Some modern optical devices are made with glass whose index of refraction changes with distance from the front surface. FIGURE P16.72 shows the index of refraction as a function of the distance into a slab of glass of thickness L. The index of refraction increases linearly from n₁ at the front surface to n₂ at the rear surface. Evaluate your expression for a 1.0-cm-thick piece of glass for which n₁ = 1.50 and n₂ = 1.60.

Verified step by step guidance

1. Step 1: Understand the problem. The index of refraction changes linearly with distance through the glass. This means the index of refraction, n(x), at a distance x from the front surface can be expressed as a linear function. The goal is to evaluate the expression for the given parameters: thickness L = 1.0 cm, n₁ = 1.50, and n₂ = 1.60.
2. Step 2: Write the linear equation for the index of refraction. Since the index of refraction changes linearly, we can express it as: $n(x)\; =\; n₁\; +\; (n₂\; -\; n₁)\; *\; (x\; /\; L)$, where x is the distance from the front surface, L is the thickness of the glass, n₁ is the index of refraction at the front surface, and n₂ is the index of refraction at the rear surface.
3. Step 3: To evaluate the expression, substitute the given values into the equation. Use L = 1.0 cm, n₁ = 1.50, and n₂ = 1.60. The equation becomes: $n(x)\; =\; 1.50\; +\; (1.60\; -\; 1.50)\; *\; (x\; /\; 1.0)$.
4. Step 4: Simplify the equation. The difference (1.60 - 1.50) is 0.10, so the equation becomes: $n(x)\; =\; 1.50\; +\; 0.10\; *\; x$. This is the linear expression for the index of refraction as a function of distance x.
5. Step 5: To evaluate the index of refraction at specific points within the glass, substitute the desired values of x (e.g., x = 0 cm for the front surface, x = 1.0 cm for the rear surface) into the simplified equation. For example, at x = 0, $n(0)\; =\; 1.50\; +\; 0.10\; *\; 0$, and at x = 1.0 cm, $n(1.0)\; =\; 1.50\; +\; 0.10\; *\; 1.0$.

Key Concepts

Index of Refraction

The index of refraction (n) is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. A higher index indicates that light travels slower in that medium, affecting how light bends when entering or exiting the material.

Linear Variation of Index of Refraction

In this context, the index of refraction varies linearly with distance, meaning it increases steadily from n₁ at the front surface to n₂ at the rear surface. This linear relationship can be expressed mathematically, allowing for the calculation of the effective index of refraction at any point within the slab, which is crucial for understanding light behavior in the material.

Snell's Law

Snell's Law describes how light refracts when it passes from one medium to another with different indices of refraction. It states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant and equal to the ratio of the indices of refraction of the two media. This principle is essential for analyzing how light behaves as it travels through the varying index of refraction in the glass slab.