# Prism

Prism chart

In the usual triangular prism, the ray changes direction twice, one time when it enters the prism and second time when it leave the prism. The prism form is such that both times the ray is bended toward the same direction, and the total deviation angle is the sum of these two deviation angles:

[[Math:c| \delta=\delta_{in} \plus \delta_{out} \ \ (1) ]]

Now, the four angle polygon ABCD has two right angles (B and D). As the sum of angles in polygon is always 360 (degrees), the bottom angle C = 180 - A (A is the angle between the two relevant prism faces at the top of the prism).

As the sum of angles in triangle is always 180:

[[Math:c| 180^o=\beta_1\plus\beta_2\plus(180^o-A) ]]

or

[[Math:c| A = \beta_1\plus\beta_2 \ \ (2) ]].

By adding (1) and (2) side by side we get

[[Math:c|\delta \plus A = \delta_{in} \plus \delta_{out}\plus\beta_1\plus\beta_2]]

Now we notice that

[[Math:c|\alpha_1=\delta_{in}\plus\beta_1]]
[[Math:c|\alpha_2=\delta_{out}\plus\beta_2]]

and finally

[[Math:c|\delta=\alpha_1\plus\alpha_2-A]]

To derive further, we need to use the Snell's law that the ratio of the sines of the angles of incidence and refraction equal to the opposite ratio of the indices of refraction:

[[Math:c|\frac{\sin\theta_1}{\sin\theta_2} = \frac{n_2}{n_1}]]

Following this law and assuming that the optical density of the air (where the prism is located) is close to 1, the optical density of the prism material can be written as

[[Math:c| n = {sin \alpha_1} \over {sin \beta_1} = {sin \alpha_2} \over {sin \beta_2} ]]

so

[[Math:c| \alpha_1 = arcsin(n \cdot sin \beta_1) ]].

As

[[Math:c| \beta_1 = A - \beta_2 ]]

so

[[Math:c| \alpha_2 = arcsin(n \cdot sin \beta_2) = arcsin(n \cdot sin (A - \beta_1) \ \ (3) ]].

Now, going from another side,

[[Math:c| n= {sin \alpha_1} \over {sin \beta_1} \to\beta_1 = arcsin({sin \alpha_1} \over n) ]]

Substituting this [[Math:c|\beta_1]] expression into (3), together, we get

[[Math:c| \alpha_2=arcsin \big ( n \cdot sin (A-arcsin{sin \alpha_1}\over n) \big ) ]]

This is already half or result as we we tie the two angles that depend mostly on our ray to A and n, the characteristics of the prism. Additional analysis can be found at [1] (spectroscopy related site).

Prism applet. The ray path is shown by the thick red line. The thin red lines are continuations showing the [[Math:c|\beta_1]] and [[Math:c|\beta_2]] angles. The demo also allows to change the refractive index of the prism material (1.0 is close to air, 1.52 is close to water, 1.6 is close to flint glass). The mathematical model should deliver correct path and deviation of the ray for the prism of this size, made of the assumed material. Android app is also available.
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## The minimum deviation

The minimum deviation angle is an important characteristic of the prism. The minimum deviation occurs when entering angle and exiting angle are the same. It allows to find the refraction coefficient of the prism. At the minimum angle,

[[Math:c| n = {sin( {\delta\plusA} \over 2 ) } \over {sin A \over 2} ]][2]

## Spectral decomposition

As the bending angles actually depends on the wavelength, the prism decomposes the white light, producing the rainbow.

## References

1. 1 ioannis.virtualcomposer2000.com. Calculating the Deviation Angle For a Prism (spectroscopy-related side)
2. 2 M.A. Peterson Minimum Deviation by a Prism (includes Java applet).