34-6 Three Proofs

34-6 Three Proofs#

Prompts

  • For the spherical mirror proof: what geometric theorem relates the angles \(\alpha\), \(\beta\), \(\gamma\), and \(\theta\)? Why is the paraxial approximation needed?

  • For the refracting surface proof: how does Snell’s law simplify for small angles? How do the triangle relations \(\theta_1 = \alpha + \beta\) and \(\beta = \theta_2 + \gamma\) arise?

  • For the thin lens proof: why can we treat the lens as two separate refracting surfaces? What role does the thin-lens approximation (\(L \approx 0\)) play?

  • What is the exterior angle theorem? How is it used in all three proofs?

  • Why is only the relation \(\beta = \text{arc}/r\) exact in the mirror and refracting-surface proofs, while \(\alpha\) and \(\gamma\) are approximate?

Lecture Notes#

Overview#

This section derives the three main formulas of geometrical optics:

  1. Spherical mirror: \(1/p + 1/i = 1/f = 2/r\)

  2. Refracting surface: \(n_1/p + n_2/i = (n_2 - n_1)/r\)

  3. Thin lens: \(1/p + 1/i = 1/f = (n-1)(1/r_1 - 1/r_2)\)

All three proofs rely on paraxial rays (small angles with the central axis), the exterior angle theorem (exterior angle = sum of opposite interior angles), and geometry of similar arcs.

Proof 1: Spherical mirror formula#

Setup: Point object O on the axis of a concave mirror; a ray from O reflects at point \(a\) and crosses the axis at image I. Angles: \(\alpha\) (at O), \(\beta\) (at C), \(\gamma\) (at I), \(\theta\) (incidence/reflection at \(a\)).

Key steps:

  1. Exterior angle theorem on triangles O\(a\)C and O\(a\)I: \(\beta = \alpha + \theta\), \(\gamma = \alpha + 2\theta\).

  2. Eliminate \(\theta\): \(\alpha + \gamma = 2\beta\).

  3. Small-angle approximation (paraxial rays): \(\alpha \approx \text{arc}/p\), \(\beta = \text{arc}/r\) (exact—arc is part of circle centered at C), \(\gamma \approx \text{arc}/i\).

  4. Substitute and use \(f = r/2\) \(\to\) \(1/p + 1/i = 1/f\).

Note

Only \(\beta = \text{arc}/r\) is exact because the arc at \(a\) is part of a circle centered at C. The arcs for \(\alpha\) and \(\gamma\) are not centered at O or I, so the relations are approximate for small angles.

Proof 2: Refracting surface formula#

Setup: Point object O in medium \(n_1\); ray refracts at point \(a\) on a spherical surface into medium \(n_2\), crossing the axis at image I. Angles: \(\theta_1\) (incidence), \(\theta_2\) (refraction), \(\alpha\), \(\beta\), \(\gamma\) as before.

Key steps:

  1. Snell’s law for small angles: \(n_1 \sin\theta_1 \approx n_2 \sin\theta_2\) \(\to\) \(n_1 \theta_1 \approx n_2 \theta_2\).

  2. Exterior angle theorem on triangles C O \(a\) and I C \(a\): \(\theta_1 = \alpha + \beta\), \(\beta = \theta_2 + \gamma\).

  3. Eliminate \(\theta_1\), \(\theta_2\): \(n_1 \alpha + n_2 \gamma = (n_2 - n_1)\beta\).

  4. Small-angle approximation: \(\alpha \approx \text{arc}/p\), \(\beta = \text{arc}/r\), \(\gamma \approx \text{arc}/i\).

  5. Substitute \(\to\) \(n_1/p + n_2/i = (n_2 - n_1)/r\).

Proof 3: Thin lens formulas#

Strategy: Treat the lens as two refracting surfaces in sequence. The image from the first surface becomes the object for the second.

Setup: Object O\('\) in air; left surface (radius \(r'\)) forms virtual image I\('\) in glass; right surface (radius \(r''\)) has I\('\) as object O\(''\) (effectively in glass) and forms final image I\(''\) in air.

Key steps:

  1. First surface (air \(\to\) glass): \(1/p' - n/i' = (n-1)/r'\) (negative \(i'\) for virtual image).

  2. Second surface (glass \(\to\) air): Object distance \(p'' = i' + L \approx i'\) when \(L \approx 0\) (thin lens). Apply refracting-surface equation with \(n_1 = n\), \(n_2 = 1\).

  3. Add the two equations: the \(n/i'\) terms cancel, yielding \(1/p' + 1/i'' = (n-1)(1/r' - 1/r'')\).

  4. Relabel \(p' \to p\), \(i'' \to i\), \(r' \to r_1\), \(r'' \to r_2\) \(\to\) thin lens equation and lens maker’s equation.

Important

The thin-lens approximation \(L \approx 0\) is essential: it lets the object distance for the second surface equal the image distance from the first, so the intermediate terms cancel when the equations are combined.

Summary#

  • Spherical mirror: Exterior angle theorem + small-angle geometry \(\to\) \(1/p + 1/i = 2/r\).

  • Refracting surface: Snell’s law (small angles) + exterior angle theorem + geometry \(\to\) \(n_1/p + n_2/i = (n_2 - n_1)/r\).

  • Thin lens: Two refracting surfaces in sequence; thin-lens approximation (\(L \approx 0\)) \(\to\) \(1/p + 1/i = (n-1)(1/r_1 - 1/r_2)\).

  • Common thread: Paraxial rays, exterior angle theorem, and arc-length approximations.