Refraction through Lenses — Notes

• Lens: a transparent medium bounded by two surfaces, at least one curved (usually spherical).
• Convex (converging): thicker at the middle — biconvex, plano-convex, concavo-convex. Concave (diverging): thinner at the middle — biconcave, plano-concave, convexo-concave.
• Lens as prisms: a convex lens behaves like prisms with bases towards the axis (light bends towards the axis); a concave lens like prisms with apexes towards the axis (light bends away).
• Centres of curvature, principal axis, optical centre O (ray through it is undeviated), principal foci F₁ and F₂, focal plane (plane through F normal to the axis), focal length f (distance OF).
• A lens has two principal foci, one on each side; for a thin lens in air both focal lengths are equal.

Principal rays: (1) parallel to the axis → through F₂; (2) through O → undeviated; (3) through F₁ → parallel to the axis. Two rays fix the image; real where rays meet, virtual where they appear to meet.

• Ray parallel to the axis diverges as if from F₁; ray through O goes straight on.
• The refracted rays never meet; produced back, they meet between F₁ and O.
• For every position of a real object the image is virtual, erect, diminished, between F₁ and O, on the object side. For an object at infinity the image is at F₁, highly diminished.
• A quick test: a lens that always gives a smaller, upright image of objects at all distances is concave.

New Cartesian sign convention: distances from O; along the incident light (left to right) positive, against it negative; heights up positive, down negative. Hence $u < 0$ for a real object; $f > 0$ convex, $f < 0$ concave; $v > 0$ real image, $v < 0$ virtual image.

Lens formula: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$. Magnification: $m = \dfrac{h_{2}}{h_{1}} = \dfrac{v}{u}$ — negative for real, inverted images; positive for virtual, erect ones.

Method: list $u$, $v$, $f$ with signs → substitute → solve → interpret the signs.

• Power $P = \dfrac{1}{f}$ with $f$ in metres; unit dioptre (D); $1\ \text{D}$ is the power of a lens of focal length 1 m. Convex: $P > 0$; concave: $P < 0$.
• Magnifying glass (simple microscope): convex lens of short focal length; object between O and F; image virtual, erect, magnified, usually at about 25 cm from the eye. Shorter focal length → greater magnification.
• Uses: convex — magnifying glass, camera, projector, microscope and telescope lenses, spectacles for long sight; concave — spectacles for short sight, peephole, lens combinations.