Reflection of light

Two laws govern every reflection, flat or curved: the angle of incidence equals the angle of reflection, and the incident ray, the normal, and the reflected ray all sit in the same plane.

A plane mirror's image is always virtual, erect, the same size as the object, laterally inverted, and sits as far behind the mirror as the object is in front of it. Curve that mirror, though, and all of that changes depending on where the object sits — which is the rest of this chapter.

Spherical mirrors

A mirror curved inward, facing its own centre, is concave; curved outward, it's convex. Every spherical mirror has a pole (P, the centre of the mirror's surface), a centre of curvature (C, the centre of the sphere it's cut from — in front of a concave mirror, behind a convex one), and a principal axis running straight through both.

Parallel rays reflecting off the mirror converge at (concave) or appear to diverge from (convex) a single point on that axis — the principal focus (F). The distance from P to F is the focal length (f), and for any mirror with a small aperture:

— the focus always sits exactly halfway between the pole and the centre of curvature.

Locating the image: four rules

Any two of these rays, drawn from the top of the object, will cross (or appear to) at the image point:

• A ray parallel to the axis reflects through F (concave) or appears to come from F (convex).
• A ray through F (concave) or heading toward F (convex) reflects back parallel to the axis.
• A ray through C reflects straight back along the same path — it hit the mirror along the normal.
• A ray hitting the pole obliquely reflects at an equal angle to the axis, same as any flat surface.

Concave mirror: image by object position

• At infinity → at F, point-sized, real, inverted
• Beyond C → between F and C, diminished, real, inverted
• At C → at C, same size, real, inverted
• Between C and F → beyond C, enlarged, real, inverted
• At F → no image forms (rays emerge parallel)
• Between P and F → behind the mirror, enlarged, virtual, erect

A convex mirror skips all that drama: wherever the object is, the image is always virtual, erect, and smaller — which is exactly why it's used as a car's rear-view mirror: a smaller image means a wider field of view, so drivers see more of what's behind them.

Sign convention, mirror formula, magnification

To turn all this into numbers, distances get signed using the New Cartesian Sign Convention.

With object distance u, image distance v, and focal length f:

Magnification compares image height to object height:

Refraction of light

A pencil in a glass of water looks bent at the surface; a coin at the bottom of a bucket looks closer than it is; a tank of water looks shallower than it really is. All three are the same thing: light bends when it crosses at an angle into a different transparent medium — refraction — because its speed changes.

Two laws describe it: the incident ray, refracted ray, and normal all lie in one plane, and (Snell's law):

That constant is the refractive index of the second medium relative to the first — and it's really just a ratio of speeds:

Light is fastest in vacuum (3×10⁸ m/s) and slows in anything denser — water (n = 1.33), crown glass (n = 1.52), diamond (n = 2.42, the highest of the common examples). A higher refractive index means optically denser — note that's not the same as mass density: kerosene is optically denser than water but weighs less.

Send light through a rectangular glass slab and it bends toward the normal going in, then away from the normal coming out — by exactly equal and opposite amounts, so the emergent ray ends up parallel to the incident ray again, just shifted sideways.

Lenses

A convex (converging) lens is thicker in the middle and brings parallel rays together at a focus. A concave (diverging) lens is thicker at the edges and spreads them apart, so its focus is where the rays only appear to come from. Same vocabulary as mirrors — optical centre, principal axis, focal length — except distances are measured from the optical centre (O), not a pole.

• Convex, object at infinity → at F₂, point-sized, real, inverted
• Convex, at 2F₁ → at 2F₂, same size, real, inverted
• Convex, between F₁ and 2F₁ → beyond 2F₂, enlarged, real, inverted
• Convex, between F₁ and O → same side as object, enlarged, virtual, erect
• Concave, any position → always virtual, erect, diminished

Sign convention carries over with one flip: convex lens focal length is positive, concave is negative. The lens formula drops a sign compared to the mirror formula:

and magnification here has no minus sign in front:

Power of a lens

A short focal length bends light sharply; a long one barely bends it at all. That bending strength is the lens's power:

measured in dioptres (D) when f is in metres — positive for convex, negative for concave. Stack lenses together (as in a camera or a pair of testing spectacles) and their powers simply add: