34-3 Spherical Refracting Surfaces#

Prompts

How does a spherical refracting surface form an image? What determines whether the image is real or virtual?
For refraction (unlike reflection): on which side of the surface do real images form? Virtual images?
What is the sign of \(r\) when the object faces a convex surface? A concave surface? How does this differ from mirrors?
State the refracting-surface equation relating \(n_1\), \(n_2\), \(p\), \(i\), and \(r\).
Object in glass (\(n_1 \approx 1.5\)) faces a convex surface into air (\(n_2 \approx 1\)). Does the ray bend toward or away from the normal? Toward or away from the central axis?

Lecture Notes#

Overview#

A spherical refracting surface (radius \(r\), center \(C\)) separates two media with indices \(n_1\) and \(n_2\). Light from object O in medium \(n_1\) refracts into medium \(n_2\).
Image type depends on \(n_1\), \(n_2\), surface curvature, and object distance.
Key difference from mirrors: Real images form on the opposite side of the surface from the object; virtual images form on the same side.
Refracting-surface equation: \(n_1/p + n_2/i = (n_2 - n_1)/r\).

Real vs virtual images#

Refraction bends rays toward the normal when entering a higher-index medium and away when entering a lower-index medium.

Real image: Refracted rays converge toward the central axis and intersect on the opposite side of the surface from the object.
Virtual image: Refracted rays diverge; backward extensions intersect on the same side as the object.

Image type	Location (relative to object)
Real	Opposite side of surface
Virtual	Same side as object

Important

This is the reverse of mirrors: for mirrors, real images are on the same side as the object; for refracting surfaces, real images are on the opposite side.

Sign convention for \(r\)#

The radius of curvature \(r\) is measured from the surface to the center of curvature \(C\):

Object faces convex surface (surface bulges toward the object): \(r > 0\).
Object faces concave surface (surface is caved in from the object’s view): \(r < 0\).

Caution

This sign convention for \(r\) is opposite to that for spherical mirrors. For mirrors, concave (caved in) has \(r > 0\); for refracting surfaces, the object faces the surface, and convex-as-seen-by-object means \(r > 0\).

Refracting-surface equation#

For paraxial rays (small angles with the central axis):

(257)#\[ \frac{n_1}{p} + \frac{n_2}{i} = \frac{n_2 - n_1}{r} \]

where \(n_1\) = index of the object side, \(n_2\) = index of the other side.

Sign convention: \(p > 0\); \(i > 0\) (real image), \(i < 0\) (virtual image).

Six cases (summary)#

The textbook (Fig. 34-12) shows six arrangements. In general:

Real images (a, b): Object far from surface; refraction directs rays toward the central axis.
Virtual images (c–f): Object near the surface, or when refraction always directs rays away from the axis; backward extensions form the image.

When \(n_2 > n_1\) (light entering denser medium): rays bend toward the normal. When \(n_2 < n_1\): rays bend away from the normal. The geometry of the surface (convex vs concave) and object distance determine whether rays converge or diverge.

Summary#

Spherical refracting surface: single interface between \(n_1\) and \(n_2\); can form real or virtual images.
Real image: opposite side of surface; virtual image: same side as object.
\(r > 0\): object faces convex surface; \(r < 0\): object faces concave surface.
Equation: \(n_1/p + n_2/i = (n_2 - n_1)/r\).