Refraction through a Curved Surface

Refraction is the bending of light as it passes from one transparent medium to another. This phenomenon is governed by Snell's Law, which states that the ratio of the sines of the angles of incidence (i) and refraction (r) is constant and is equal to the ratio of the velocities in the two media, or equivalently, to the inverse ratio of the refractive indices.

When light refracts through a curved surface, such as that of a lens or a bubble, the situation becomes more complex due to the varying angles of incidence across the surface. The refraction at a curved surface forms the basis of lens optics and is crucial in understanding how lenses form images.

Refraction at a Spherical Surface

Consider a spherical surface separating two media with refractive indices $n_1$ and $n_2$. Let $R$ be the radius of curvature of the spherical surface. If $R$ is positive, the center of curvature is on the same side as the light is coming from; if negative, it is on the opposite side.

Refractive Index and Snell's Law

The refractive index ($n$) of a medium is a measure of how much the speed of light is reduced inside the medium compared to the speed of light in a vacuum. Snell's Law for refraction at a curved surface can be written as:

$$ n_1 \sin(i) = n_2 \sin(r) $$

The Lensmaker's Formula

For a thin lens, the relationship between the object distance ($u$), the image distance ($v$), and the radius of curvature ($R$) can be described by the Lensmaker's Formula:

$$ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} $$

This formula is derived by considering the geometry of the situation and applying Snell's Law at each point on the curved surface.

Table of Differences and Important Points

Property	Plane Surface	Curved Surface
Shape	Flat	Spherical or part of a sphere
Refraction	Uniform across the surface	Varies with the angle of incidence
Image Formation	No change in image size	Can magnify or reduce the image
Formula	$n_1/u + n_2/v = 0$ (for plane surface)	$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$ (for curved surface)
Focal Point	Not applicable	Exists and depends on the curvature of the surface

Examples

Example 1: Refraction through a Convex Surface

Consider a convex spherical surface with a radius of curvature of 20 cm, separating air ($n_1 = 1.00$) and glass ($n_2 = 1.50$). An object is placed 30 cm from the surface in air.

Using the Lensmaker's Formula:

$$ \frac{1.50}{v} - \frac{1.00}{-30 \text{ cm}} = \frac{1.50 - 1.00}{20 \text{ cm}} $$

Solving for $v$ gives us the image distance in the glass medium.

Example 2: Refraction through a Concave Surface

Now consider a concave surface with the same radii and refractive indices. If an object is placed at the same distance, the calculation would be similar, but the sign of $R$ would be negative, indicating that the center of curvature is on the opposite side.

$$ \frac{1.50}{v} - \frac{1.00}{-30 \text{ cm}} = \frac{1.50 - 1.00}{-20 \text{ cm}} $$

Again, solving for $v$ gives us the image distance, but the nature of the image (real or virtual) may differ due to the concavity of the surface.

Conclusion

Refraction through a curved surface is a fundamental concept in optics, essential for understanding how lenses work. The curvature of the surface introduces complexity in the refraction process, leading to the formation of images with varying sizes and properties. By applying the Lensmaker's Formula and considering the geometry of the situation, one can predict the behavior of light as it passes through curved surfaces, which is crucial for designing optical instruments such as glasses, cameras, and telescopes.