two plane surfaces inclined at an angle. These surfaces are known as refracting surfaces, and the angle between them is known as the refracting angle or the angle of prism $(A)$.
Consider a prism of refracting angle $A$ having $AB$ and $AC$ as refracting surfaces as shown in the figure,
A light incident at point $P$ undergoes refraction twice and emerges at point $Q$ as shown in the figure.
1. Relation between angle of prism $(A)$, $r_1$ and $r_2$
$$\begin{equation} \begin{aligned} \angle APR + \angle AQR = 180^\circ \quad ...(i) \\ \angle PAQ + \angle PRQ = 180^\circ \\\end{aligned} \end{equation} $$ or $$A + \angle PRQ = 180^\circ \quad ...(ii)$$
Sum of all angles in a triangle is $180^\circ $. So,
$${r_1} + {r_2} + \angle PRQ = 180^\circ \quad ...(iii)$$
$${r_1} + {r_2} = A\quad ...(iv)$$
2. Angle of deviation
The angle between the incident ray and the emergent ray is known as the angle of deviation.
It is represented by $\delta $.
We can write,
$$\begin{equation} \begin{aligned} \angle SPQ = i - {r_1} \\ \angle SQP = e - {r_2} \\\end{aligned} \end{equation} $$
In $\Delta SPQ$,
$$\angle SPQ + \angle SQP = \delta $$
i.e. sum of two interior angle is equal to opposite exterior angle of a triangle.
$$\begin{equation} \begin{aligned} (i - {r_1}) + (e - {r_2}) = \delta \\ \delta = (i + e) - ({r_1} + {r_2}) \\\end{aligned} \end{equation} $$
As $\left( {{r_1} + {r_2} = A} \right)$ So,
$$\delta = i + e - A\quad ...(v)$$
3. When $A$ and $i_1$ are small
From snell's law we can write,
$$\sin i = \mu \sin r$$
As $i$ is small then $r_1$ will also be small. So,
$$i = \mu {r_1}\quad ...(vi)$$
Similarly,
$$e = \mu {r_2}\quad ...(vii)$$
From equation $(v)$, $(vi)$ and $(vii)$ we get,
$$\begin{equation} \begin{aligned} \delta = \mu {r_1} + \mu {r_2} - A \\ \delta = \mu ({r_1} + {r_2}) - A \\\end{aligned} \end{equation} $$
As $\left( {{r_1} + {r_2} = A} \right)$ So,
$$\delta = \left( {\mu - 1} \right)A\quad ...(viii)$$
4. Minimum deviation
The angle of deviation varies with angle of incidence $i$.
From experiment, variation between angle of deviation and angle of incidence is plotted as shown in the figure.
So, the minimum angle of deviation $\left( {{\delta _m}} \right)$ is obtained when $i=e$.
Using the formula,
$$\delta = i + e - A$$
$$i = e,\quad \delta = {\delta _m}$$ So, $$\begin{equation} \begin{aligned} {\delta _m} = 2i - A \\ i = \left( {\frac{{{\delta _m} + A}}{2}} \right)\quad ...\left( {ix} \right) \\\end{aligned} \end{equation} $$
Also, $$\mu = \frac{{\sin i}}{{\sin r}}$$
As, ${r_1} + {r_2} = A$ & $r_1=r_2 = r$
So, $$r = \left( {\frac{A}{2}} \right)$$
Therefore, $$\mu = \frac{{\sin i}}{{\sin \left( {\frac{A}{2}} \right)}}\quad ...(x)$$
From the equation $(ix)$ ad $(x)$ we get,
$$\mu = \frac{{\sin \left( {\frac{{{\delta _m} + A}}{2}} \right)}}{{\sin \left( {\frac{A}{2}} \right)}}$$
where,
$A$: Angle of prism
${\delta _m}$: Angle of minimum deviation
5. Condition of no emergence
It is the condition in which the ray of light entering the face $AB$ does not come out of the face $AC$ for any value of the angle of incidence. This happens because the total internal reflection takes place on face $AC$.
$$\sin i = \mu \sin {r_1}$$ and $$\mu \sin {r_2} = \sin 90^\circ $$ or $$\sin {r_2} = \frac{1}{\mu }$$
6. Dispersion of light
It is the phenomenon of splitting of white light into its 7 constituent colour on passing through a prism.
This is because different colours have different wavelengths.
According to Cauchy's formula, $$\mu = A + \frac{B}{{{\lambda ^2}}} + \frac{C}{{{\lambda ^4}}}$$
where $A$, $B$ and $C$ are arbitary constants.
So, the refractive index of material of prism for different colours is different.
As, $$\delta = \left( {\mu - 1} \right)A$$
The above equation is only valid when $i$ and $A$ are very small.
Therefore, different colours deviates through different angles on passing through the prism. This is the cause of dispersion.
So, deviation of red, yellow and violet light is given by,
$$\begin{equation} \begin{aligned} {\delta _r} = \left( {{\mu _r} - 1} \right)A \\ {\delta _y} = \left( {{\mu _y} - 1} \right)A \\ {\delta _v} = \left( {{\mu _v} - 1} \right)A \\\end{aligned} \end{equation} $$
7. Angular dispersion
The difference in deviation between any two colours is known as angular dispersion.
Mathematically angular dispersion of light is given as,
$${\delta _V} - {\delta _R} = \left( {{\mu _V} - {\mu _R}} \right)A$$
8. Average deviation
Mean of the deviation of red and violet right.
Mathermatically, $$\delta = \frac{{{\delta _V} + {\delta _R}}}{2}$$
