Why does light bend at an interface? (Snell's Law Intuition)

Imagine light as a wavefront (a line of synchronized oscillators). When this wavefront hits a glass surface at an angle, one end of the wavefront enters the glass first and slows down (since v = $\frac{c}{n}$ in glass, n > 1). The other end is still in air, still moving fast. This creates a "pivot" — one side slows, the other doesn't — and the entire wavefront rotates toward the normal. This is Snell's law emerging from the wave nature of light, but in ray optics, we just accept the result: $n_{1}$ sin θ_{1} = $n_{2}$ sin θ_{2}.

Why does a concave mirror converge light?

The concave surface curves toward the incident light. When parallel rays hit different parts of the curved surface, the normals at each point are angled differently. By the law of reflection (angle of incidence = angle of reflection from each local normal), the reflected rays from the outer parts of the mirror are angled more steeply than those from the central part — and they all converge at the focal point F. It's geometry: a spherical surface redirects rays to meet at a common point (approximately, for paraxial rays).

Why is the magnification formula different for mirrors and lenses?

For a mirror, the image is on the same side as the object (for a real image). The coordinate system has the object at negative u and the image also at negative v (for a real image). The magnification m = $\frac{h_i}{h_o}$. By similar triangles in the geometry: m = −v/u. The extra negative arises from the geometry of reflection — the image is flipped relative to the object.

For a lens, the real image is on the other side from the object. The object is at negative u, the image is at positive v (for a real image). By similar triangles: m = $\frac{v}{u}$. No extra negative is needed because the geometric relationship already accounts for the inversion through the "positive v" region.

Why does TIR have a critical angle?

Snell's law: $n_{1}$ sin θ_{1} = $n_{2}$ sin θ_{2}. When $n_{1}$ > $n_{2}$ and we increase θ_{1}, the value of sin θ_{2} = ($n_{1}$/$n_{2}$) sin θ_{1} increases. At some θ_{1} = θ_c, sin θ_{2} = 1 (θ_{2} = 90°, refracted ray grazes surface). For θ_{1} > θ_c, sin θ_{2} would need to exceed 1 — which is impossible for a real angle. So no refracted ray can exist. All energy must return into the denser medium: total internal reflection. The critical angle is exactly the angle at which Snell's law pushes the refracted ray to the limit of its existence.