Question

Derive prism formula for a thin prism.

Answer 1

Hint A thin prism is a transparent, triangular prism which has a very small apex angle, which is less than${10}^{\circ }$. This prism is always set at the angle of minimum deviation, when both of these small quantities are rounded off and the equation is rearranged, the prism formula for a thin prism is derived.

Complete step by step answer:
The prism formula for a prism in general is given by-
$\mu ={\displaystyle \frac{\mathrm{Sin}\left[{\displaystyle \frac{A+{\delta }_m}{2}}\right]}{\mathrm{Sin}\left({\displaystyle \frac{A}{2}}\right)}}$
Here, $\mu$is the refractive index of the glass (or the material from which the prism is constructed.)
A is the apex angle of the prism.
${\delta }_m$ is the minimum deviation produced in the given prism, the minimum deviation occurs when the angle of incidence is equal to the angle of emergence.
For a thin prism, the apex angle A is very small.
This implies that the refraction edge of the prism is also small. Lesser the refraction of the light, lesser will be the deviation it undergoes. Therefore in a thin prism the angle of deviation is also very small.
We know that, when the angle made in a triangle is very small, the sine of that angle can be approximated to equal to that angle. (in radians)
$\mathrm{Sin}{\theta }_{\theta \to 0}=\theta$
Applying this for A and ${\delta }_m$ in the prism formula, we get-
$\mu ={\displaystyle \frac{{\displaystyle \frac{\left(A+{\delta }_m\right)}{2}}}{\left({\displaystyle \frac{A}{2}}\right)}}$
On rearranging this equation we get-
$\mu ={\displaystyle \frac{A+{\delta }_m}{A}}$
Or
$\mu A=A+{\delta }_m$
$A(\mu -1)={\delta }_m$
This is the required prism formula for a thin prism.

Note The thin prism formula is used when the angle of prism is given less than ${10}^{\circ }$ in the numerical problems, this reduces the efforts required in calculation and also gets solved easily. The relation between the minimum deviation angle and the apex angle can be readily found using the thin prism formula.