Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Angle of Refraction

The angle between the refracted ray and the normal to the surface at the point of refraction, determined by Snell's Law.
📜

The statement of the theorem

Let r1\mathbf{r}_1 and r2\mathbf{r}_2 be the position vectors of the incident and refracted rays, respectively, intersecting the interface Σ\Sigma at point P\mathbf{P}. Let n\mathbf{n} be the unit normal vector to Σ\Sigma at P\mathbf{P}. Define the incident direction vector k1=r1′−r1\mathbf{k}_1 = \mathbf{r}_1' - \mathbf{r}_1 and the refracted direction vector k2=r2′−r2\mathbf{k}_2 = \mathbf{r}_2' - \mathbf{r}_2, where r1′\mathbf{r}_1' and r2′\mathbf{r}_2' are points infinitesimally further along the rays. The angle of incidence θ1\theta_1 and the angle of refraction θ2\theta_2 are defined by the directional cosines: cos⁡(θ1)=∣k1⋅n∣∣k1∣\cos(\theta_1) = \frac{|\mathbf{k}_1 \cdot \mathbf{n}|}{|\mathbf{k}_1|} and cos⁡(θ2)=∣k2⋅n∣∣k2∣\cos(\theta_2) = \frac{|\mathbf{k}_2 \cdot \mathbf{n}|}{|\mathbf{k}_2|}. The relationship between the angles and the refractive indices n1n_1 and n2n_2 of the media is given by Snell's Law, which mandates the equality of the tangential components of the wave vector: n1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2).