Question

Write the formula of the angle of minimum deviation produced by a thin prism.

Answer 1

Hint: Start by using the equation of prism $$\mu ={\displaystyle \frac{\sin \left[{\displaystyle \frac{(A+D)}{2}}\right]}{\sin {\displaystyle \frac{A}{2}}}}$$ . Since the angles are small $$\sin \left({\displaystyle \frac{A+D}{2}}\right)={\displaystyle \frac{A+D}{2}}$$ and $$\sin \left({\displaystyle \frac{A}{2}}\right)={\displaystyle \frac{A}{2}}$$ the equation now becomes $$D=A\left(\mu -1\right)$$ . Then use the snell’s law $$\mu ={\displaystyle \frac{\sin i}{\sin r_1}}$$ and since angles are small $$\sin i=i$$ , $$\sin r_1=r_1$$ , $$\sin r_2=r_2$$ and $$\sin e=e$$ . Use these values to obtain the equation $$u\approx {\displaystyle \frac{i}{r_1}}$$ and $$\mu \approx {\displaystyle \frac{e}{r_2}}$$ . Then use simple geometry to obtain the equation $$(i+e)-\left(r_1+r_2\right)=\mathrm{\angle }D$$ . Then put the value $$r_1+r_2=\mathrm{\angle }A$$ in the previous equation to reach the solution.

Complete answer:
The angle of deviation is the angle that the emerging ray coming out of a prism makes with the incident ray coming in the prism (indicated as angle D in the diagram below). This angle of deviation decreases with an increase in the angle of incidence, but only up to a minimum angle called the angle of minimum deviation (indicated by $$D_m$$ ).

The refractive index of the material of the prism is calculated by using the following formula
$$\mu ={\displaystyle \frac{\sin \left[{\displaystyle \frac{(A+D)}{2}}\right]}{\sin {\displaystyle \frac{A}{2}}}}$$ (Equation 1)
$$\mu =$$ The refractive index of the material of the prism
$$A=$$ The angle of the prism
$$D=$$ Angle of deviation
For thin prisms or small angle prisms, as the angles become very small, the sine of the angle nearly equals the angle itself, i.e. $$\sin \left({\displaystyle \frac{A+D}{2}}\right)={\displaystyle \frac{A+D}{2}}$$ and $$\sin \left({\displaystyle \frac{A}{2}}\right)={\displaystyle \frac{A}{2}}$$ .
Putting these values in equation 1, we get
$$\mu \approx {\displaystyle \frac{\left({\displaystyle \frac{A+D}{2}}\right)}{\left({\displaystyle \frac{A}{2}}\right)}}$$
$$\mu ={\displaystyle \frac{A+D}{A}}$$
$$\mu A=A+D$$
$$D=A\left(\mu -1\right)$$
Now, for the angle of minimum deviation this equation becomes
$$D_m=A\left(\mu -1\right)$$
We know by Snell’s law for the incident ray
$$\mu ={\displaystyle \frac{\sin i}{\sin r_1}}$$
Since the angles are small, so $$\sin i=i$$ and $$\sin r_1=r_1$$ .
So, $$u\approx {\displaystyle \frac{i}{r_1}}$$
We know by Snell’s law for the emergent ray
$${\displaystyle \frac{1}{\mu }}={\displaystyle \frac{\sin r_2}{\sin e}}$$
$$\mu ={\displaystyle \frac{\sin e}{\sin r_2}}$$
Since the angles are small, so $$\sin e=e$$ and $$\sin r_2=r_2$$ .
So, $$\mu \approx {\displaystyle \frac{e}{r_2}}$$
We know,
$$\mathrm{\angle }AMB=\mathrm{\angle }OMP=i$$ (Vertically opposite Angles)
$$\mathrm{\angle }CND=\mathrm{\angle }PNO=e$$ (Vertically opposite angles)
In $$\mathrm{\Delta }OMN$$ ,
$$\mathrm{\angle }OMN+\mathrm{\angle }MNO=\mathrm{\angle }QON$$ ( $$\mathrm{\angle }QON$$ is an external angle to the triangle $$\mathrm{\Delta }OMN$$ )
$$(\mathrm{\angle }OMP-\mathrm{\angle }NMP)+(\mathrm{\angle }PNO-\mathrm{\angle }MNP)=\mathrm{\angle }QON$$
$$i-r_1+e-r_2=\mathrm{\angle }D$$
$$(i+e)-\left(r_1+r_2\right)=\mathrm{\angle }D$$ (Equation 1)
In $$\mathrm{\Delta }PMN$$ the sum of all the angles is $${180}^{\circ }$$
$$\mathrm{\angle }OMN+\mathrm{\angle }MNO+\mathrm{\angle }MPN={180}^{\circ }$$
$$({90}^{\circ }-\mathrm{\angle }NMP)+\left({90}^{\circ }-\mathrm{\angle }MNP\right)+\mathrm{\angle }A={180}^{\circ }$$
$$\left({90}^{\circ }-r_1\right)+\left({90}^{\circ }-r_2\right)+\mathrm{\angle }A=180$$
$$r_1+r_2=\mathrm{\angle }A$$
Substituting this value in equation 1, we get
$$\left(i+e\right)-\left(\mathrm{\angle }A\right)=\mathrm{\angle }D$$
For, the angle of minimum deviation, $$i=e$$ , so equation 2 becomes
$$2i-\mathrm{\angle }A=\mathrm{\angle }{D}_{min}$$
$$i={\displaystyle \frac{A+D_{min}}{2}}$$

Note:
The prism is optical equipment that is used to observe the dispersion of white light. The prism makes use of the fact that light travels with different speeds in different mediums. The prism is normally made out of glass, the edges of the prism should be perfect during the manufacturing of the glass prisms.