Silvered Lenses: Mirror and Lens Combined in One Device

A silvered lens is a lens where one surface is coated with silver to act as a mirror. Light refracts through the lens, reflects off the silvered surface, and refracts again on way out. The device therefore behaves like a mirror with power contributed by both lens and mirror.

This clever combination appears in some telescopes, searchlights, and school experiments. It also creates interesting exam problems because focal length is not just that of lens or mirror alone, but their combined effect.

Key Concepts

Silvered surface: One side of lens polished and silvered so it reflects. If silvered surface is curved, it acts as spherical mirror with focal length R divided by 2.

Effective power: Light passes lens twice, so lens power counts twice, plus mirror power once.

Effective focal length F: Overall system behaves like mirror of focal length F, forming images on same side as object.

• System forms real images in front, like concave mirror, in most cases
• If plane surface silvered, mirror is plane, power zero, lens double pass dominates
• Sign convention same as mirror for final image

Formulas

1 divided by F = 2 times (1 divided by f_lens) plus 1 divided by f_mirror. Here f_lens is focal length of lens alone in metres or cm, f_mirror is focal length of silvered surface as mirror (R divided by 2), F is effective focal length. Powers add: P_effective = 2 P_lens + P_mirror.

For surface radius R, mirror focal length f_m = R divided by 2 with sign (concave negative in Cartesian mirror convention). Assumes thin lens and paraxial rays.

If silvered at convex surface etc., use correct sign for mirror.

Worked Example

A plano-convex lens (flat one side, convex radius 20 cm) refractive index 1.5, silvered at plane surface. Find effective focal length. First lens focal length: For plano-convex, 1 divided by f_lens = (n minus 1) times (1 divided by R1 minus 1 divided by R2). R1 = 20 cm, R2 infinite, so 1 divided by f = 0.5 times 1 divided by 20 = 1 divided by 40, f = 40 cm.

1. Lens power effect doubled: 2 divided by f_lens = 2 divided by 40 = 1 divided by 20 per cm.
2. Plane mirror f_m infinite, 1 divided by f_m = 0.
3. 1 divided by F = 1 divided by 20 + 0, so F = 20 cm.
4. Sign: system acts as concave mirror of 20 cm focal length (in front), so object at 20 cm forms image at same point inverted same size.

If silvered at curved surface instead, mirror power adds, giving shorter F. Students verify by extra term 1 divided by 10 cm.

Common Misconceptions

• Silvered lens is just mirror: Lens refraction twice contributes more than mirror sometimes. Ignore lens and answer will be wrong.
• Light passes lens once: It passes twice — entry and after reflection. That doubling is key formula factor 2.
• Image forms behind system: Because final is reflection, image forms on same side as object, like mirror. Transmission image does not exist.

Real-World Applications

• Reflecting telescopes use silvered surfaces combined with lenses in catadioptric designs for compact size
• Some searchlights and headlamps use silvered lens to direct beam efficiently
• School laboratory experiments measure refractive index using silvered lens method
• Camera telephoto designs fold light path with reflective surfaces to reduce length

Silvered lens combines refraction twice and reflection once into one effective mirror. Its focal length from 1 divided by F = 2 divided by f_lens + 1 divided by f_mirror predicts image neatly.