Magnification, power of a lens and lenses in contact

Magnification and power

The linear magnification of a lens is m = h′/h = v/u. Watch the sign: a mirror has m = −v/u, but a lens has m = +v/u. The power of a lens measures how strongly it converges or diverges light. It is defined as P = 1/f, with f in metres. Its SI unit is the dioptre (D): a lens of focal length 1 m has a power of 1 D. A convex lens has positive power, and a concave lens has negative power. A shorter focal length means a stronger lens and a larger power. Opticians write spectacle prescriptions in dioptres.

Worked example: thin lens formula and magnification

Derivation: two thin lenses in contact

Two thin lenses A and B, of focal lengths f₁ and f₂, touch each other on a common axis. A point object O is at distance u.

1. 1
   Lens A alone forms an image I₁ at distance v₁: 1/v₁ − 1/u = 1/f₁. ...(1)
2. 2
   I₁ acts as the object for lens B. The lenses are thin and touching, so I₁ is also at distance v₁ from B.
3. 3
   Lens B forms the final image I at distance v: 1/v − 1/v₁ = 1/f₂. ...(2)
4. 4
   Add (1) and (2): 1/v − 1/u = 1/f₁ + 1/f₂.
5. 5
   A single equivalent lens of focal length f would form the same image: 1/v − 1/u = 1/f.
6. 6★
   So, 1/f = 1/f₁ + 1/f₂, and in powers, P = P₁ + P₂. For several lenses, 1/f = 1/f₁ + 1/f₂ + 1/f₃ + ...
7. 7
   Magnification: m₁ = v₁/u and m₂ = v/v₁, so the total m = m₁ × m₂ = v/u.

★ key step

How to draw the diagram for lenses in contact

• Draw a principal axis and two thin convex lenses, A and B, touching each other.
• Mark the object O on the axis on the left.
• Draw a ray from O to lens A. After A, continue it as a dotted line to I₁, the image that A alone would form.
• At lens B, bend the ray more steeply so that it meets the axis at I, closer to the lenses than I₁.
• Label u (from O to the lenses), v₁ (to I₁), v (to I), f₁ and f₂. Put arrows on the rays.

Worked example: lenses in contact

Activity: refractive index of a liquid using a convex lens and a plane mirror

The liquid between the lens and the mirror forms a plano-concave liquid lens in contact with the glass lens.

1. 1
   Put a plane mirror on the base of a stand and a convex lens on it. Move a needle above the lens until the needle and its inverted image coincide with no parallax.
2. 2
   Light now returns along its own path from the plane mirror, so the needle is at the focus. Measure f₁, the focal length of the glass lens.
3. 3
   Put a few drops of the liquid on the mirror and replace the lens. Repeat to find F, the focal length of the glass lens and liquid lens together.
4. 4
   Find the liquid lens from 1/f₂ = 1/F − 1/f₁. It comes out negative, because F is longer than f₁.
5. 5
   Measure the radius R of the lens face touching the liquid with a spherometer. For the plano-concave liquid lens, 1/|f₂| = (n − 1)/R, so n = 1 + R/|f₂|.
6. 6
   Result (example): f₁ = 20 cm, F = 30 cm and R = 20 cm give 1/f₂ = 1/30 − 1/20 = −1/60. So f₂ = −60 cm and n = 1 + 20/60 ≈ 1.33, the value for water.