Prism formulae
Prism Formulae
A prism is a transparent optical element with flat, polished surfaces that refract light. It is typically triangular in shape. Prisms can disperse light into its constituent colors (a spectrum), reflect it, or split it into components with different polarizations.
Basic Parameters of a Prism
Before diving into the formulas, let's understand the basic parameters of a prism:
- Refractive Index (n): This is a measure of how much the speed of light is reduced inside the material of the prism compared to the speed of light in a vacuum.
- Angle of Prism (A): This is the angle between the two plane faces of the prism which are inclined to each other.
- Angle of Incidence (i): The angle between the incident ray and the normal at the point of incidence on the first face of the prism.
- Angle of Refraction (r): The angle between the refracted ray inside the prism and the normal at the point of refraction.
- Angle of Emergence (e): The angle between the emergent ray and the normal at the point of emergence on the second face of the prism.
- Angle of Deviation (δ): The angle between the direction of the incident ray and the direction of the emergent ray.
Prism Formulae
Here are some of the key formulas related to prisms:
Snell's Law at Each Surface
At the first surface:
$$ n_1 \sin(i) = n_2 \sin(r) $$
At the second surface:
$$ n_2 \sin(r') = n_1 \sin(e) $$
Where:
- ( n_1 ) is the refractive index of the medium from which light is incident (usually air, so ( n_1 \approx 1 )).
- ( n_2 ) is the refractive index of the prism material.
- ( r' ) is the angle of refraction at the second surface, which is equal to the angle of incidence at the second surface due to symmetry.
Angle of Deviation
The angle of deviation ( \delta ) can be found using the relationship:
$$ \delta = i + e - A $$
Minimum Deviation
When the angle of incidence is such that the angle of deviation is minimum (( \delta_{min} )), the light ray passes symmetrically through the prism, and the angles of incidence and emergence are equal (( i = e )). In this case, the angle of refraction ( r ) is equal to the angle of the prism ( A ), and Snell's Law gives us:
$$ n = \frac{\sin\left(\frac{\delta_{min} + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} $$
Dispersion of Light
Dispersion occurs when different wavelengths of light are refracted by different amounts. The refractive index ( n ) is a function of wavelength ( \lambda ), so each component of the incident white light will have a different angle of deviation.
Examples and Comparison
Let's consider an example to illustrate these concepts:
Example 1: Calculating the Angle of Deviation
Suppose we have a prism with an angle ( A = 60^\circ ) and a refractive index ( n = 1.5 ). If the angle of incidence ( i ) is ( 45^\circ ), what is the angle of deviation?
First, we use Snell's Law to find the angle of refraction ( r ):
$$ 1 \cdot \sin(45^\circ) = 1.5 \cdot \sin(r) $$
Solving for ( r ), we get:
$$ r \approx 28.1^\circ $$
Now, assuming the light ray passes symmetrically through the prism, we have ( r' = r ) and ( e = i ). Using the angle of deviation formula:
$$ \delta = i + e - A = 45^\circ + 45^\circ - 60^\circ = 30^\circ $$
Comparison Table
Here is a comparison table highlighting the differences between some key parameters:
| Parameter | Symbol | Description | Example Value |
|---|---|---|---|
| Refractive Index | ( n ) | Measure of light bending due to the prism material | 1.5 |
| Angle of Prism | ( A ) | Angle between the prism's two inclined faces | ( 60^\circ ) |
| Angle of Incidence | ( i ) | Angle between the incident ray and the normal | ( 45^\circ ) |
| Angle of Refraction | ( r ) | Angle between the refracted ray and the normal inside prism | ( 28.1^\circ ) |
| Angle of Emergence | ( e ) | Angle between the emergent ray and the normal | ( 45^\circ ) |
| Angle of Deviation | ( \delta ) | Angle between the incident and emergent rays | ( 30^\circ ) |
Understanding these formulas and parameters is crucial for solving problems related to prisms in optics. By applying Snell's Law and the deviation formula, one can analyze how light behaves as it passes through a prism, which is essential knowledge for both academic studies and practical applications in optical devices.