Let the x z-plane be the boundary between two transparent...

Let the $x z$-plane be the boundary between two transparent media. Medium 1 in $z \geq 0$ has a refractive index of $\sqrt{2}$ and medium 2 with $z<0$ has a refractive index $\sqrt{3} .$ A ray of light in medium 1 given by the vector $\mathbf{A}=6 \sqrt{3} \hat{\mathbf{i}}+8 \sqrt{3} \hat{\mathbf{j}}-10 \hat{\mathbf{k}}$ is incident on the plane of separation. The angle of refraction in medium 2 is (a) $45^{*}$ (b) $60^{\circ}$ (c) 75 (d) $30^{*}$

Let the $x z$-plane be the boundary between two transparent media. Medium 1 in $z \geq 0$ has a refractive index of $\sqrt{2}$ and medium 2 with $z<0$ has a refractive index $\sqrt{3} .$ A ray of light in medium 1 given by the vector $\mathbf{A}=6 \sqrt{3} \hat{\mathbf{i}}+8 \sqrt{3} \hat{\mathbf{j}}-10 \hat{\mathbf{k}}$ is incident on
the plane of separation. The angle of refraction in medium 2 is
(a) $45^{*}$
(b) $60^{\circ}$
(c) 75
(d) $30^{*}$

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Key Concepts

Snell's Law

Snell's Law governs the behavior of light as it passes from one medium to another with different refractive indices. It relates the sine of the incident angle to the sine of the refraction angle using the ratio of the refractive indices of the two media, ensuring the continuity of the phase of the electromagnetic wave at the interface.

Refractive Index

The refractive index is a measure of how much light slows down as it travels through a medium compared to its speed in vacuum. Variations in refractive index between two media cause the bending or refraction of light at their interface.

Vector Decomposition in Optical Interfaces

When a ray of light is described by a vector, it can be decomposed into components parallel and perpendicular to the interface. This decomposition is essential for determining the angle of incidence, which in turn is used in Snell's Law to compute the angle of refraction.

Determination of Angles from Vectors

The angle of incidence is found by analyzing the vector's component perpendicular to the interface, often using the dot product with the normal vector. This angle, defined relative to the line normal to the interface, is crucial for applying Snell's Law to predict how the light ray will refract upon entering the second medium.

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